diff --git a/-9AyT4oBgHgl3EQf3vkG/content/tmp_files/2301.00772v1.pdf.txt b/-9AyT4oBgHgl3EQf3vkG/content/tmp_files/2301.00772v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0702c9182f07058523264e37bff462fe6b0026f --- /dev/null +++ b/-9AyT4oBgHgl3EQf3vkG/content/tmp_files/2301.00772v1.pdf.txt @@ -0,0 +1,3502 @@ +1 +A Unified Visual Information Preservation +Framework for Self-supervised Pre-training in +Medical Image Analysis +Hong-Yu Zhou, Student Member, IEEE, Chixiang Lu, Chaoqi Chen, Sibei Yang, +and Yizhou Yu, Fellow, IEEE, +Abstract—Recent advances in self-supervised learning (SSL) in computer vision are primarily comparative, whose goal is to preserve +invariant and discriminative semantics in latent representations by comparing siamese image views. However, the preserved high-level +semantics do not contain enough local information, which is vital in medical image analysis (e.g., image-based diagnosis and tumor +segmentation). To mitigate the locality problem of comparative SSL, we propose to incorporate the task of pixel restoration for explicitly +encoding more pixel-level information into high-level semantics. We also address the preservation of scale information, a powerful tool +in aiding image understanding but has not drawn much attention in SSL. The resulting framework can be formulated as a multi-task +optimization problem on the feature pyramid. Specifically, we conduct multi-scale pixel restoration and siamese feature comparison in +the pyramid. In addition, we propose non-skip U-Net to build the feature pyramid and develop sub-crop to replace multi-crop in 3D +medical imaging. The proposed unified SSL framework (PCRLv2) surpasses its self-supervised counterparts on various tasks, +including brain tumor segmentation (BraTS 2018), chest pathology identification (ChestX-ray, CheXpert), pulmonary nodule detection +(LUNA), and abdominal organ segmentation (LiTS), sometimes outperforming them by large margins with limited annotations. Codes +and models are available at https://github.com/RL4M/PCRLv2. +Index Terms—Medical image analysis, Self-supervised learning, Transfer Learning, Context restoration, Feature pyramid. +! +1 +INTRODUCTION +I +T is usual to acquire a substantial amount of manually +labeled data before training deep neural networks. This +condition is easy to meet in natural images, where labor +costs and labeling difficulties are tolerable. In medical image +analysis, however, credible annotations are mainly derived +from domain experts’ diagnoses, which are challenging to +obtain due to the rarity of the target disease, the need to safe- +guard patient privacy, and the scarcity of medical resources. +Against this background, self-supervised learning (SSL) has +been widely accepted as a viable technique to learn medical +image representations without specialistic annotations. We +usually deploy SSL in the pre-training stage to obtain well- +transferable features, which can be transferred to various +downstream tasks for performance boosting. +Recent advances in SSL are mostly based on compar- +ative learning [8], [10], [15], [17]. The rationale behind is +to learn transferable latent representations with invariant +and discriminative semantics by maximizing the mutual +information between a pair of siamese images. One potential +problem of these comparative methods is that they mainly +focus on encoding high-level global semantics in representa- +• +Hong-Yu Zhou, Chixiang Lu, Chaoqi Chen, and Yizhou Yu are with the +Department of Computer Science, The University of Hong Kong, Hong +Kong. Email: {whuzhouhongyu, luchixiang, cqchen1994}@gmail.com, +yizhouy@acm.org. +• +Sibei Yang is with ShanghaiTech University and Shanghai Engineering +Research Center of Intelligent Vision and Imaging, Shanghai, China. +Email: yangsb@shanghaitech.edu.cn. +• +First two authors contributed equally. +• +Corresponding author: Sibei Yang and Yizhou Yu. +tions but ignore the preservation of pixel-level information1. +However, in medical image analysis, the latter type of +information usually plays a vital role. For instance, in chest +pathology detection, radiologists or clinicians are required +to point out small lesions from a chest X-ray according to +their textures. Sometimes, these areas of pathologies are so +hard to identify that even medical experts have to check +pixel-level details to tell where the lesions are. Another +typical example lies in brain tumor segmentation, where the +segmentation error of one voxel may cause irreparable harm +to patients in brain surgeries, such as a permanent damage +to the cochlear nerve when trying to remove the acoustic +neuroma. +An intuitive way to preserve pixel-level information in +learned features is to restore the pixel-level content from +latent representations directly. This methodology, known +as context restoration [29], has already been adopted as a +surrogate task in pretext-based SSL for natural [23], [29], [44] +and medical images [7], [49]. Specifically, these approaches +first apply various data augmentation strategies to a given +image to generate a corrupted input, based on which deep +models are trained to restore original pixels. In this way, +we explicitly require the latent representations to preserve +information closely related to pixels. Although pure pixel- +based features are not as transferable as those from com- +parative SSL [17], [48], we hypothesize it is still beneficial +to explicitly preserve pixel-level information and global se- +1. In 3D medical images, we often use “voxel” to denote the same +concept as the pixel does in 2D images. For simplicity, we use “pixel” +to denote the smallest addressable element in both 2D and 3D images +in the rest of this paper. +arXiv:2301.00772v1 [cs.CV] 2 Jan 2023 + +2 +ℱ# +ℱ$ +ℱ% +ℱ" +ℱ! +Pixels +Semantics +Scales +Fig. 1. Motivation illustration. We propose a unified SSL framework +to simultaneously preserve information in visual representations from +perspectives of pixels, semantics, and scales. {F1, F2, F3, F4, F5} de- +note different levels in the feature pyramid, given an input image. Our +approach restores uncorrupted inputs from the feature maps directly +to preserve pixel-level details. In order to retain the global semantic +information, our method compares siamese one-dimensional represen- +tations. Last but not the least, the proposed methodology conducts pixel +restoration and feature comparison at different scales. The rationale +behind is to introduce multi-scale self-supervised latent representations, +making them more transferable to various downstream tasks. +mantics, especially in medical image analysis where details +matter a lot. +Besides semantics and pixels, introducing multi-scale +representations has been proven to be quite helpful in +aiding image understanding [12], [24], [26], [27], [32], [39]. +The common practice of these methods is to construct a +feature pyramid during training, testing, or both stages. +Then, various tasks, such as detection, and segmentation, +can be conducted on the basis of multi-scale features. The +goal of building the feature pyramid is to endow image +representations with the ability to recognize objects at dif- +ferent scales, which is also consistent with the law of human +cognition [31]. However, the preservation of visual informa- +tion at multiple scales is rarely mentioned in SSL. Thus, it +is unclear whether introducing multi-scale self-supervised +representations provides a stronger transfer learning ability. +In Figure 1, we illustrate the motivation of the proposed +unified visual information preservation framework for SSL. +The introduced framework addresses the preservation of +information in self-supervised visual representations from +three aspects: pixels, semantics, and scales. Firstly, to re- +tain pixel-level information in latent representations, our +framework involves a reconstruction branch in the self- +supervised model to rebuild uncorrupted images from cor- +rupted inputs. Specifically, we ask the self-supervised model +to restore pixels from feature maps of randomly corrupted +inputs during training. As a result, information closely +associated with pixels can be explicitly encoded into the +latent representations. In practice, this type of information +would enhance the ability of self-supervised representations +to recognize and differentiate textures. Apart from pixel- +level information, preserving invariant and discriminative +semantics in visual representations is also necessary. To- +wards this end, we adopt the existing comparative SSL +to encode invariant semantic information by comparing +high-level representations of siamese image patches [10]. +We empirically found the siamese SSL not only produces +comparably (sometimes more) transferable medical image +representations but also is much easier to implement in com- +parison to the typical contrastive manner [17]. Last but not +the least, the proposed unified framework introduces multi- +scale latent representations by conducting pixel restoration +and feature comparison in a range of scales. To achieve this +goal, we propose a non-skip U-Net (nsUNet) that constructs a +feature pyramid upon the U-shape architecture [32]. In prac- +tice, nsUNet effectively avoids the production of shortcut +solutions when performing the context restoration task. On +the basis of nsUNet, we conduct pixel-level context restora- +tion and siamese feature comparison in each level (i.e., scale) +of the feature pyramid. In this way, the proposed framework +helps improve the ability of self-supervised representations +to recognize objects (e.g., lesions and organs in medical +images) at different sizes and scales. +We summarize the contributions of this paper as follows: +• +We present an information preservation framework +for advancing SSL in medical image analysis. In this +framework, we unify the preservation of visual infor- +mation in latent representations from three aspects: +pixels, semantics, and scales. Towards this end, pixel +restoration and feature comparison are conducted at +different feature scales. +• +We introduce non-skip U-Net (nsUNet) to construct +the feature pyramid. Compared to the typical U- +shape models in medical imaging [11], [32], nsUNet +maintains more feature scales and eliminates the us- +age of the widely adopted skip connections to avoid +shortcut solutions to pixel restoration. +• +Inspired by multi-crop [5], we propose sub-crop to +compare global volumes against local volumes. In +order to mitigate the problem of the reduced mutual +information between global and local views in 3D +space, sub-crop restricts the cropping of local views +within the 3D minimum bounding box of global +views. Experiments on 3D medical images found that +sub-crop is more effective than multi-crop in various +downstream tasks. +• +We conduct extensive and comprehensive experi- +ments to validate the effectiveness of the proposed +framework. We show that the unification of pixels, +semantics, and scales can provide impressive perfor- +mance under the pre-training/fine-tuning protocol. +Specifically, the proposed framework outperforms +both self-supervised and supervised counterparts in +chest pathology classification, pulmonary nodule de- +tection, abdominal organ segmentation, and brain +tumor segmentation by substantial margins. +The conference version of this paper (PCRLv1) was pre- +sented in [47], which demonstrates the benefits of incor- +porating more pixel-level information besides the invariant +and discriminative semantics obtained by contrastive learn- +ing. In this paper, we made significant and substantial modi- +fications to PCRLv1, and we name the improved framework +as PCRLv2 (i.e., Preservational Comparative Representation +Learning). The modifications and improvements in PCRLv2 +include but are not limited to (i) Besides local pixel-level +and global semantic information, scale information is also + +3 +× +× +× +× +× +× +× +× +R +R +R +R +R +𝑥 +𝑥! +𝑥" +t! +t" +R +R +R +R +R +R Pixel restoration +Candidate scale +Chosen scale +nsUNet +Siamese nsUNet +𝑥! +# +t! +# +𝑥" +# +t" +# +Global aug. +Global aug. Local aug. +Local aug. +(a) Multi-scale pixel restoration +× +× +× +× +× +× +× +× +C +C +C +C +C +C Feature comparison +Candidate scale +Chosen scale +nsUNet +Siamese nsUNet +𝑥! +𝑥" +𝑥! +# +𝑥" +# +𝑥 +t! +t" +t! +# +t" +# +Global aug. +Global aug. Local aug. +Local aug. +(b) Multi-scale feature comparison +Fig. 2. The overall structure of PCRLv2. PCRLv2 performs self-supervised visual learning on siamese feature pyramids. To achieve this goal, we +propose non-skip U-Net (nsUNet). nsUNet consists of five feature scales and removes the skip connections to prevent network optimizers from +finding shortcut solutions to context restoration. On the basis of nsUNet, we propose to decouple the preservation of pixel-level, semantic, and +scale information into two tasks: (a) multi-scale pixel restoration; (b) multi-scale feature comparison. The rationale behind is to incorporate pixel +details and semantics into features at different scales. During the training stage, we randomly choose a feature scale from the feature pyramid, +on top of which we conduct pixel restoration and feature comparison. x denotes a batch of input images. t1 and t2 stand for two distinct global +augmentations, while t′ +1 and t′ +2 denote the successive local augmentations. +preserved in self-supervised visual representations. The +motivation behind is that although multiple feature scales +have been considered in various vision tasks, they have +not drawn much attention in SSL. PCRLv2 shows that +introducing multi-scale latent representations can boost the +transfer learning performance of SSL in downstream tasks. +(ii) PCRLv2 simplifies the attentional pixel restoration and +hybrid feature contrast operations of PCRLv1 into a con- +cise multi-task optimization problem. As a result, PCRLv2 +is simpler and easier to implement while achieving bet- +ter performance, thus more practical. (iii) Compared to +PCRLv1 that relies on the plain U-Net architecture [32], +PCRLv2 conducts SSL on top of a new backbone, i.e., non- +skip U-Net (nsUNet). There are two inherent advantages +of nsUNet. First, the feature pyramid of nsUNet allows +performing multi-scale pixel-level context restoration and +semantic feature comparison. As a result, the unification +of pixels, semantics, and scales produces more transferable +visual representations. Second, nsUNet can effectively avoid +the production of shortcut solutions, providing obvious +performance gains over the use of the typical skip con- +nections. (iv) We integrate the idea of multi-crop [5] in +PCRLv2. Moreover, in 3D medical imaging, we propose sub- +crop to produce reliable local views with increased mutual +information by randomly cropping multiple local volumes +within the 3D minimum bounding box of global views. In +practice, we found that the proposed sub-crop has better +pre-training performance than multi-crop. (v) In 5 classifica- +tion/segmentation tasks, PCRLv2 provides more transfer- +able pre-trained visual representations, not only surpass- +ing previous self-supervised and supervised counterparts +by substantial margins but also obviously outperforming +PCRLv1 in all experiments. +2 +RELATED WORK +This section reviews related work in comparative SSL, +including contrastive and non-contrastive methods, and +lists SSL approaches that use context restoration as the +pretext task. In the third part, we collect papers that +emphasize the incorporation of multi-scale features in SSL. +Comparative SSL methodologies. One of the core ideas +behind comparative SSL is to extract and encode invari- +ant and discriminative semantics into representations via +feature-level comparison. Hjelm et al. [20] proposed Deep In- +foMax to maximize the mutual information between global +and local feature vectors of the same input image using +InfoNCE [28]. Bachman et al. [3] augmented InfoMax by +conducting a global-local comparison on feature vectors of +independently-augmented versions of each input. Tian et +al. [36] increased the number of augmented views of each +input and extended InfoNCE to multiple views. He et al. [17] +presented Momentum Contrast (MoCo), which comprises +a momentum encoder to maintain the consistency among +positive and negative feature vectors. Different from [3], +[20], MoCo performs InfoNCE on top of global feature +vectors only. Compared to MoCo, SimCLR removes the +momentum architecture and defines InfoNCE on the output +of a MLP with one hidden layer. Inspired by SimCLR, +Chen et al. [9] proposed MoCov2, which improves MoCo +with an additional MLP head and more augmentations. +SwAV [5] replaces the feature vectors in InfoNCE with +cluster assignments and introduces the multi-crop strategy +to increase the number of views of an image with affordable +computational overhead. Grill et al. [15] proposed BYOL +(bootstrap your own latent), which eliminates the use of +InfoNCE in SSL by distilling semantics from positive pairs +only. Based on BYOL, Chen et al. [10] further removed the +restriction of the momentum architecture and introduced +a simple siamese learning framework named SimSiam. In +practice, SimSiam produces comparable results to MoCov2 +in various downstream tasks. Recently, Zbontar et al. [42] +simplified SimSiam by measuring the cross-correlation ma- +trix between the siamese global feature vectors and trying +to make this matrix close to the identity. +Comparative SSL, especially InfoNCE-based method- +ology, has also been widely adopted in medical image + +4 +analysis. Zhou et al. [48] proposed to integrate mixup [43] +into MoCov2, increasing the diversity of both positive and +negative samples in InfoNCE. Taleb et al. [34] developed 3D +versions of existing SSL techniques and compared 2D and +3D SSL approaches on downstream tasks. Azizi et al. [2] +incorporated multi-instance learning into SimCLR, which +helps utilize multiple views of each patient. Around the +same time, Vu et al. [37] developed a method to select posi- +tive pairs coming from views of the same patient and used +this strategy to improve MoCov2. There are also a number +of approaches [6], [40], [41] that tailored comparative SSL +for semi-supervised medical image segmentation. +However, the methodologies mentioned above fail +to +address +the +importance +of +integrating +pixel-level +information into the high-level representations with rich +semantics, which is the primary focus of the proposed +PCRL. +Context restoration for preserving pixel-level information. +Restoring original context has been treated as an important +pretext task in SSL. Pathak et al. [29] first time conducted +self-supervised feature learning by recovering masked input +images. Larsson et al. [23] and Zhang et al. [44] performed +SSL on pixels via predicting RGB color values. For medical +images, Chen et al. [7] extended the approach in [29] with +swapped image patches. Zhou et al. [49] showed that adding +more augmentations to input images brings benefits to SSL. +Tao et al. [35] presented a volume-wise context transforma- +tion for 3D medical images. Different from the approaches +mentioned above, Henaff [19] proposed to predict the next +context feature vectors following an auto-regressive manner. +We can see that context restoration is more prevalent in +medical imaging than in natural images from the above. +The underlying reason is that medical imaging tasks +require more pixel-level information to make fine-grained +yet accurate decisions. On the other hand, we observe +that comparative SSL can produce representations with +richer semantics. Thus, it can be beneficial to build a SSL +framework that simultaneously integrates pixel-level and +semantic information. As far as we are concerned, none +of these context restoration based approaches incorporate +such a combination. +Multi-scale features in SSL. Although multi-scale features +have not drawn much attention in existing SSL research, +it has already been treated as an implicit yet effective +regularization method for SSL in some methodologies. Deep +InfoMax [20] contrasts high-level feature vectors with low- +level feature maps using InfoNCE. To improve Deep Info- +Max, Bachman et al. [3] proposed to contrast global and +local feature vectors on multiple levels. In medical image +analysis, preserving scale information becomes essential, +as pathologies may show different characteristics on dif- +ferent scales. In [6], a local contrastive loss is introduced +to learn distinctive representations of local regions that are +helpful to per-pixel segmentation. At the same time, global +feature vectors are used to distill discriminative semantics +for classification tasks. A similar idea has also been used +in image registration [25] and one-shot segmentation [46], +where global and local feature vectors are employed to +provide information on semantics and position, respectively. +However, most of these methods only perform SSL on +two scales, i.e., one global and one local, which cannot fully +capture multi-scale information. Besides, although these +approaches emphasize the benefit of introducing local in- +formation to SSL, they do not exploit pixel-level information +that is helpful to encode locality. In contrast, this paper pro- +poses a unified framework that can simultaneously preserve +semantic, pixel-level, and scale information. +3 +METHODOLOGY +We provide an overview of PCRLv2 in Fig. 2. Suppose x +denotes a batch of input images. We introduce cascaded +augmentations to distort x in global and local views, respec- +tively. To be specific, the first-stage augmentations (t1 and t2 +in Fig. 2) mainly consist of global transformations, such as +flip and rotation, whose goal is to distort the semantics of +input images from a global perspective. In comparison, the +second-stage augmentations (t′ +1 and t′ +2 in Fig. 2) comprise +local pixel-level transformations, such as random noise and +gaussian blur, which are leveraged to perturb the local +semantics. After two-stage augmentations, the finally aug- +mented images x′ +1 and x′ +2 are passed to siamese networks +to perform pixel restoration and feature comparison, while +the results of applying t1 and t2 to x, i.e., x1 and x2, serve +as the ground truth targets for the pixel restoration task (as +shown in Fig. 2a). +We perform SSL on the feature pyramid to encode multi- +scale visual representations. Following the standard practice +in medical image processing, we build feature pyramids +using a U-shape model named non-skip U-Net (nsUNet). +Compared to the typical U-Net architecture [11], [32], +nsUNet has more feature scales and completely removes +skip connections, both of which we empirically found help- +ful in producing better pre-trained representations. During +the training stage, one scale is first randomly chosen from all +five feature scales, after which we conduct pixel restoration +and feature comparison on the siamese feature maps at the +chosen scale. After the pre-training stage, we fine-tune the +encoder of nsUNet on various downstream tasks. +3.1 +Feature pyramid in non-skip U-Net +U-Net and its series [11], [22], [32] have been known in med- +ical imaging for their abilities to handle image segmentation +tasks. The most distinctive characteristic of these models is +the skip connection that connects equal-resolution low- and +high-level feature maps. The critical insight is to recover the +spatial information lost in down-sampling operations of the +encoder network, such as strided pooling or convolution. +U-shape models use a feature pyramid to progressively +incorporate multi-scale details brought by skip connections +into high-level semantics, making the U-shape architecture +an ideal choice for conducting context restoration. +In this paper, we explore the potential of U-shape ar- +chitecture in SSL from two perspectives: deeply fusing +semantic and pixel-level information by removing the skip +connections and introducing multi-scale latent representa- +tions by conducting SSL on the feature pyramid. For the +first perspective, we empirically found that skip connec- +tions provide shortcuts for context restoration, as the low- +level feature maps contain rich, high-resolution pixel-level + +5 +× +× +× +× +𝐻 +2 × 𝑊 +2 +𝐻×𝑊 +𝐻 +4 × 𝑊 +4 +𝐻 +8 × 𝑊 +8 +𝐻 +16 × 𝑊 +16 +𝐻 +32 × 𝑊 +32 +Skip feature maps +Feature +hierarchy +Down-sampling +Up-sampling +× +No skip connection +Conv+BN ++ReLU +×2 +Conv+BN ++ReLU +×2 +Conv+BN ++ReLU +×2 +Conv+BN ++ReLU +×2 +Conv+BN ++ReLU +×2 +Fig. 3. The architecture of non-skip U-Net (nsUNet). In comparison +to previous U-Net series, nsUNet removes skip connections, and the +associated skip feature maps to prevent shortcut solutions to the pixel +restoration and feature comparison tasks. Besides, nsUNet consists of +five levels of feature maps (denoted with different colors), where two self- +supervised tasks are further conducted. Note that this is a 2D illustration +of nsUNet. +details. This characteristic does contribute to the restoration +of context. However, it may prevent the high-level latent +representations (with rich semantics) from incorporating +more pixel-level information because the task of providing +pixel-level details is assigned to low-level feature maps. To +address this point, we remove the skip connections in U- +shape architecture and propose non-skip U-Net (nsUNet). +nsUNet relies on high-level representations without any +skip connections to restore pixel-level details. In this way, +the semantic and pixel-level information can be deeply +fused. Meanwhile, the inherent multi-scale feature maps of +nsUNet offer the opportunity to construct a feature pyra- +mid, on top of which SSL can be conducted in multiple +scales simultaneously. +Fig. 3 presents the architecture of nsUNet. The feature +pyramid in nsUNet comprises five levels, ranging from low +resolution (the down-sampling rate is 32) to full resolution +(no down-sampling). For 2D input data, we use ResNet- +18 [18] as the encoder, while for 3D input volumes, we +build the encoder following [11]. As illustrated in Fig. 3, the +decoder of nsUNet maintains a shared architecture across +all pyramid levels, which can be summarized as: +Fi = Conv-BN-ReLU(Conv-BN-ReLU(Up(Fi−1)), +(1) +where i ∈ {1, 2, 3, 4, 5}. F0 denotes the output of the bottle- +neck block, which has the lowest spatial resolution (down- +sampling rate=32). Up represents the up-sampling opera- +tion. Conv-BN-ReLU stands for a sequence of operations, +including convolution (kernel size=3), batch normalization +(BN), and ReLU activation. As a result, the bag of feature +maps {F1, F2, F3, F4, F5} is then forwarded to following +task-dependent heads to perform pixel restoration and fea- +ture comparison, respectively and simultaneously. +3.2 +Multi-scale pixel restoration +As the name implies, multi-scale pixel restoration aims to +preserve pixel-level and scale information in latent visual +representations simultaneously. To achieve this goal, we ask +the network to recover the exact pixel-level details across +different scales, where each pair of siamese feature maps +share one pixel restoration head. In contrast, PCRLv1 only +restores pixel details at the full resolution, which inevitably +loses multi-scale properties in learned representations. +As shown in Fig. 4a, the input images x′ +1 and x′ +2 are +intentionally corrupted via various pixel-level augmenta- +tions, such as guassian blur and random noise. For each +training iteration, we first randomly choose a feature scale +Fi from {F1, F2, F3, F4, F5}. Then, we pass Fi to the pixel +restoration head f R +i (·) for the i-th scale, whose internal +processing procedure can be summarized as: +f R +i (Fi) = Conv(Conv-BN-ReLU(Fi)), +(2) +where all convolution layers use a kernel size of 3 and a +stride of 2. Similarly, we apply the shared pixel restoration +head to the paired siamese feature map Fs +i to acquire the +prediction output f R +i (Fs +i ): +f R +i (Fs +i ) = Conv(Conv-BN-ReLU(Fs +i )), +(3) +Lastly, we employ the mean square error (MSE) loss to +measure the reconstruction errors between f R +i (Fi) and x1. +For the siamese feature pyramid, we apply MSE loss to +f R +i (Fs +i ) and x2. The cost function LR of the pixel restoration +task in each training iteration (with mini-batch optimiza- +tion) is as follows: +LR = +N +� +j=1, +∀i∈H +1[i==j] [MSE(f R +i (Fi), x1) + MSE(f R +i (Fs +i ), x2)], +(4) +where N = 5 denotes the number of scales in each feature +pyramid. H = {1, 2, 3, 4, 5} stands for the scale index. +1[i==j] is an indicator function, which is equal to 1 when +i==j is true (otherwise, 0). The explanation of LR can be +summarized as: (i) randomly choose a feature scale Fi +from all five scales; (ii) pass Fi and its siamese feature +map Fs +i to the shared task head f R +i (·); (iii) calculate the +MSE loss between the outputs of f R +i (·) and uncorrupted +images {x1, x2}. By reconstructing the same targets x1/x2 +across different feature scales, LR can encode the pixel-level +information into multi-scale latent visual representations. +3.3 +Multi-scale feature comparison +PCRLv1 employs a hybrid way to conduct contrastive +learning with the help of the momentum encoder [17] +and mixup [43]. However, this contrastive deployment is +complex, making PCRLv1 heavy, thus troublesome to im- +plement and improve. To address these issues, PCRLv2 +replaces the hybrid contrastive strategies in PCRLv1 with +the multi-scale comparison. Inspired by [10], multi-scale +comparison conducts SSL with siamese learning, whose +key operation is to attract the same image’s siamese views. +Different from [10] that conducts feature comparison on one +scale, we propose to preserve the discriminative semantics +across different feature scales, which forces the model to +preserve multi-scale self-supervised representations. In the +following, we provide technical details of performing the +multi-scale comparison. + +6 +Conv +BN +ReLU +Conv +Conv +BN +ReLU +Conv +𝑥! +𝑥# +Shared +𝑥! +" +… +𝑥# +" +… +Siamese scale +Chosen scale +(a) Architectural details of the pixel restoration head +Siamese scale +Chosen scale +GAP +GAP +BN +FC +BN +ReLU +FC +BN +FC +BN +ReLU +FC +Predictor +Predictor +Shared +𝑥! +" +… +𝑥# +" +… +(b) Architectural details of the feature comparison head +Fig. 4. Architectural details of the pixel restoration and feature comparison heads. Conv, BN, GAP, and FC denote the convolution, batch +normalization, global average pooling, and fully-connected layers, respectively. The kernel size of all convolution layers is 3, and the convolution +stride is set to 1. Note that each pair of siamese feature maps share one pixel restoration head and one feature comparison head, while different +feature scales employ distinct task heads. +Given the feature maps at a randomly chosen scale Fi, +we pass them through a global average pooling layer and +a shared batch normalization layer (as shown in Fig. 4b) to +acquire 1D representations vi: +vi = BN(GAP(Fi)). +(5) +We can get vs +i by processing the siamese feature maps Fs +i in +a similar way. +Next, we forward vi to the shared predictor fP(·), whose +architecture is displayed in Fig. 4b and can be summarized +as: +fP(vi) = FC(FC-BN-ReLU(vi)). +(6) +where FC denotes the fully-connected layer. FC-BN-ReLU +stands for a sequence of layers, which are the fully- +connected layer, batch normalization layer, and ReLU ac- +tivation. Similarly, we can acquire fP(vs +i ) by passing vs +i to +the same predictor. +We measure the similarity between siamese feature vec- +tors with the cosine similarity: +cos(vi, fP(vs +i )) = +vi +∥vi∥2 +· +fP(vs +i ) +∥fP(vs +i )∥2 +, +(7) +where || · ||2 denotes the L2 normalization. Symmetrically, +we calculate cos(fP(vi), vs +i ) as follows: +cos(fP(vi), vs +i )) = +fP(vi) +∥fP(vi)∥2 +· +vs +i +∥vs +i ∥2 +. +(8) +Finally, the cost function LC of multi-scale feature compari- +son can be summarized as: +LC = +N +� +j=1, +∀i∈H +−1 +21[i==j] [cos(sg(vi), fP(vs +i )) ++ cos(fP(vi), sg(vs +i ))]. +(9) +N += +5 denotes the number of feature scales. H += +{1, 2, 3, 4, 5} stands for the scale index. Following [10], we +apply the stop-gradient operation (denoted as sg) in Eq. 9 +to prevent the network optimizer from finding shortcut +solutions. +Minimizing LC requires the model to maximize the +similarity between siamese latent features across all feature +scales. In this way, scale invariance can be implicitly incor- +porated into the preserved latent semantics. +1 +2 +1 +2 +3 +4 +5 +6 +7 +8 +Randomly crop two +global patches with +an IoU constraint +Find the minimum 3D bounding box +Randomly crop +local patches +1 +2 +3 +4 +5 +6 +7 +8 +3D Global views +3D Local views +Fig. 5. Illustration of sub-crop. Given a 3D local volume, we first randomly +crop two large patches, where an intersection over union (IoU) constraint +is applied to guarantee that two patches are partly overlapped. These +two large patches are considered as x1 and x2 in Fig. 2 and will be +passed to the siamese architecture to conduct the following multi-scale +pixel restoration and feature comparison tasks. To acquire local views, +we compute the minimum 3D bounding box of two large patches, after +which random crop is applied to extract multiple local patches. Finally, +we reshape these local patches to a fixed size and forward them to the +network to extract local representations. +3.4 +From multi-crop to sub-crop +Multi-crop [5] has been known as a helpful strategy to im- +prove SSL performance in natural images, which increases +the number of input views by sampling several standard +resolution crops and more low-resolution crops from the +original input. One key insight behind multi-crop is to +capture relations between parts of a scene or an object, while +low-resolution views ensure a controllable increase in the +computational cost. +When applied to medical images, multi-crop works well +in 2D X-ray data but leads to the non-convergence of the +model in 3D volume data (such as CT and MRI). After +careful investigation, we found the root of this problem +lies in the contradiction between the limited input size and +many candidate crops in three-dimensional space. Specif- + +7 +ically, on the one hand, we cannot afford large-sized 3D +inputs because processing them with 3D deep models often +costs dramatic GPU memory. On the other hand, if we +overly reduce the size of 3D inputs, the sampled views +would be too dispersed to guarantee the model capture the +local-global associations. +To mitigate the above issue, we introduce sub-crop to +replace multi-crop in 3D medical images. The core idea of +sub-crop is straightforward: reducing the sampling space. As +illustrated in Fig. 5, sub-crop mainly consists of three steps: +(i) randomly crop two extensive global views with an IoU +constraint; (ii) find the minimum 3D bounding box over the +cropped global patches; (iii) randomly crop multiple local +patches within the 3D bounding box. There are two critical +operations in sub-crop: the constraint of IoU on global views +and the sampling of local patches within the minimum +bounding box. In practice, the first operation guarantees the +global-global association by ensuring the overlap between +large patches larger than a fixed threshold. The second +operation mitigates the disperse problem of local views and +helps the model to discover local-global relations. +3.5 +Overall training objective +After applying multi-crop/sub-crop to medical images, we +can acquire two global views {g1, g2} and ˆN local views +{l1, l2, ..., l ˆ +N}. For clarification, we denote the associated in- +puts in notations of loss functions. For instance, LC(g1, g2) +means we calculate LC on top of the extracted siamese +representations of two global views, where g1 and g2 can +be regarded as a pair of siamese images. At last, the overall +training objective of PCRLv2 can be formalized as follows: +LTotal(g1, g2, l1, ..., l ˆ +N) =LR(g1, g2) + LC(g1, g2) ++ +� +m∈{1,2} +ˆ +N +� +k=1 +LC(lk, gm). +(10) +There are three terms in LTotal: LR(g1, g2), LC(g1, g2), and +� +m∈{1,2} +� ˆ +N +k=1 LC(lk, gm). The first term is designed to +preserve pixel-level details in multi-scale learned repre- +sentations. The second term addresses the importance of +encoding multi-scale semantics into latent features. The last +term aims to capture the multi-scale global-local semantic +relations. +3.6 +Short discussion: PCRLv2 vs. PCRLv1 +Simpler. PCRLv1 combines the context restoration and +comparative SSL via transformation-conditioned attention +and cross-model mixup. These two components make +the framework heavy, less intuitive, and not easy to +implement. Compared to PCRLv1, PCRLv2 exploits a +simpler yet more intuitive design to incorporate pixel-level +and semantic information via multi-scale learning. As +aforementioned, PCRLv2 can be formulated as a simple +multi-task optimization problem whose objective function +maximizes the preservation of multi-level information in +latent visual representations. These characteristics make it +easier for both implementation and potential expansion. +Faster. PCRLv1 makes heavy use of mixup (to both inputs +and features) in its implementation, which is found to +deliver performance gains. In PCRLv2, we eliminate mixup +strategies and cut the training time in half. In addition, +PCRLv2 requires less running memory in GPUs during +the training stage, making it more practical in real-world +scenarios. +4 +EXPERIMENTS +In this section, we first conduct thorough ablation studies to +investigate the influence of different modules in PCRLv2. +Then, we evaluate the effectiveness of PCRLv2 on both +2D and 3D medical imaging tasks, including chest pathol- +ogy classification, pulmonary nodule detection, abdominal +organ segmentation, and brain tumor segmentation. For +model evaluation, we follow the pre-training (on source +data)→fine-tuning (on target data) protocol and employ two +settings, which are semi-supervised learning and transfer +learning. In the first setting, the source and target data come +from the same dataset. Specifically, we first pre-train the +model using all training data without labels, and then fine- +tune the pre-trained model with limited annotations. As for +transfer learning (the second setting), we pre-train and fine- +tune the model on different datasets. Different from semi- +supervised learning, we fine-tune the pre-trained model +with both limited and full annotations in transfer learning. +4.1 +Datasets +NIH ChestX-ray (2D) [38] is made up of 112,120 X- +ray +scans +from +30,805 +patients. +There +are +fourteen +different chest pathologies in NIH ChestX-ray, including +atelectasis, cardiomegaly, consolidation, edema, effusion, +emphysema, fibrosis, hernia, infiltration, mass, nodule, +pleural thickening, pneumonia, and pneumothorax. The +labels of radiographs were automatically extracted from +associated +radiology +reports +using +natural +language +process (NLP) techniques. We use NIH ChestX-ray in +semi-supervised learning in our experiments and treat it as +the target dataset in transfer learning. +CheXpert (2D) [21] involves 224,316 chest radiographs +from 65,240 patients for the presence of 14 common +chest +radiographic +observations: +no +finding, +enlarged +cardio, cardiomegaly, lung opacity, lung Lesion, edema, +consolidation, +pneumonia, +atelectasis, +pneumothorax, +pleural +effusion, +pleural +other, +fracture, +and +support +devices. Similar to NIH ChestX-ray, an NLP labeler was +developed to detect the presence of 14 observations in +radiology +reports +automatically. +In +practice, CheXpert +serves as the source data in transfer learning. +LUNA (3D) [33] was collected for the automatic detection +of pulmonary nodules, which involves 888 annotated +thoracic computed tomography (CT) scans. LUNA is a +cherry-picked subset of LIDC-IDRI [1], which excludes +scans with a slice thickness greater than 3mm, inconsistent +slice spacing, or missing slices. In the 888 scans, a total of +5,855 annotations were made by the radiologists, where +only nodules ≥ 3mm are categorized as relevant lesions, + +8 +and at least one radiologist checks each nodule. On LUNA, +we perform semi-supervised learning and transfer learning +experiments. For transfer learning, LUNA is mainly used +for self-supervised pre-training. +LiTS (3D) [4] releases 131 abdominal CT Volumes and +associated annotations for training and validation. There +are two types of labels in LiTS: the liver and tumor. In this +paper, we only utilize the ground truth masks of the liver +to evaluate the effectiveness of various SSL algorithms. The +task on LiTS is abdominal organ segmentation, where LiTS +is used for fine-tuning in transfer learning. +BraTS (3D) has been known as a series of challenges in brain +tumor segmentation. In this paper, we perform experiments +on the released 351 magnetic resonance imaging (MRI) scans +of BraTS 2018. There are three classes in BraTS: whole tumor +(WT), tumor core (TC), and enhancing tumor (ET). Similar +to the role of LiTS, BraTS serves as the target data in transfer +learning. +4.2 +Baselines +A variety of SSL baselines are included in our extensive +experiments, which can be roughly divided into three +categories: 2D specific methods, 3D specific approaches, +and generic (2D & 3D) methodologies. Details of baselines +in each category are listed below. +2D specific SSL methodologies consist of ImageNet-based +pre-training (IN) [14], Comparing to Learn (C2L) [48], +and Simple Siamese Learning (SimSiam) [10]. IN is the +most widely adopted pre-training methodology, which +conducts supervised pre-training on one of the biggest +natural image datasets, i.e., ImageNet [14]. C2L is a recently +proposed SSL approach based on momentum contrast (i.e., +MoCov1 [17] and MoCov2 [9]). SimSiam is a simple siamese +SSL framework that eliminates the barrier of negative +samples in contrastive learning and the use of a momentum +encoder in BYOL [15]. Besides, we compare PCLRv2 against +SimSiam to highlight the significance of the preserved +pixel-level information and multi-scale features. +3D specific SSL methodologies include Rubik’s cube++ [35] +and 3D-CPC [34]. Rubik’s cube++ is the most recent SSL +approach built on top of context restoration for 3D +medical images. It adopts a volume-wise transformation +for context permutation. In comparison, 3D-CPC is based +on contrastive predictive encoding [19], a variation of +contrastive learning, and demonstrates the most superior +performance among different SSL approaches investigated +in [34]. +Generic SSL methodologies involve train from scratch (TS), +Model Genesis (MG) [49], TransVW [16], and PCRLv1 [47] +(the conference version of our approach). MG resorts to ag- +gressive augmentations to generate corrupted input images, +based on which the model is asked to restore the original in- +puts. TransVW improves MG by appending an intermediate +classification head to encode anatomical patterns explicitly. +PCRLv1 first proposes simultaneously preserving semantic +and pixel-level information in SSL. +4.3 +Implementation details +Dataset pre-processing for pre-training. On NIH ChestX- +ray and CheXpert, each input image is resized to 224×224 +after random crop. On LUNA, we randomly crop a +volume from the whole CT scan with a random size +from {64×64×32, 96×96×64, 96×96×96, 112×112×64}. +Each cropped volume is then resized to 64×64×32. Each +voxel’s Hounsfield Unit (HU) in the crop is truncated to +[-1000,1000]. If a voxel’s HU is lower than -150, we regard +it as a background voxel. In practice, if over 85% voxels +within a crop belong to the background, we would not use +this crop in pre-training. +Dataset pre-processing for fine-tuning. For NIH ChestX- +ray and CheXpert, we follow the same pre-processing +procedures as in the pre-training stage. On LUNA, we +randomly crop a volume for each training iteration, and the +size of each crop is 48×48×48. On LiTS, we first localize +the liver and expand the target volume by 30 slices on +each axis. After random crop, the size of each crop is +256×256×64. Unlike LUNA, we truncate the HU of each +voxel to [-200, 200]. For BraTS, the size of each random crop +is 112×112×112×4. +Data augmentation and multi-crop/sub-crop. As shown in +Fig. 2, there are two types of augmentations, i.e., global and +local augmentations. Specifically, for 2D tasks, the global +augmentation includes random crop, random horizontal +flip, and random rotation. The local augmentation involves +random grayscale, gaussian blur, and cutout. In comparison, +for 3D tasks, the global augmentation consists of random +flip and random affine. Local augmentation strategies are +applied, including Gaussian blur, random noise, random +gamma, and random swap. Note that all 3D augmentations +are implemented following [30]. As for multi-crop in 2D +tasks, we resort to the scale factor of random crop2 to +generate global and local views. Specifically, we set the +range of scale to [0.3, 1] to generate two global views. For six +local views, the scale range is set to [0.05, 0.3]. Both global +and local views are resized to 224×224. As for sub-crop +in 3D tasks, we randomly sample two global views with +a random size from {64×64×32, 96×96×64, 96×96×96, +112×112×64}. The IoU constraint (i.e., threshold) between +two global views is 0.3. Then, we find the minimum +bounding box of global views, from which six local views +are randomly cropped, each with a random size from +{8×8×8, 16×16×16, 32×32×16, 32×32×32}. After random +crop, all 3D global views are resized to 64×64×32, while all +local views are resized to 16×16×16. +Training and evaluation details. We use stochastic gradient +descent (SGD) with momentum as the default optimizer, +where the momentum is set to 0.9. The initial learning rate +is 1e-2, and we employ the cosine annealing strategy for +learning rate decay. We set the weight decay to 1e-5. The +number of training epochs is 240. The batch sizes of 2D pre- +training and fine-tuning (on NIH ChestX-ray or CheXpert) +are 256 and 512, respectively. As for 3D pre-training, the +2. https://pytorch.org/vision/main/generated/torchvision. +transforms.RandomResizedCrop.html + +9 +0 +15 +Epochs +-0.5 +-1.5 +Training MSE loss (log10) +w/ skip +w/o skip +Fig. 6. Influence of skip connections in pixel restoration. We display the +loss curve of mean square error (MSE) in the first 15 epoches. +TABLE 1 +Impact of skip connections on chest pathology identification (NIH +ChestX-ray), brain tumor segmentation (BraTS), and abdominal organ +segmentation (LiTS). On NIH, We use 95% unlabeled training data for +pre-training, while the rest 5% data with labels are used for fine-tuning. +On BraTS and LiTS, we use 10% labeled data for fine-tuning. +Datasets +w/o skip +w/ skip +Gain +NIH +76.6 +75.4 +1.2 +BraTS +73.0 +71.5 +1.5 +LiTS +79.0 +77.6 +1.4 +batch size (on LUNA) is 32. For 3D fine-tuning tasks, the +batch sizes on LUNA, LiTS, and BraTS are 32, 4, and +4, respectively. The evaluation metric on NIH ChestX-ray, +CheXpert, and LUNA is AUROC (Area Under the Receiver +Operating Characteristics). For segmentation tasks on LiTS +and BraTS, we use Dice similarity as the evaluation metric. +We use 70%, 10%, and 20% of the whole dataset to build the +training, validation, and test sets. In particular, for semi- +supervised learning, we construct the pre-training set by +removing a specific amount of data from the entire training +set. At the same time, the remainder is used as the training +set for fine-tuning. Binary cross-entropy loss is used for the +fine-tuning of NIH ChestX-ray, CheXpert, and LUNA, while +Dice loss is used for the fine-tuning of LiTS and BraTS. +4.4 +Ablation studies +Impact of skip connections on pixel restoration. In Fig. 6, +we present the mean square error (MSE) loss (cf. Eq. 4) +curves during the training stage. We see that the MSE +loss, with skip connections, decreases rapidly in the first +15 training epochs. In comparison, the proposed nsUNet +(w/o skip) slows down the decreasing rate of MSE loss. +These phenomena are consistent with the role of skip +connections, which bridges the gap between low-level +pixel details and high-level latent semantics. The existence +of skip connections makes it easier to restore pixels +by incorporating pixel-level details from low-level but +high-resolution feature maps. However, nsUNet removes +skip connections, avoiding shortcut solutions to context +restoration. Although this design makes it harder to restore +pixels (higher loss values in Fig. 6), it helps encode pixel- +level information into high-level semantic representations. +ℱ# +ℱ$ +ℱ% +ℱ" +ℱ! +ℱ# +& +ℱ$ +& +ℱ% +& +ℱ" +& +ℱ" +& +(a) Pairwise +ℱ# +ℱ$ +ℱ% +ℱ" +ℱ! +ℱ# +& +ℱ$ +& +ℱ% +& +ℱ" +& +ℱ" +& +(b) Cross-scale +Fig. 7. Two choices of how to conduct siamese feature comparison for +multiple feature scales. Here, we primarily consider pairwise feature +comparison and cross-scale feature comparison. +Such advantage can be verified by the performance gains in +Table 1, where removing skip connections brings over 1% +improvement to chest pathology identification, brain tumor +segmentation, and abdominal organ segmentation. +How to conduct siamese feature comparison for multiple +feature scales? We illustrate two intuitive choices in Fig. 7. +Besides the adopted pairwise comparison manner (Fig. 7a), +another obvious choice is to compare siamese features +following a crossed way (a similar strategy was used +in [3]). As shown in Fig. 7b, the cross-scale comparison +aggressively compares siamese features across all feature +scales. The motivation behind is to introduce multi-scale +latent representations by coupling features across different +scales. Table 2 reports the experimental results of pairwise +and cross-scale siamese feature comparison. We find that +cross-scale feature comparison slightly deteriorates the +performance of semi-supervised pathology identification +by 0.6 percents. The underlying reason might be that the +features in each scale maintains distinct characteristics, +and neglecting these discrepancies can lead to degenerate +feature representations. +Investigation of different modules in PCRLv2. In Table 3, +we study and report the impact of different modules on +the whole tumor (WT) and enhancing tumor (ET) classes of +BraTS. Note that in practice, most instances of WT are much +larger than instances from ET, making ET instances harder +to segment. Besides, we also present the transfer learning +results on NIH ChestX-ray. +TABLE 2 +Results of pairwise and crossed siamese feature comparison +(semi-supervised learning on NIH ChestX-ray). The ratio of unlabeled +to labeled data is 9.5:0.5. +Pairwise +Crossed [3] +Gain +Mean AUROC +76.6 +76.0 +0.6 +First of all, we investigate the influence of pixel restora- +tion (row 0) and feature comparison (row 1), respectively. +We directly reconstruct the full resolution uncorrupted im- +ages for the pixel restoration task while siamese feature +comparison is conducted on the last-layer output of the +encoder. Comparing row 0 with row 1, we see that the +context restoration task is more advantageous in segmenta- +tion of small tumor regions (i.e., ET) while the comparative +SSL is more capable of dealing with large tumor regions +(i.e., WT) and chest pathologies. Such comparison shows + +10 +TABLE 3 +Impact of different modules in PCRLv2. Res. and Comp. denote the tasks of pixel restoration and feature comparison, respectively. S (N) means +there are N scales included. MC and SC stand for the multi-crop and proposed sub-crop strategies, respectively. WT and ET denote classes of the +whole tumor and enhancing tumor in BraTS, respectively. In most cases, instances from WT are much larger (in size) than those of ET. We +performed these experiments by first using LUNA for self-supervised pre-training, and then we fine-tune the pre-trained model on BraTS using +10% labeled data. NIH denotes the transfer learning on chest pathology identification, where we use CheXpert for pre-training and fine-tune the +pre-trained model with 50% labeled data from NIH ChestX-ray. +# +Res. +Comp. +S (3) +S (5) +MC +SC +WT (BraTS) +ET (BraTS) +NIH +0 +✓ +74.2 +64.9 +78.2 +1 +✓ +76.4 +63.8 +78.5 +2 +✓ +✓ +76.2 +64.6 +80.9 +3 +✓ +✓ +✓ +76.9 +66.1 +81.5 +4 +✓ +✓ +✓ +77.2 +66.8 +82.0 +5 +✓ +✓ +✓ +✓ +fail +fail +82.5 +6 +✓ +✓ +✓ +✓ +77.7 +67.2 +82.7 +that semantic information preservation may be more helpful +to the detection of large objects, while segmenting small +objects requires the incorporation of pixel-level information. +In row 2, we can already acquire noticeable performance +gains by directly combining pixel restoration and feature +comparison. +Next, we show that multi-scale representations benefit +both pixel restoration and feature comparison tasks. By +conducting both tasks on 3 scales, we observe a 0.7-percent +improvement on WT, a 1.5-percent gain on ET, and a +0.6-percent improvement on chest pathology classification. +These results show that introducing multiple scales is more +helpful to the segmentation of small regions. Moreover, +by increasing the number of scales from 3 to 5, we can +improve the accuracy of all three tasks consistently. Not +surprisingly, ET benefits the most from the introduction of +multiple scales, indicating the necessity of utilizing multi- +scale representations in medical image segmentation. +Last but not the least, we investigate the significance of +multi-crop (row 4) and sub-crop (row 5). We empirically +found that directly applying multi-crop to 3D medical vol- +umes leads to the failure of model training. The underlying +reason might be that it is difficult for cropped global and +local views to maintain clear spatial relations in the 3D +space as in the 2D space. In contrast, sub-crop can provide +consistent performance gains on both types of tumor regions +by successfully preserving the spatial relations in latent +representations. When applying sub-crop to 2D X-rays, +we observe a marginal improvement over multi-crop. The +underlying reason is that sub-crop is proposed to handle +dispersed sampled views in a 3D space to guarantee the +model captures local-global relations. However, in a 2D +space, the sampled views usually (partly) overlap. +4.5 +Semi-supervised chest pathology identification +Table 4 presents the experimental results of applying semi- +supervised learning on NIH ChestX-ray. Specifically, we use +a specific amount of the training set (denoted as the labeling +ratio in Table 4) as labeled data while the remaining training +data is used for self-supervised pre-training. +From Table 4, we see that self-supervised pre-training +can dramatically boost the performance compared to train +from scratch (TS), which verify the necessity of conduct- +ing pre-training in medical imaging. Comparing MG with +TransVW, they show similar performance in different label- +ing ratios. Such comparison is easy to explain as TransVW +is built upon MG, and both are based on context restora- +tion. TransVW performs slightly better than MG, as it +incorporates an additional classification head to encode +more semantics. Compared to context restoration based +methods, comparative methodologies (C2L and SimSiam) +display better overall and class-specific results, especially +in small labeling ratios. The underlying reason might be +that semantic information is more critical than pixel-level +information in chest pathology detection. As for C2L and +SimSiam, C2L performs better when the amount of labeled +data is quite limited. However, SimSiam gradually produces +better diagnosis results as the labeling ratio increases. +After incorporating the semantic, pixel-level, and scale +information into a unified framework, PCRLv2 outperforms +various SSL baselines in different labeling ratios signifi- +cantly. It surpasses the previous conference version by clear +margins, i.e., PCRLv1. Particularly, PCRLv2 seems to have +more advantages in small labeling ratios. For instance, when +the labeling ratio is 5%, PCRLv2 outperforms PCRLv1 by +2.5 percents on average, which verifies the significance of +multi-scale latent representations. +4.6 +Semi-supervised pulmonary nodule detection +In Table 5, we report the experimental results of semi- +supervised pulmonary nodule detection. Interestingly, we +observe narrowed performance gaps between TS and SSL +baselines than those reported in Table 4. One possible ex- +planation is that the task of detecting pulmonary nodules +is less sensitive to the amount of labeled data. Among +all SSL baselines, Cube++ gives better performance when +utilizing small amounts of labeled data, while 3D-CPC is +more advantageous in large labeling ratios. In addition, we +see TransVW quickly catching up with MG and Cube++ as +the labeling ratio increases. +PCRLv1 outperforms previous SSL approaches in dif- +ferent labeling ratios by large margins. After incorporat- +ing multi-scale latent representations, PCRLv2 consistently +surpasses PCRLv1 in a range of labeling ratios. When the +baseline SSL methods show similar performance as the +labeling ratio increases, PCRLv2 can still provide impressive +improvements over PCRLv1 and previous SSL approaches. + +11 +TABLE 4 +Semi-supervised chest pathology identification (on NIH ChestX-ray). The labeling ratio denotes the amount of data with labels in the training set +that is used for fine-tuning while the remaining data in the training set is used for self-supervised pre-training. The best results are bolded. +Labeling ratio +Methodology +Mean +Atelectasis +Cardiomegaly +Effusion +Infiltration +Mass +Nodule +Pneumonia +Pneumothorax +Consolidation +Edema +Emphysema +Fibrosis +Pleural Thick. +Hernia +5% +TS +61.8 +58.8 +72.0 +68.8 +51.5 +63.8 +49.2 +57.4 +67.4 +61.5 +71.0 +62.7 +58.1 +60.0 +63.1 +MG [49] +66.4 +63.4 +74.1 +72.9 +53.5 +67.2 +54.3 +59.9 +71.3 +66.5 +77.0 +65.8 +64.5 +62.8 +76.2 +TransVW [16] +66.5 +64.2 +72.9 +72.2 +54.8 +69.4 +55.7 +59.6 +71.0 +64.8 +77.4 +66.6 +63.6 +62.8 +75.6 +C2L [48] +71.7 +69.9 +77.9 +76.2 +59.1 +73.4 +60.0 +64.5 +76.2 +71.4 +80.3 +76.1 +69.9 +68.4 +80.4 +SimSiam [10] +71.7 +68.9 +79.3 +77.8 +58.7 +73.0 +61.0 +65.4 +76.2 +72.1 +81.7 +75.1 +69.6 +68.1 +76.8 +PCRLv1 [47] +74.1 +70.1 +80.3 +79.3 +61.8 +76.8 +64.6 +68.6 +77.2 +72.8 +83.7 +77.4 +71.3 +72.7 +80.8 +PCRLv2 +76.6 +75.7 +81.0 +80.3 +64.0 +76.8 +68.7 +70.7 +83.2 +77.5 +87.8 +79.2 +72.5 +73.2 +81.8 +10% +TS +68.1 +65.8 +77.6 +74.4 +57.1 +69.4 +54.8 +63.0 +72.9 +68.3 +78.8 +68.2 +64.3 +66.4 +72.5 +MG [49] +70.0 +67.1 +78.1 +76.1 +57.2 +72.8 +57.5 +63.3 +75.5 +70.9 +79.5 +68.8 +67.4 +68.0 +77.6 +TransVW [16] +70.2 +66.6 +78.9 +74.9 +58.4 +71.2 +59.5 +64.8 +72.6 +70.4 +79.4 +70.7 +67.2 +68.3 +79.5 +C2L [48] +74.1 +72.3 +81.7 +79.9 +60.2 +74.6 +62.7 +67.6 +78.7 +73.9 +83.5 +78.2 +72.8 +69.8 +81.4 +SimSiam [10] +74.0 +71.2 +81.4 +78.9 +60.2 +75.5 +63.2 +67.3 +78.7 +73.2 +83.5 +77.7 +72.5 +71.8 +80.8 +PCRLv1 [47] +76.2 +73.6 +82.9 +81.2 +64.7 +77.1 +66.7 +69.7 +79.8 +74.5 +86.9 +78.8 +75.6 +74.2 +81.1 +PCRLv2 +78.2 +77.2 +84.3 +84.4 +67.4 +77.5 +68.9 +71.6 +84.4 +77.8 +89.0 +79.3 +76.1 +74.0 +82.4 +20% +TS +71.5 +68.9 +80.7 +77.5 +60.2 +73.6 +58.7 +66.2 +76.1 +71.7 +82.9 +72.2 +69.0 +68.7 +74.7 +MG [49] +73.9 +71.9 +83.0 +80.0 +62.3 +75.2 +62.2 +67.5 +79.0 +73.3 +83.6 +73.4 +71.0 +70.6 +81.4 +TransVW [16] +74.3 +71.6 +82.5 +80.1 +62.3 +76.7 +62.8 +69.2 +78.2 +73.5 +83.8 +75.4 +72.2 +71.2 +80.3 +C2L [48] +76.4 +74.2 +83.9 +81.7 +63.8 +77.3 +64.7 +70.3 +81.5 +75.5 +86.0 +80.2 +75.2 +73.4 +81.8 +SimSiam [10] +76.5 +73.8 +84.0 +81.4 +63.2 +78.2 +64.7 +69.6 +82.1 +76.2 +86.4 +80.7 +75.0 +73.9 +81.7 +PCRLv1 [47] +78.8 +75.4 +86.2 +83.6 +65.1 +79.9 +69.6 +72.0 +82.3 +79.9 +88.3 +82.6 +76.5 +75.9 +81.9 +PCRLv2 +79.9 +78.1 +87.2 +85.9 +68.2 +80.5 +69.9 +72.5 +85.3 +80.4 +89.2 +83.1 +77.5 +77.0 +83.5 +30% +TS +73.4 +70.6 +81.9 +79.1 +61.6 +75.5 +60.7 +68.8 +78.3 +72.7 +84.3 +74.1 +70.3 +70.9 +78.9 +MG [49] +76.1 +74.3 +84.4 +82.1 +63.6 +78.3 +64.4 +69.6 +81.2 +75.8 +85.6 +75.9 +73.6 +73.6 +82.8 +TransVW [16] +76.7 +74.9 +84.1 +81.9 +64.9 +79.0 +65.3 +70.9 +80.3 +76.2 +86.5 +78.6 +74.5 +74.2 +82.1 +C2L [48] +77.5 +74.3 +84.8 +82.6 +64.6 +78.3 +66.3 +71.5 +83.0 +76.8 +87.6 +81.3 +76.5 +74.4 +82.9 +SimSiam [10] +78.0 +75.4 +85.1 +82.9 +65.0 +79.4 +67.0 +71.4 +83.4 +77.4 +87.8 +82.8 +76.1 +75.5 +82.7 +PCRLv1 [47] +79.0 +75.5 +86.6 +83.8 +65.9 +80.7 +70.2 +72.8 +82.9 +80.4 +88.9 +83.3 +76.6 +76.5 +81.9 +PCRLv2 +81.1 +78.4 +87.6 +86.6 +69.6 +82.8 +72.0 +74.0 +86.2 +81.0 +89.9 +84.4 +79.5 +79.0 +84.6 +40% +TS +75.4 +72.6 +83.6 +81.5 +62.9 +77.3 +63.3 +70.1 +80.3 +74.9 +85.5 +76.4 +72.5 +73.0 +81.8 +MG [49] +77.3 +75.4 +86.0 +83.3 +65.1 +79.0 +65.1 +70.8 +82.1 +77.0 +87.3 +76.7 +74.8 +74.9 +83.5 +TransVW [16] +77.6 +75.0 +85.1 +82.7 +65.2 +79.7 +66.5 +72.0 +81.0 +76.7 +87.2 +79.2 +75.5 +76.5 +83.7 +C2L [48] +79.0 +76.0 +86.1 +84.3 +66.0 +80.0 +67.9 +72.5 +84.1 +78.5 +88.5 +83.7 +77.9 +76.6 +83.8 +SimSiam [10] +79.4 +76.7 +86.7 +84.7 +67.0 +80.9 +69.0 +73.1 +84.4 +78.9 +88.9 +83.5 +77.7 +76.6 +83.4 +PCRLv1 [47] +79.9 +76.7 +87.1 +84.9 +67.1 +82.7 +72.2 +73.3 +83.6 +80.6 +89.2 +83.8 +77.3 +76.9 +83.2 +PCRLv2 +81.5 +78.7 +87.8 +87.0 +69.8 +83.2 +72.5 +74.7 +86.3 +81.2 +90.2 +84.9 +80.0 +79.4 +85.0 +TABLE 5 +Semi-supervised pulmonary nodule detection (on LUNA). The labeling +ratio indicates how much data from the training set with labels is utilized +for fine-tuning while the rest of the data is used for pre-training. Best +results are bolded. +Methodology +Labeling ratio +10% +20% +30% +40% +TS +78.4 +83.0 +85.7 +87.5 +MG [49] +80.2 +85.0 +87.5 +90.3 +TransVW [16] +79.3 +84.5 +87.9 +90.5 +Cube++ [35] +81.4 +85.2 +87.9 +90.0 +3D-CPC [34] +80.2 +85.2 +88.3 +90.6 +PCRLv1 [47] +84.4 +87.5 +89.8 +92.2 +PCRLv2 +85.5 +88.3 +90.3 +93.1 +4.7 +Transfer learning on chest pathology identification +In Table 6, we validate the transferable ability of visual +representations provided by different pre-training method- +ologies. Specifically, we compare PCRLv2 against train from +scratch, ImageNet-based pre-training (IN), different SSL +baselines, and PCRLv1. +Comparing MG/TransVW with IN, we see context +restoration based SSL maintains the limited transferable +ability. This phenomenon becomes more apparent when the +target domain has quite limited annotations. The underlying +reason is that semantic information plays a crucial role in +transfer learning. In contrast, the significant performance +gains brought by C2L and SimSiam again verify the effec- +tiveness of comparative SSL. C2L and SimSiam still cannot +outperform IN by significant margins, especially when con- +sidering that IN is more advantageous when the labeling +ratio is 10%. +After integrating the benefits of context restoration +based and comparative SSL, PCRLv1 is already capable of +outperforming previous SSL methodologies by observable +margins. Furthermore, by exploiting multi-scale semantic +and pixel-level information, PCRLv2 achieves consistent +improvements over PCRLv1 in overall and class-specific +results in different labeling ratios. +4.8 +Transfer learning on brain tumor segmentation +We report the experimental results of applying transfer +learning to brain tumor segmentation in Table 7, where +we use LUNA dataset for self-supervised pre-training and +fine-tune the pre-trained model with different amounts of +labeled data. + +12 +TABLE 6 +Transfer learning on chest pathology identification. We pre-train the model using data from CheXpert (without labels). Then, we fine-tune the +pre-trained model on NIH ChestX-ray with different amounts of labeled data (denotes as different labeling ratios). The best results are bolded. +Labeling ratio +Methodology +Mean +Atelectasis +Cardiomegaly +Effusion +Infiltration +Mass +Nodule +Pneumonia +Pneumothorax +Consolidation +Edema +Emphysema +Fibrosis +Pleural Thick. +Hernia +10% +TS +68.1 +67.6 +63.3 +76.8 +57.5 +71.5 +61.8 +64.2 +76.2 +69.8 +80.2 +72.4 +62.8 +68.0 +61.1 +IN [28] +73.5 +73.3 +68.7 +81.6 +63.0 +76.6 +67.3 +70.0 +81.3 +75.6 +85.9 +78.5 +68.6 +72.5 +65.9 +MG [49] +70.1 +69.9 +65.6 +79.2 +59.4 +72.9 +64.3 +67.0 +77.9 +72.0 +82.3 +75.8 +65.9 +69.6 +59.4 +TransVW [16] +69.7 +69.4 +64.3 +78.2 +59.5 +72.6 +63.1 +67.2 +77.2 +70.9 +83.0 +75.3 +65.8 +68.9 +60.2 +C2L [48] +73.1 +72.5 +68.0 +81.3 +62.4 +75.8 +67.2 +70.2 +80.6 +74.8 +85.4 +78.4 +68.3 +72.2 +66.1 +SimSiam [10] +72.5 +71.9 +67.5 +81.2 +61.7 +75.9 +66.6 +69.6 +79.8 +74.2 +84.8 +77.6 +67.7 +71.8 +64.5 +PCRLv1 [47] +75.8 +75.4 +70.6 +84.2 +65.5 +78.9 +69.6 +72.7 +83.5 +77.6 +88.5 +80.8 +71.3 +74.8 +67.6 +PCRLv2 +77.2 +76.8 +72.0 +85.6 +66.8 +80.2 +71.0 +74.0 +84.8 +78.9 +89.8 +82.2 +72.6 +76.2 +69.7 +20% +TS +71.4 +71.8 +73.1 +78.4 +59.6 +72.5 +64.5 +66.6 +77.7 +71.7 +82.0 +75.5 +69.8 +68.9 +68.2 +IN [14] +76.2 +75.9 +78.3 +82.9 +64.2 +77.8 +68.8 +70.7 +83.0 +76.4 +87.2 +80.0 +75.3 +73.9 +73.1 +MG [49] +73.8 +73.9 +75.4 +80.2 +61.9 +74.9 +66.5 +68.3 +80.0 +74.0 +85.1 +78.1 +72.8 +71.5 +71.3 +TransVW [16] +73.8 +73.0 +75.5 +80.1 +62.3 +75.6 +66.7 +68.6 +80.2 +74.0 +85.2 +77.5 +72.9 +71.5 +69.4 +C2L [48] +77.0 +76.5 +78.9 +83.4 +65.0 +78.6 +69.8 +71.8 +83.5 +77.2 +88.1 +80.8 +76.0 +74.2 +73.5 +SimSiam [10] +76.6 +76.6 +78.7 +83.3 +64.6 +77.9 +69.2 +71.6 +83.1 +76.9 +87.8 +80.5 +75.5 +73.8 +73.6 +PCRLv1 [47] +77.5 +77.3 +79.7 +84.3 +65.7 +78.9 +70.3 +72.8 +83.8 +77.6 +88.6 +81.1 +76.5 +74.8 +74.3 +PCRLv2 +79.4 +79.0 +81.3 +85.9 +67.3 +80.8 +72.1 +74.0 +86.0 +79.4 +90.3 +83.1 +78.4 +76.7 +76.6 +30% +TS +73.5 +71.7 +79.7 +79.9 +60.5 +76.5 +68.4 +66.8 +79.2 +72.8 +83.4 +76.9 +71.4 +70.5 +71.3 +IN [14] +78.5 +77.2 +84.6 +84.3 +66.2 +80.8 +73.0 +72.3 +84.0 +78.0 +88.5 +82.0 +76.8 +75.3 +76.0 +MG [49] +75.6 +74.1 +81.8 +81.0 +63.3 +77.9 +70.1 +69.0 +80.9 +74.8 +85.4 +79.7 +73.6 +72.6 +74.2 +TransVW [16] +75.7 +74.8 +81.4 +81.0 +63.6 +77.7 +69.9 +69.8 +80.9 +75.4 +86.0 +79.3 +73.9 +72.3 +73.8 +C2L [48] +78.6 +77.1 +84.5 +84.5 +66.1 +81.1 +73.0 +72.5 +84.0 +78.1 +88.3 +82.1 +76.8 +75.5 +76.8 +SimSiam [10] +78.3 +77.0 +84.4 +84.1 +65.7 +80.7 +72.7 +72.2 +83.9 +77.9 +88.1 +82.1 +76.6 +75.2 +75.6 +PCRLv1 [47] +79.9 +78.5 +85.8 +85.6 +67.4 +82.3 +74.2 +73.8 +85.5 +79.4 +89.7 +83.5 +78.1 +76.7 +78.1 +PCRLv2 +80.5 +79.1 +86.4 +86.2 +68.0 +82.8 +74.8 +74.3 +86.0 +80.0 +90.3 +84.1 +78.6 +77.2 +79.2 +40% +TS +75.4 +72.6 +80.0 +81.0 +62.5 +76.9 +69.2 +68.0 +80.7 +74.7 +85.1 +79.5 +74.0 +71.0 +79.8 +IN [14] +79.0 +76.7 +84.2 +84.3 +66.3 +80.7 +73.6 +72.3 +84.7 +78.5 +88.6 +83.4 +77.4 +75.0 +79.7 +MG [49] +76.5 +74.1 +81.3 +81.7 +63.9 +77.9 +71.1 +70.1 +82.5 +76.1 +85.6 +80.6 +74.5 +73.1 +77.9 +TransVW [16] +77.3 +75.2 +82.4 +82.4 +64.4 +79.0 +71.4 +70.5 +83.2 +76.7 +86.6 +82.0 +75.8 +73.6 +78.4 +C2L [48] +79.1 +76.9 +84.3 +84.5 +66.4 +80.8 +73.4 +72.2 +84.8 +78.3 +88.6 +83.4 +77.2 +75.4 +80.6 +SimSiam [10] +78.9 +76.7 +83.9 +84.1 +66.6 +80.4 +73.1 +72.1 +84.7 +78.1 +88.4 +83.4 +77.2 +74.8 +80.5 +PCRLv1 [47] +80.8 +78.5 +86.0 +86.2 +68.2 +82.4 +75.2 +74.0 +86.6 +80.2 +90.2 +85.1 +79.0 +76.9 +82.1 +PCRLv2 +81.5 +79.2 +86.6 +86.9 +68.9 +83.0 +75.8 +74.6 +87.2 +80.8 +90.9 +85.8 +79.7 +77.6 +83.4 +50% +TS +77.5 +75.2 +82.0 +82.0 +64.5 +79.6 +71.8 +71.3 +82.9 +75.8 +86.6 +80.9 +76.1 +75.5 +80.3 +IN +79.5 +77.2 +84.5 +84.4 +66.6 +81.4 +73.6 +73.0 +84.6 +78.2 +89.1 +82.7 +77.9 +77.3 +82.0 +MG [49] +77.6 +75.0 +82.8 +82.8 +64.8 +79.5 +71.8 +71.6 +82.3 +75.7 +86.7 +81.5 +76.2 +75.7 +79.5 +TransVW [16] +77.3 +74.5 +81.9 +82.4 +64.8 +78.8 +71.5 +71.3 +82.4 +75.7 +86.8 +80.4 +75.7 +74.9 +80.6 +C2L [48] +79.8 +77.6 +84.7 +84.5 +67.0 +81.6 +73.6 +73.4 +84.7 +78.5 +89.0 +83.1 +78.4 +78.0 +82.6 +SimSiam [10] +80.0 +77.7 +84.9 +84.8 +67.1 +81.7 +74.0 +73.5 +84.7 +78.3 +89.5 +83.6 +78.8 +77.7 +83.2 +PCRLv1 [47] +81.2 +78.7 +86.1 +86.3 +68.3 +82.8 +75.4 +74.5 +86.8 +80.4 +90.5 +85.3 +79.5 +78.2 +83.5 +PCRLv2 +82.5 +80.0 +87.4 +87.3 +69.6 +84.1 +76.4 +76.1 +87.4 +81.0 +91.8 +85.9 +81.0 +80.4 +86.1 +100% +TS +80.9 +77.7 +86.1 +85.1 +67.7 +84.2 +73.3 +73.9 +84.9 +78.7 +89.4 +85.4 +79.4 +78.5 +87.6 +IN +80.8 +77.8 +86.3 +84.7 +67.3 +83.6 +73.0 +74.1 +84.9 +78.8 +89.5 +85.7 +79.6 +78.2 +87.0 +MG [49] +80.8 +77.8 +86.3 +84.7 +67.3 +83.6 +73.0 +74.1 +84.9 +78.8 +89.5 +85.7 +79.6 +78.2 +87.0 +TransVW [16] +81.2 +77.9 +86.4 +85.3 +67.6 +84.3 +73.8 +74.4 +85.1 +79.3 +89.8 +86.2 +80.0 +78.6 +88.8 +C2L [48] +81.4 +78.2 +87.0 +85.3 +68.3 +84.8 +73.7 +74.8 +85.5 +79.6 +90.1 +86.3 +80.0 +78.6 +88.1 +SimSiam [10] +81.6 +78.3 +87.2 +85.5 +68.3 +84.9 +74.2 +74.7 +85.7 +79.6 +90.1 +86.2 +80.2 +79.1 +89.1 +PCRLv1 [47] +83.0 +79.8 +88.5 +87.1 +69.7 +86.1 +75.6 +76.1 +87.0 +81.2 +91.6 +87.7 +81.7 +80.4 +90.2 +PCRLv2 +84.0 +80.7 +89.3 +87.9 +70.5 +87.0 +76.4 +77.0 +87.9 +82.0 +92.5 +88.6 +82.6 +81.3 +91.6 +TABLE 7 +Transfer learning on brain tumor segmentation (on BraTS). WT, TC, and ET stand for the whole tumor, tumor core, and enhancing tumor. For all +SSL approaches, we use LUNA for pre-training, and then fine-tune the pre-trained model on BraTS with varying amounts of labeled data. Best +results are bolded. +Methodology +10% +20% +30% +40% +100% +Mean +WT +TC +ET +Mean +WT +TC +ET +Mean +WT +TC +ET +Mean +WT +TC +ET +Mean +WT +TC +ET +TS +66.6 +71.2 +66.7 +62.1 +72.7 +78.5 +74.3 +65.5 +76.7 +81.8 +77.9 +70.6 +77.1 +82.3 +78.3 +70.9 +81.5 +86.8 +82.8 +75.1 +MG [49] +69.6 +72.4 +71.4 +65.1 +75.5 +80.4 +77.3 +68.9 +79.6 +84.2 +80.6 +74.1 +80.4 +85.3 +82.0 +74.0 +82.4 +87.1 +83.6 +76.6 +TransVW [16] +70.3 +74.6 +71.7 +64.6 +75.6 +79.9 +75.4 +71.5 +79.1 +83.8 +79.9 +73.6 +80.8 +85.8 +82.1 +74.5 +82.3 +87.1 +83.3 +76.5 +Cube++ [35] +69.0 +74.5 +70.6 +61.9 +74.9 +80.7 +75.9 +68.1 +79.3 +84.0 +79.4 +74.5 +79.7 +84.5 +80.0 +74.6 +82.2 +87.2 +82.4 +77.0 +3D-CPC [34] +70.1 +76.7 +70.5 +63.1 +75.9 +81.6 +75.6 +70.5 +79.4 +84.6 +79.9 +73.7 +81.2 +86.5 +81.8 +75.3 +82.9 +88.0 +83.3 +77.4 +PCRLv1 [47] +71.6 +76.9 +73.1 +65.2 +77.6 +81.4 +79.1 +72.7 +81.1 +84.9 +82.2 +76.6 +83.3 +87.5 +84.6 +78.2 +85.0 +89.0 +86.2 +80.2 +PCRLv2 +73.0 +77.7 +74.3 +67.2 +78.8 +83.2 +79.4 +74.0 +82.1 +85.1 +82.7 +78.7 +84.1 +87.9 +84.5 +80.1 +85.6 +89.4 +85.9 +81.7 + +13 +TransVW +PCRLv1 +PCRLv2 +GT +TransVW +PCRLv1 +PCRLv2 +GT +10% +10% +10% +20% +20% +20% +TransVW +PCRLv1 +PCRLv2 +GT +10% +20% +30% +TransVW +PCRLv1 +PCRLv2 +GT +10% +20% +30% +b +c +Atelectasis +TransVW +PCRLv1 +PCRLv2 +Effusion +Infiltration +Mass +Nodule +Pneumonia +TransVW +PCRLv1 +PCRLv2 +a +Fig. 8. Visual interpretation of the transfer learning on chest pathology identification (a), and segmentation results of brain tumor (b) and liver (c). We +mainly compare PCRLv2 against PCRLv1 and TransVW. Red boxes in the top figure a denote the ground-truth (GT) annotations from radiologists. +In figure b, we present the segmentation results of the enhancing tumor (ET) from BraTS when the labeling ratios are 10% and 20%. Similarly in +the bottom figure, we display the liver segmentation results in three different labeling ratios (10%, 20%, and 30%). + +PORTABLEPORTABLEPORTABLE14 +TABLE 8 +Transfer learning on abdominal organ segmentation (on LiTS). We use +LUNA for pre-training, and fine-tune the pre-trained model on LiTS with +different amounts of labeled data. Best results are bolded. +Methodology +Labeling ratio +10% +20% +30% +40% +100% +TS +71.1 +77.2 +84.1 +87.3 +90.7 +MG [49] +73.3 +79.5 +84.3 +87.9 +91.3 +TransVW [16] +73.8 +79.3 +85.5 +88.2 +91.4 +Cube++ [35] +74.2 +79.3 +84.5 +88.2 +91.8 +3D-CPC [34] +74.8 +80.2 +85.6 +88.9 +91.9 +PCRLv1 [47] +77.3 +83.5 +87.8 +90.1 +93.7 +PCRLv2 +79.0 +86.5 +89.3 +90.9 +94.5 +Nodule +Infiltrate +Atelectasis +Fig. 9. Failure case analysis on chest pathology identification. Red boxes +stand for the lesion areas delineated by radiologists. Images are from +NIH ChestX-ray. +Somewhat surprisingly, we find 3D-CPC does not out- +perform context restoration based SSL (MG, TransVW, and +Cube++) as obviously as those in Tables 4, 5, and 7. +This comparison is consistent with our intuition: pixel- +level information matters a lot in medical image segmen- +tation. Again, PCRLv1 and PCRLv2 outperform previous +SSL methodologies in all three classes by large margins. +Compared to PCRLv1, PCRLv2 is more advantageous in +segmenting the enhancing tumor (ET) regions, which are +often smaller than WT and TC, and thus harder to segment. +The performance gains on ET again verify the effectiveness +of multi-scale latent representations, which advances the +segmentation of small objects. +4.9 +Transfer learning on liver segmentation +In Table 8, we present the results of liver segmentation. +There exist three observable phenomena. First, we see that +all SSL approaches provide substantial performance gains +over train from scratch. Second, we find the comparative +methodology, i.e., 3D-CPC, achieves comparable segmen- +tation performance to traditional context restoration based +SSL. This phenomenon verifies the necessity of utilizing +pixel-level information in medical image segmentation (sim- +ilar results also appear in Table 7). Last but not the least, +PCRLv2 consistently outperforms PCRLv1 in all labeling +ratios, which again validates the effectiveness of introducing +multiple scales into SSL. +4.10 +Visual analysis +In Fig. 8, we visually analyze the experimental results of +transfer learning with limited annotations on chest pathol- +ogy identification (10%), brain tumor segmentation (10% +and 20%), and liver segmentation (10%, 20%, and 30%). +Here, we compare PCRLv2 against generic SSL methodolo- +gies. Considering TransVW was developed on top of MG, +we exclude MG and compare PCRLv2 against PCRLv1 and +TransVW. +Fig. 8a presents the visual interpretation of chest pathol- +ogy diagnoses using CAM [45] on six different pathologies. +We find that TransVW fails to capture the correct location of +lesions on atelectasis, infiltration, nodule, and pneumonia. +In comparison, PCRLv1 can generate more interpretable +diagnosis results but still yields inconsistent predictions on +infiltration and nodule. By integrating multi-scale latent rep- +resentations, PCRLv2 can capture the small lesion areas on +infiltration and nodule, resulting in centralized yet accurate +diagnosis results. +In Fig. 8b and Fig. 8c, we visualize the segmentation +results of the enhancing tumor (ET) on BraTS and liver +on LiTS. Compared to TransVW and PCRLv1, PCRLv2 +reduces the false positive predictions and contains richer +fine-grained details. We believe such superiority of PCRLv2 +can be attributed to the integration of multi-scale pixel-level +and semantic information. +We also provide some failure examples in Fig. 9. One +common characteristic of these detection results is that they +include high-confidence predictions outside the lung area. +However, in daily clinical practice, such anomalies should +not be located outside the lung area. Similar phenomena +have been reported in [13], where the authors summarized +them as “shortcuts” that are common in learning systems +based on neural networks. To mitigate this problem in self- +supervised learning, we can add commonsense knowledge +to pre-trained models. Besides, it is also necessary to de- +velop more powerful machine learning tools for model +interpretation in various downstream tasks. +5 +CONCLUSION +We present a unified visual information preservation frame- +work for self-supervised learning in medical imaging. This +framework aims to encode the pixel-level, semantic, and +scale information into latent representations simultaneously. +To achieve this goal, we conduct multi-scale pixel restora- +tion and feature comparison on the feature pyramid, which +non-skip U-Net supports. The proposed PCRLv2 outper- +forms previous self-supervised pre-training approaches by +large margins and yields consistent improvements over +its conference version (PCRLv1) on four well-established +datasets in both quantitative and qualitative validation. 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Springer, 2020. +[49] Zongwei Zhou, Vatsal Sodha, Jiaxuan Pang, Michael B Gotway, +and Jianming Liang. +Models genesis. +Medical Image Analysis, +67:101840, 2021. + diff --git a/-9AyT4oBgHgl3EQf3vkG/content/tmp_files/load_file.txt b/-9AyT4oBgHgl3EQf3vkG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d2ee803507baaa4abf5f060cb0b7242755d8dd66 --- /dev/null +++ b/-9AyT4oBgHgl3EQf3vkG/content/tmp_files/load_file.txt @@ -0,0 +1,2298 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf,len=2297 +page_content='1 A Unified Visual Information Preservation Framework for Self-supervised Pre-training in Medical Image Analysis Hong-Yu Zhou, Student Member, IEEE, Chixiang Lu, Chaoqi Chen, Sibei Yang, and Yizhou Yu, Fellow, IEEE, Abstract—Recent advances in self-supervised learning (SSL) in computer vision are primarily comparative, whose goal is to preserve invariant and discriminative semantics in latent representations by comparing siamese image views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, the preserved high-level semantics do not contain enough local information, which is vital in medical image analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', image-based diagnosis and tumor segmentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To mitigate the locality problem of comparative SSL, we propose to incorporate the task of pixel restoration for explicitly encoding more pixel-level information into high-level semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We also address the preservation of scale information, a powerful tool in aiding image understanding but has not drawn much attention in SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The resulting framework can be formulated as a multi-task optimization problem on the feature pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specifically, we conduct multi-scale pixel restoration and siamese feature comparison in the pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In addition, we propose non-skip U-Net to build the feature pyramid and develop sub-crop to replace multi-crop in 3D medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The proposed unified SSL framework (PCRLv2) surpasses its self-supervised counterparts on various tasks, including brain tumor segmentation (BraTS 2018), chest pathology identification (ChestX-ray, CheXpert), pulmonary nodule detection (LUNA), and abdominal organ segmentation (LiTS), sometimes outperforming them by large margins with limited annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Codes and models are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='com/RL4M/PCRLv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Index Terms—Medical image analysis, Self-supervised learning, Transfer Learning, Context restoration, Feature pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 1 INTRODUCTION I T is usual to acquire a substantial amount of manually labeled data before training deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' This condition is easy to meet in natural images, where labor costs and labeling difficulties are tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In medical image analysis, however, credible annotations are mainly derived from domain experts’ diagnoses, which are challenging to obtain due to the rarity of the target disease, the need to safe- guard patient privacy, and the scarcity of medical resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Against this background, self-supervised learning (SSL) has been widely accepted as a viable technique to learn medical image representations without specialistic annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We usually deploy SSL in the pre-training stage to obtain well- transferable features, which can be transferred to various downstream tasks for performance boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Recent advances in SSL are mostly based on compar- ative learning [8], [10], [15], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The rationale behind is to learn transferable latent representations with invariant and discriminative semantics by maximizing the mutual information between a pair of siamese images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' One potential problem of these comparative methods is that they mainly focus on encoding high-level global semantics in representa- Hong-Yu Zhou, Chixiang Lu, Chaoqi Chen, and Yizhou Yu are with the Department of Computer Science, The University of Hong Kong, Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Email: {whuzhouhongyu, luchixiang, cqchen1994}@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='com, yizhouy@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Sibei Yang is with ShanghaiTech University and Shanghai Engineering Research Center of Intelligent Vision and Imaging, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Email: yangsb@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' First two authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Corresponding author: Sibei Yang and Yizhou Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' tions but ignore the preservation of pixel-level information1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, in medical image analysis, the latter type of information usually plays a vital role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For instance, in chest pathology detection, radiologists or clinicians are required to point out small lesions from a chest X-ray according to their textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Sometimes, these areas of pathologies are so hard to identify that even medical experts have to check pixel-level details to tell where the lesions are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Another typical example lies in brain tumor segmentation, where the segmentation error of one voxel may cause irreparable harm to patients in brain surgeries, such as a permanent damage to the cochlear nerve when trying to remove the acoustic neuroma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' An intuitive way to preserve pixel-level information in learned features is to restore the pixel-level content from latent representations directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' This methodology, known as context restoration [29], has already been adopted as a surrogate task in pretext-based SSL for natural [23], [29], [44] and medical images [7], [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specifically, these approaches first apply various data augmentation strategies to a given image to generate a corrupted input, based on which deep models are trained to restore original pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In this way, we explicitly require the latent representations to preserve information closely related to pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Although pure pixel- based features are not as transferable as those from com- parative SSL [17], [48], we hypothesize it is still beneficial to explicitly preserve pixel-level information and global se- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In 3D medical images, we often use “voxel” to denote the same concept as the pixel does in 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For simplicity, we use “pixel” to denote the smallest addressable element in both 2D and 3D images in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='00772v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='CV] 2 Jan 2023 2 ℱ# ℱ$ ℱ% ℱ" ℱ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Pixels Semantics Scales Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Motivation illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We propose a unified SSL framework to simultaneously preserve information in visual representations from perspectives of pixels, semantics, and scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' {F1, F2, F3, F4, F5} de- note different levels in the feature pyramid, given an input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Our approach restores uncorrupted inputs from the feature maps directly to preserve pixel-level details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In order to retain the global semantic information, our method compares siamese one-dimensional represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Last but not the least, the proposed methodology conducts pixel restoration and feature comparison at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The rationale behind is to introduce multi-scale self-supervised latent representations, making them more transferable to various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' mantics, especially in medical image analysis where details matter a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Besides semantics and pixels, introducing multi-scale representations has been proven to be quite helpful in aiding image understanding [12], [24], [26], [27], [32], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The common practice of these methods is to construct a feature pyramid during training, testing, or both stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Then, various tasks, such as detection, and segmentation, can be conducted on the basis of multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The goal of building the feature pyramid is to endow image representations with the ability to recognize objects at dif- ferent scales, which is also consistent with the law of human cognition [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, the preservation of visual informa- tion at multiple scales is rarely mentioned in SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Thus, it is unclear whether introducing multi-scale self-supervised representations provides a stronger transfer learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In Figure 1, we illustrate the motivation of the proposed unified visual information preservation framework for SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The introduced framework addresses the preservation of information in self-supervised visual representations from three aspects: pixels, semantics, and scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Firstly, to re- tain pixel-level information in latent representations, our framework involves a reconstruction branch in the self- supervised model to rebuild uncorrupted images from cor- rupted inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specifically, we ask the self-supervised model to restore pixels from feature maps of randomly corrupted inputs during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As a result, information closely associated with pixels can be explicitly encoded into the latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In practice, this type of information would enhance the ability of self-supervised representations to recognize and differentiate textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Apart from pixel- level information, preserving invariant and discriminative semantics in visual representations is also necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To- wards this end, we adopt the existing comparative SSL to encode invariant semantic information by comparing high-level representations of siamese image patches [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We empirically found the siamese SSL not only produces comparably (sometimes more) transferable medical image representations but also is much easier to implement in com- parison to the typical contrastive manner [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Last but not the least, the proposed unified framework introduces multi- scale latent representations by conducting pixel restoration and feature comparison in a range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To achieve this goal, we propose a non-skip U-Net (nsUNet) that constructs a feature pyramid upon the U-shape architecture [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In prac- tice, nsUNet effectively avoids the production of shortcut solutions when performing the context restoration task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On the basis of nsUNet, we conduct pixel-level context restora- tion and siamese feature comparison in each level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', scale) of the feature pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In this way, the proposed framework helps improve the ability of self-supervised representations to recognize objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', lesions and organs in medical images) at different sizes and scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We summarize the contributions of this paper as follows: We present an information preservation framework for advancing SSL in medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In this framework, we unify the preservation of visual infor- mation in latent representations from three aspects: pixels, semantics, and scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Towards this end, pixel restoration and feature comparison are conducted at different feature scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We introduce non-skip U-Net (nsUNet) to construct the feature pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Compared to the typical U- shape models in medical imaging [11], [32], nsUNet maintains more feature scales and eliminates the us- age of the widely adopted skip connections to avoid shortcut solutions to pixel restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Inspired by multi-crop [5], we propose sub-crop to compare global volumes against local volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In order to mitigate the problem of the reduced mutual information between global and local views in 3D space, sub-crop restricts the cropping of local views within the 3D minimum bounding box of global views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Experiments on 3D medical images found that sub-crop is more effective than multi-crop in various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We conduct extensive and comprehensive experi- ments to validate the effectiveness of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We show that the unification of pixels, semantics, and scales can provide impressive perfor- mance under the pre-training/fine-tuning protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specifically, the proposed framework outperforms both self-supervised and supervised counterparts in chest pathology classification, pulmonary nodule de- tection, abdominal organ segmentation, and brain tumor segmentation by substantial margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The conference version of this paper (PCRLv1) was pre- sented in [47], which demonstrates the benefits of incor- porating more pixel-level information besides the invariant and discriminative semantics obtained by contrastive learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In this paper, we made significant and substantial modi- fications to PCRLv1, and we name the improved framework as PCRLv2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', Preservational Comparative Representation Learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The modifications and improvements in PCRLv2 include but are not limited to (i) Besides local pixel-level and global semantic information, scale information is also 3 × × × × × × × × R R R R R 𝑥 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 𝑥" t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' t" R R R R R R Pixel restoration Candidate scale Chosen scale nsUNet Siamese nsUNet 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' # t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' # 𝑥" # t" # Global aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Global aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Local aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Local aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (a) Multi-scale pixel restoration × × × × × × × × C C C C C C Feature comparison Candidate scale Chosen scale nsUNet Siamese nsUNet 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 𝑥" 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' # 𝑥" # 𝑥 t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' t" t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' # t" # Global aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Global aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Local aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Local aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (b) Multi-scale feature comparison Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The overall structure of PCRLv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' PCRLv2 performs self-supervised visual learning on siamese feature pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To achieve this goal, we propose non-skip U-Net (nsUNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' nsUNet consists of five feature scales and removes the skip connections to prevent network optimizers from finding shortcut solutions to context restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On the basis of nsUNet, we propose to decouple the preservation of pixel-level, semantic, and scale information into two tasks: (a) multi-scale pixel restoration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (b) multi-scale feature comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The rationale behind is to incorporate pixel details and semantics into features at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' During the training stage, we randomly choose a feature scale from the feature pyramid, on top of which we conduct pixel restoration and feature comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' x denotes a batch of input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' t1 and t2 stand for two distinct global augmentations, while t′ 1 and t′ 2 denote the successive local augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' preserved in self-supervised visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The motivation behind is that although multiple feature scales have been considered in various vision tasks, they have not drawn much attention in SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' PCRLv2 shows that introducing multi-scale latent representations can boost the transfer learning performance of SSL in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (ii) PCRLv2 simplifies the attentional pixel restoration and hybrid feature contrast operations of PCRLv1 into a con- cise multi-task optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As a result, PCRLv2 is simpler and easier to implement while achieving bet- ter performance, thus more practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (iii) Compared to PCRLv1 that relies on the plain U-Net architecture [32], PCRLv2 conducts SSL on top of a new backbone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', non- skip U-Net (nsUNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' There are two inherent advantages of nsUNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' First, the feature pyramid of nsUNet allows performing multi-scale pixel-level context restoration and semantic feature comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As a result, the unification of pixels, semantics, and scales produces more transferable visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Second, nsUNet can effectively avoid the production of shortcut solutions, providing obvious performance gains over the use of the typical skip con- nections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (iv) We integrate the idea of multi-crop [5] in PCRLv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Moreover, in 3D medical imaging, we propose sub- crop to produce reliable local views with increased mutual information by randomly cropping multiple local volumes within the 3D minimum bounding box of global views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In practice, we found that the proposed sub-crop has better pre-training performance than multi-crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (v) In 5 classifica- tion/segmentation tasks, PCRLv2 provides more transfer- able pre-trained visual representations, not only surpass- ing previous self-supervised and supervised counterparts by substantial margins but also obviously outperforming PCRLv1 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 2 RELATED WORK This section reviews related work in comparative SSL, including contrastive and non-contrastive methods, and lists SSL approaches that use context restoration as the pretext task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In the third part, we collect papers that emphasize the incorporation of multi-scale features in SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Comparative SSL methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' One of the core ideas behind comparative SSL is to extract and encode invari- ant and discriminative semantics into representations via feature-level comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Hjelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [20] proposed Deep In- foMax to maximize the mutual information between global and local feature vectors of the same input image using InfoNCE [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Bachman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [3] augmented InfoMax by conducting a global-local comparison on feature vectors of independently-augmented versions of each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [36] increased the number of augmented views of each input and extended InfoNCE to multiple views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [17] presented Momentum Contrast (MoCo), which comprises a momentum encoder to maintain the consistency among positive and negative feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Different from [3], [20], MoCo performs InfoNCE on top of global feature vectors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Compared to MoCo, SimCLR removes the momentum architecture and defines InfoNCE on the output of a MLP with one hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Inspired by SimCLR, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [9] proposed MoCov2, which improves MoCo with an additional MLP head and more augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' SwAV [5] replaces the feature vectors in InfoNCE with cluster assignments and introduces the multi-crop strategy to increase the number of views of an image with affordable computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Grill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [15] proposed BYOL (bootstrap your own latent), which eliminates the use of InfoNCE in SSL by distilling semantics from positive pairs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Based on BYOL, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [10] further removed the restriction of the momentum architecture and introduced a simple siamese learning framework named SimSiam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In practice, SimSiam produces comparable results to MoCov2 in various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Recently, Zbontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [42] simplified SimSiam by measuring the cross-correlation ma- trix between the siamese global feature vectors and trying to make this matrix close to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Comparative SSL, especially InfoNCE-based method- ology, has also been widely adopted in medical image 4 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [48] proposed to integrate mixup [43] into MoCov2, increasing the diversity of both positive and negative samples in InfoNCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Taleb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [34] developed 3D versions of existing SSL techniques and compared 2D and 3D SSL approaches on downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Azizi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [2] incorporated multi-instance learning into SimCLR, which helps utilize multiple views of each patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Around the same time, Vu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [37] developed a method to select posi- tive pairs coming from views of the same patient and used this strategy to improve MoCov2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' There are also a number of approaches [6], [40], [41] that tailored comparative SSL for semi-supervised medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, the methodologies mentioned above fail to address the importance of integrating pixel-level information into the high-level representations with rich semantics, which is the primary focus of the proposed PCRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Context restoration for preserving pixel-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Restoring original context has been treated as an important pretext task in SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [29] first time conducted self-supervised feature learning by recovering masked input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Larsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [23] and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [44] performed SSL on pixels via predicting RGB color values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For medical images, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [7] extended the approach in [29] with swapped image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [49] showed that adding more augmentations to input images brings benefits to SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [35] presented a volume-wise context transforma- tion for 3D medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Different from the approaches mentioned above, Henaff [19] proposed to predict the next context feature vectors following an auto-regressive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We can see that context restoration is more prevalent in medical imaging than in natural images from the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The underlying reason is that medical imaging tasks require more pixel-level information to make fine-grained yet accurate decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On the other hand, we observe that comparative SSL can produce representations with richer semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Thus, it can be beneficial to build a SSL framework that simultaneously integrates pixel-level and semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As far as we are concerned, none of these context restoration based approaches incorporate such a combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Multi-scale features in SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Although multi-scale features have not drawn much attention in existing SSL research, it has already been treated as an implicit yet effective regularization method for SSL in some methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Deep InfoMax [20] contrasts high-level feature vectors with low- level feature maps using InfoNCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To improve Deep Info- Max, Bachman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' [3] proposed to contrast global and local feature vectors on multiple levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In medical image analysis, preserving scale information becomes essential, as pathologies may show different characteristics on dif- ferent scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In [6], a local contrastive loss is introduced to learn distinctive representations of local regions that are helpful to per-pixel segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' At the same time, global feature vectors are used to distill discriminative semantics for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' A similar idea has also been used in image registration [25] and one-shot segmentation [46], where global and local feature vectors are employed to provide information on semantics and position, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, most of these methods only perform SSL on two scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', one global and one local, which cannot fully capture multi-scale information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Besides, although these approaches emphasize the benefit of introducing local in- formation to SSL, they do not exploit pixel-level information that is helpful to encode locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In contrast, this paper pro- poses a unified framework that can simultaneously preserve semantic, pixel-level, and scale information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3 METHODOLOGY We provide an overview of PCRLv2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Suppose x denotes a batch of input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We introduce cascaded augmentations to distort x in global and local views, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To be specific, the first-stage augmentations (t1 and t2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 2) mainly consist of global transformations, such as flip and rotation, whose goal is to distort the semantics of input images from a global perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In comparison, the second-stage augmentations (t′ 1 and t′ 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 2) comprise local pixel-level transformations, such as random noise and gaussian blur, which are leveraged to perturb the local semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' After two-stage augmentations, the finally aug- mented images x′ 1 and x′ 2 are passed to siamese networks to perform pixel restoration and feature comparison, while the results of applying t1 and t2 to x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', x1 and x2, serve as the ground truth targets for the pixel restoration task (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We perform SSL on the feature pyramid to encode multi- scale visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Following the standard practice in medical image processing, we build feature pyramids using a U-shape model named non-skip U-Net (nsUNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Compared to the typical U-Net architecture [11], [32], nsUNet has more feature scales and completely removes skip connections, both of which we empirically found help- ful in producing better pre-trained representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' During the training stage, one scale is first randomly chosen from all five feature scales, after which we conduct pixel restoration and feature comparison on the siamese feature maps at the chosen scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' After the pre-training stage, we fine-tune the encoder of nsUNet on various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 Feature pyramid in non-skip U-Net U-Net and its series [11], [22], [32] have been known in med- ical imaging for their abilities to handle image segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The most distinctive characteristic of these models is the skip connection that connects equal-resolution low- and high-level feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The critical insight is to recover the spatial information lost in down-sampling operations of the encoder network, such as strided pooling or convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' U-shape models use a feature pyramid to progressively incorporate multi-scale details brought by skip connections into high-level semantics, making the U-shape architecture an ideal choice for conducting context restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In this paper, we explore the potential of U-shape ar- chitecture in SSL from two perspectives: deeply fusing semantic and pixel-level information by removing the skip connections and introducing multi-scale latent representa- tions by conducting SSL on the feature pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For the first perspective, we empirically found that skip connec- tions provide shortcuts for context restoration, as the low- level feature maps contain rich, high-resolution pixel-level 5 × × × × 𝐻 2 × 𝑊 2 𝐻×𝑊 𝐻 4 × 𝑊 4 𝐻 8 × 𝑊 8 𝐻 16 × 𝑊 16 𝐻 32 × 𝑊 32 Skip feature maps Feature hierarchy Down-sampling Up-sampling × No skip connection Conv+BN +ReLU ×2 Conv+BN +ReLU ×2 Conv+BN +ReLU ×2 Conv+BN +ReLU ×2 Conv+BN +ReLU ×2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The architecture of non-skip U-Net (nsUNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In comparison to previous U-Net series, nsUNet removes skip connections, and the associated skip feature maps to prevent shortcut solutions to the pixel restoration and feature comparison tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Besides, nsUNet consists of five levels of feature maps (denoted with different colors), where two self- supervised tasks are further conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Note that this is a 2D illustration of nsUNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' This characteristic does contribute to the restoration of context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, it may prevent the high-level latent representations (with rich semantics) from incorporating more pixel-level information because the task of providing pixel-level details is assigned to low-level feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To address this point, we remove the skip connections in U- shape architecture and propose non-skip U-Net (nsUNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' nsUNet relies on high-level representations without any skip connections to restore pixel-level details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In this way, the semantic and pixel-level information can be deeply fused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Meanwhile, the inherent multi-scale feature maps of nsUNet offer the opportunity to construct a feature pyra- mid, on top of which SSL can be conducted in multiple scales simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3 presents the architecture of nsUNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The feature pyramid in nsUNet comprises five levels, ranging from low resolution (the down-sampling rate is 32) to full resolution (no down-sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For 2D input data, we use ResNet- 18 [18] as the encoder, while for 3D input volumes, we build the encoder following [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3, the decoder of nsUNet maintains a shared architecture across all pyramid levels, which can be summarized as: Fi = Conv-BN-ReLU(Conv-BN-ReLU(Up(Fi−1)), (1) where i ∈ {1, 2, 3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' F0 denotes the output of the bottle- neck block, which has the lowest spatial resolution (down- sampling rate=32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Up represents the up-sampling opera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Conv-BN-ReLU stands for a sequence of operations, including convolution (kernel size=3), batch normalization (BN), and ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As a result, the bag of feature maps {F1, F2, F3, F4, F5} is then forwarded to following task-dependent heads to perform pixel restoration and fea- ture comparison, respectively and simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 Multi-scale pixel restoration As the name implies, multi-scale pixel restoration aims to preserve pixel-level and scale information in latent visual representations simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To achieve this goal, we ask the network to recover the exact pixel-level details across different scales, where each pair of siamese feature maps share one pixel restoration head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In contrast, PCRLv1 only restores pixel details at the full resolution, which inevitably loses multi-scale properties in learned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4a, the input images x′ 1 and x′ 2 are intentionally corrupted via various pixel-level augmenta- tions, such as guassian blur and random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For each training iteration, we first randomly choose a feature scale Fi from {F1, F2, F3, F4, F5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Then, we pass Fi to the pixel restoration head f R i (·) for the i-th scale, whose internal processing procedure can be summarized as: f R i (Fi) = Conv(Conv-BN-ReLU(Fi)), (2) where all convolution layers use a kernel size of 3 and a stride of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Similarly, we apply the shared pixel restoration head to the paired siamese feature map Fs i to acquire the prediction output f R i (Fs i ): f R i (Fs i ) = Conv(Conv-BN-ReLU(Fs i )), (3) Lastly, we employ the mean square error (MSE) loss to measure the reconstruction errors between f R i (Fi) and x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For the siamese feature pyramid, we apply MSE loss to f R i (Fs i ) and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The cost function LR of the pixel restoration task in each training iteration (with mini-batch optimiza- tion) is as follows: LR = N � j=1, ∀i∈H 1[i==j] [MSE(f R i (Fi), x1) + MSE(f R i (Fs i ), x2)], (4) where N = 5 denotes the number of scales in each feature pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' H = {1, 2, 3, 4, 5} stands for the scale index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 1[i==j] is an indicator function, which is equal to 1 when i==j is true (otherwise, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The explanation of LR can be summarized as: (i) randomly choose a feature scale Fi from all five scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (ii) pass Fi and its siamese feature map Fs i to the shared task head f R i (·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (iii) calculate the MSE loss between the outputs of f R i (·) and uncorrupted images {x1, x2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' By reconstructing the same targets x1/x2 across different feature scales, LR can encode the pixel-level information into multi-scale latent visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 Multi-scale feature comparison PCRLv1 employs a hybrid way to conduct contrastive learning with the help of the momentum encoder [17] and mixup [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, this contrastive deployment is complex, making PCRLv1 heavy, thus troublesome to im- plement and improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To address these issues, PCRLv2 replaces the hybrid contrastive strategies in PCRLv1 with the multi-scale comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Inspired by [10], multi-scale comparison conducts SSL with siamese learning, whose key operation is to attract the same image’s siamese views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Different from [10] that conducts feature comparison on one scale, we propose to preserve the discriminative semantics across different feature scales, which forces the model to preserve multi-scale self-supervised representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In the following, we provide technical details of performing the multi-scale comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 6 Conv BN ReLU Conv Conv BN ReLU Conv 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 𝑥# Shared 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' " … 𝑥# " … Siamese scale Chosen scale (a) Architectural details of the pixel restoration head Siamese scale Chosen scale GAP GAP BN FC BN ReLU FC BN FC BN ReLU FC Predictor Predictor Shared 𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' " … 𝑥# " … (b) Architectural details of the feature comparison head Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Architectural details of the pixel restoration and feature comparison heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Conv, BN, GAP, and FC denote the convolution, batch normalization, global average pooling, and fully-connected layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The kernel size of all convolution layers is 3, and the convolution stride is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Note that each pair of siamese feature maps share one pixel restoration head and one feature comparison head, while different feature scales employ distinct task heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Given the feature maps at a randomly chosen scale Fi, we pass them through a global average pooling layer and a shared batch normalization layer (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4b) to acquire 1D representations vi: vi = BN(GAP(Fi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (5) We can get vs i by processing the siamese feature maps Fs i in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Next, we forward vi to the shared predictor fP(·), whose architecture is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4b and can be summarized as: fP(vi) = FC(FC-BN-ReLU(vi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (6) where FC denotes the fully-connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' FC-BN-ReLU stands for a sequence of layers, which are the fully- connected layer, batch normalization layer, and ReLU ac- tivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Similarly, we can acquire fP(vs i ) by passing vs i to the same predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We measure the similarity between siamese feature vec- tors with the cosine similarity: cos(vi, fP(vs i )) = vi ∥vi∥2 fP(vs i ) ∥fP(vs i )∥2 , (7) where || · ||2 denotes the L2 normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Symmetrically, we calculate cos(fP(vi), vs i ) as follows: cos(fP(vi), vs i )) = fP(vi) ∥fP(vi)∥2 vs i ∥vs i ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (8) Finally, the cost function LC of multi-scale feature compari- son can be summarized as: LC = N � j=1, ∀i∈H −1 21[i==j] [cos(sg(vi), fP(vs i )) + cos(fP(vi), sg(vs i ))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (9) N = 5 denotes the number of feature scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' H = {1, 2, 3, 4, 5} stands for the scale index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Following [10], we apply the stop-gradient operation (denoted as sg) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 9 to prevent the network optimizer from finding shortcut solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Minimizing LC requires the model to maximize the similarity between siamese latent features across all feature scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In this way, scale invariance can be implicitly incor- porated into the preserved latent semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 1 2 1 2 3 4 5 6 7 8 Randomly crop two global patches with an IoU constraint Find the minimum 3D bounding box Randomly crop local patches 1 2 3 4 5 6 7 8 3D Global views 3D Local views Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Illustration of sub-crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Given a 3D local volume, we first randomly crop two large patches, where an intersection over union (IoU) constraint is applied to guarantee that two patches are partly overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' These two large patches are considered as x1 and x2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 2 and will be passed to the siamese architecture to conduct the following multi-scale pixel restoration and feature comparison tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To acquire local views, we compute the minimum 3D bounding box of two large patches, after which random crop is applied to extract multiple local patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Finally, we reshape these local patches to a fixed size and forward them to the network to extract local representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 From multi-crop to sub-crop Multi-crop [5] has been known as a helpful strategy to im- prove SSL performance in natural images, which increases the number of input views by sampling several standard resolution crops and more low-resolution crops from the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' One key insight behind multi-crop is to capture relations between parts of a scene or an object, while low-resolution views ensure a controllable increase in the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' When applied to medical images, multi-crop works well in 2D X-ray data but leads to the non-convergence of the model in 3D volume data (such as CT and MRI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' After careful investigation, we found the root of this problem lies in the contradiction between the limited input size and many candidate crops in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specif- 7 ically, on the one hand, we cannot afford large-sized 3D inputs because processing them with 3D deep models often costs dramatic GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On the other hand, if we overly reduce the size of 3D inputs, the sampled views would be too dispersed to guarantee the model capture the local-global associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To mitigate the above issue, we introduce sub-crop to replace multi-crop in 3D medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The core idea of sub-crop is straightforward: reducing the sampling space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 5, sub-crop mainly consists of three steps: (i) randomly crop two extensive global views with an IoU constraint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (ii) find the minimum 3D bounding box over the cropped global patches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (iii) randomly crop multiple local patches within the 3D bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' There are two critical operations in sub-crop: the constraint of IoU on global views and the sampling of local patches within the minimum bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In practice, the first operation guarantees the global-global association by ensuring the overlap between large patches larger than a fixed threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The second operation mitigates the disperse problem of local views and helps the model to discover local-global relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 Overall training objective After applying multi-crop/sub-crop to medical images, we can acquire two global views {g1, g2} and ˆN local views {l1, l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', l ˆ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For clarification, we denote the associated in- puts in notations of loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For instance, LC(g1, g2) means we calculate LC on top of the extracted siamese representations of two global views, where g1 and g2 can be regarded as a pair of siamese images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' At last, the overall training objective of PCRLv2 can be formalized as follows: LTotal(g1, g2, l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', l ˆ N) =LR(g1, g2) + LC(g1, g2) + � m∈{1,2} ˆ N � k=1 LC(lk, gm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' (10) There are three terms in LTotal: LR(g1, g2), LC(g1, g2), and � m∈{1,2} � ˆ N k=1 LC(lk, gm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The first term is designed to preserve pixel-level details in multi-scale learned repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The second term addresses the importance of encoding multi-scale semantics into latent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The last term aims to capture the multi-scale global-local semantic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 Short discussion: PCRLv2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' PCRLv1 Simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' PCRLv1 combines the context restoration and comparative SSL via transformation-conditioned attention and cross-model mixup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' These two components make the framework heavy, less intuitive, and not easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Compared to PCRLv1, PCRLv2 exploits a simpler yet more intuitive design to incorporate pixel-level and semantic information via multi-scale learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As aforementioned, PCRLv2 can be formulated as a simple multi-task optimization problem whose objective function maximizes the preservation of multi-level information in latent visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' These characteristics make it easier for both implementation and potential expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' PCRLv1 makes heavy use of mixup (to both inputs and features) in its implementation, which is found to deliver performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In PCRLv2, we eliminate mixup strategies and cut the training time in half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In addition, PCRLv2 requires less running memory in GPUs during the training stage, making it more practical in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4 EXPERIMENTS In this section, we first conduct thorough ablation studies to investigate the influence of different modules in PCRLv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Then, we evaluate the effectiveness of PCRLv2 on both 2D and 3D medical imaging tasks, including chest pathol- ogy classification, pulmonary nodule detection, abdominal organ segmentation, and brain tumor segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For model evaluation, we follow the pre-training (on source data)→fine-tuning (on target data) protocol and employ two settings, which are semi-supervised learning and transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In the first setting, the source and target data come from the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specifically, we first pre-train the model using all training data without labels, and then fine- tune the pre-trained model with limited annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As for transfer learning (the second setting), we pre-train and fine- tune the model on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Different from semi- supervised learning, we fine-tune the pre-trained model with both limited and full annotations in transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 Datasets NIH ChestX-ray (2D) [38] is made up of 112,120 X- ray scans from 30,805 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' There are fourteen different chest pathologies in NIH ChestX-ray, including atelectasis, cardiomegaly, consolidation, edema, effusion, emphysema, fibrosis, hernia, infiltration, mass, nodule, pleural thickening, pneumonia, and pneumothorax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The labels of radiographs were automatically extracted from associated radiology reports using natural language process (NLP) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We use NIH ChestX-ray in semi-supervised learning in our experiments and treat it as the target dataset in transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' CheXpert (2D) [21] involves 224,316 chest radiographs from 65,240 patients for the presence of 14 common chest radiographic observations: no finding, enlarged cardio, cardiomegaly, lung opacity, lung Lesion, edema, consolidation, pneumonia, atelectasis, pneumothorax, pleural effusion, pleural other, fracture, and support devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Similar to NIH ChestX-ray, an NLP labeler was developed to detect the presence of 14 observations in radiology reports automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In practice, CheXpert serves as the source data in transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' LUNA (3D) [33] was collected for the automatic detection of pulmonary nodules, which involves 888 annotated thoracic computed tomography (CT) scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' LUNA is a cherry-picked subset of LIDC-IDRI [1], which excludes scans with a slice thickness greater than 3mm, inconsistent slice spacing, or missing slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In the 888 scans, a total of 5,855 annotations were made by the radiologists, where only nodules ≥ 3mm are categorized as relevant lesions, 8 and at least one radiologist checks each nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On LUNA, we perform semi-supervised learning and transfer learning experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For transfer learning, LUNA is mainly used for self-supervised pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' LiTS (3D) [4] releases 131 abdominal CT Volumes and associated annotations for training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' There are two types of labels in LiTS: the liver and tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In this paper, we only utilize the ground truth masks of the liver to evaluate the effectiveness of various SSL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The task on LiTS is abdominal organ segmentation, where LiTS is used for fine-tuning in transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' BraTS (3D) has been known as a series of challenges in brain tumor segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In this paper, we perform experiments on the released 351 magnetic resonance imaging (MRI) scans of BraTS 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' There are three classes in BraTS: whole tumor (WT), tumor core (TC), and enhancing tumor (ET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Similar to the role of LiTS, BraTS serves as the target data in transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 Baselines A variety of SSL baselines are included in our extensive experiments, which can be roughly divided into three categories: 2D specific methods, 3D specific approaches, and generic (2D & 3D) methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Details of baselines in each category are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 2D specific SSL methodologies consist of ImageNet-based pre-training (IN) [14], Comparing to Learn (C2L) [48], and Simple Siamese Learning (SimSiam) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' IN is the most widely adopted pre-training methodology, which conducts supervised pre-training on one of the biggest natural image datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', ImageNet [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' C2L is a recently proposed SSL approach based on momentum contrast (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', MoCov1 [17] and MoCov2 [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' SimSiam is a simple siamese SSL framework that eliminates the barrier of negative samples in contrastive learning and the use of a momentum encoder in BYOL [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Besides, we compare PCLRv2 against SimSiam to highlight the significance of the preserved pixel-level information and multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 3D specific SSL methodologies include Rubik’s cube++ [35] and 3D-CPC [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Rubik’s cube++ is the most recent SSL approach built on top of context restoration for 3D medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' It adopts a volume-wise transformation for context permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In comparison, 3D-CPC is based on contrastive predictive encoding [19], a variation of contrastive learning, and demonstrates the most superior performance among different SSL approaches investigated in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Generic SSL methodologies involve train from scratch (TS), Model Genesis (MG) [49], TransVW [16], and PCRLv1 [47] (the conference version of our approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' MG resorts to ag- gressive augmentations to generate corrupted input images, based on which the model is asked to restore the original in- puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' TransVW improves MG by appending an intermediate classification head to encode anatomical patterns explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' PCRLv1 first proposes simultaneously preserving semantic and pixel-level information in SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 Implementation details Dataset pre-processing for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On NIH ChestX- ray and CheXpert, each input image is resized to 224×224 after random crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On LUNA, we randomly crop a volume from the whole CT scan with a random size from {64×64×32, 96×96×64, 96×96×96, 112×112×64}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Each cropped volume is then resized to 64×64×32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Each voxel’s Hounsfield Unit (HU) in the crop is truncated to [-1000,1000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' If a voxel’s HU is lower than -150, we regard it as a background voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In practice, if over 85% voxels within a crop belong to the background, we would not use this crop in pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Dataset pre-processing for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For NIH ChestX- ray and CheXpert, we follow the same pre-processing procedures as in the pre-training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On LUNA, we randomly crop a volume for each training iteration, and the size of each crop is 48×48×48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On LiTS, we first localize the liver and expand the target volume by 30 slices on each axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' After random crop, the size of each crop is 256×256×64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Unlike LUNA, we truncate the HU of each voxel to [-200, 200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For BraTS, the size of each random crop is 112×112×112×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Data augmentation and multi-crop/sub-crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 2, there are two types of augmentations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', global and local augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specifically, for 2D tasks, the global augmentation includes random crop, random horizontal flip, and random rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The local augmentation involves random grayscale, gaussian blur, and cutout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In comparison, for 3D tasks, the global augmentation consists of random flip and random affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Local augmentation strategies are applied, including Gaussian blur, random noise, random gamma, and random swap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Note that all 3D augmentations are implemented following [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As for multi-crop in 2D tasks, we resort to the scale factor of random crop2 to generate global and local views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specifically, we set the range of scale to [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3, 1] to generate two global views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For six local views, the scale range is set to [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Both global and local views are resized to 224×224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As for sub-crop in 3D tasks, we randomly sample two global views with a random size from {64×64×32, 96×96×64, 96×96×96, 112×112×64}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The IoU constraint (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', threshold) between two global views is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Then, we find the minimum bounding box of global views, from which six local views are randomly cropped, each with a random size from {8×8×8, 16×16×16, 32×32×16, 32×32×32}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' After random crop, all 3D global views are resized to 64×64×32, while all local views are resized to 16×16×16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Training and evaluation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We use stochastic gradient descent (SGD) with momentum as the default optimizer, where the momentum is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The initial learning rate is 1e-2, and we employ the cosine annealing strategy for learning rate decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We set the weight decay to 1e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The number of training epochs is 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The batch sizes of 2D pre- training and fine-tuning (on NIH ChestX-ray or CheXpert) are 256 and 512, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As for 3D pre-training, the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='org/vision/main/generated/torchvision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='RandomResizedCrop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='html 9 0 15 Epochs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 Training MSE loss (log10) w/ skip w/o skip Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Influence of skip connections in pixel restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We display the loss curve of mean square error (MSE) in the first 15 epoches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' TABLE 1 Impact of skip connections on chest pathology identification (NIH ChestX-ray), brain tumor segmentation (BraTS), and abdominal organ segmentation (LiTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On NIH, We use 95% unlabeled training data for pre-training, while the rest 5% data with labels are used for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' On BraTS and LiTS, we use 10% labeled data for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Datasets w/o skip w/ skip Gain NIH 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 BraTS 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 LiTS 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 batch size (on LUNA) is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For 3D fine-tuning tasks, the batch sizes on LUNA, LiTS, and BraTS are 32, 4, and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The evaluation metric on NIH ChestX-ray, CheXpert, and LUNA is AUROC (Area Under the Receiver Operating Characteristics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For segmentation tasks on LiTS and BraTS, we use Dice similarity as the evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We use 70%, 10%, and 20% of the whole dataset to build the training, validation, and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In particular, for semi- supervised learning, we construct the pre-training set by removing a specific amount of data from the entire training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' At the same time, the remainder is used as the training set for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Binary cross-entropy loss is used for the fine-tuning of NIH ChestX-ray, CheXpert, and LUNA, while Dice loss is used for the fine-tuning of LiTS and BraTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 Ablation studies Impact of skip connections on pixel restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 6, we present the mean square error (MSE) loss (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4) curves during the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We see that the MSE loss, with skip connections, decreases rapidly in the first 15 training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In comparison, the proposed nsUNet (w/o skip) slows down the decreasing rate of MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' These phenomena are consistent with the role of skip connections, which bridges the gap between low-level pixel details and high-level latent semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The existence of skip connections makes it easier to restore pixels by incorporating pixel-level details from low-level but high-resolution feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, nsUNet removes skip connections, avoiding shortcut solutions to context restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Although this design makes it harder to restore pixels (higher loss values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 6), it helps encode pixel- level information into high-level semantic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' ℱ# ℱ$ ℱ% ℱ" ℱ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' ℱ# & ℱ$ & ℱ% & ℱ" & ℱ" & (a) Pairwise ℱ# ℱ$ ℱ% ℱ" ℱ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' ℱ# & ℱ$ & ℱ% & ℱ" & ℱ" & (b) Cross-scale Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Two choices of how to conduct siamese feature comparison for multiple feature scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Here, we primarily consider pairwise feature comparison and cross-scale feature comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Such advantage can be verified by the performance gains in Table 1, where removing skip connections brings over 1% improvement to chest pathology identification, brain tumor segmentation, and abdominal organ segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' How to conduct siamese feature comparison for multiple feature scales?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We illustrate two intuitive choices in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Besides the adopted pairwise comparison manner (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 7a), another obvious choice is to compare siamese features following a crossed way (a similar strategy was used in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 7b, the cross-scale comparison aggressively compares siamese features across all feature scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The motivation behind is to introduce multi-scale latent representations by coupling features across different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Table 2 reports the experimental results of pairwise and cross-scale siamese feature comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We find that cross-scale feature comparison slightly deteriorates the performance of semi-supervised pathology identification by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 percents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The underlying reason might be that the features in each scale maintains distinct characteristics, and neglecting these discrepancies can lead to degenerate feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Investigation of different modules in PCRLv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In Table 3, we study and report the impact of different modules on the whole tumor (WT) and enhancing tumor (ET) classes of BraTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Note that in practice, most instances of WT are much larger than instances from ET, making ET instances harder to segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Besides, we also present the transfer learning results on NIH ChestX-ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' TABLE 2 Results of pairwise and crossed siamese feature comparison (semi-supervised learning on NIH ChestX-ray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The ratio of unlabeled to labeled data is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Pairwise Crossed [3] Gain Mean AUROC 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 First of all, we investigate the influence of pixel restora- tion (row 0) and feature comparison (row 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We directly reconstruct the full resolution uncorrupted im- ages for the pixel restoration task while siamese feature comparison is conducted on the last-layer output of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Comparing row 0 with row 1, we see that the context restoration task is more advantageous in segmenta- tion of small tumor regions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', ET) while the comparative SSL is more capable of dealing with large tumor regions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', WT) and chest pathologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Such comparison shows 10 TABLE 3 Impact of different modules in PCRLv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' and Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' denote the tasks of pixel restoration and feature comparison, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' S (N) means there are N scales included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' MC and SC stand for the multi-crop and proposed sub-crop strategies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' WT and ET denote classes of the whole tumor and enhancing tumor in BraTS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In most cases, instances from WT are much larger (in size) than those of ET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We performed these experiments by first using LUNA for self-supervised pre-training, and then we fine-tune the pre-trained model on BraTS using 10% labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' NIH denotes the transfer learning on chest pathology identification, where we use CheXpert for pre-training and fine-tune the pre-trained model with 50% labeled data from NIH ChestX-ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' # Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' S (3) S (5) MC SC WT (BraTS) ET (BraTS) NIH 0 ✓ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 1 ✓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 2 ✓ ✓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 3 ✓ ✓ ✓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 4 ✓ ✓ ✓ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 5 ✓ ✓ ✓ ✓ fail fail 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 6 ✓ ✓ ✓ ✓ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 that semantic information preservation may be more helpful to the detection of large objects, while segmenting small objects requires the incorporation of pixel-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In row 2, we can already acquire noticeable performance gains by directly combining pixel restoration and feature comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Next, we show that multi-scale representations benefit both pixel restoration and feature comparison tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' By conducting both tasks on 3 scales, we observe a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7-percent improvement on WT, a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5-percent gain on ET, and a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6-percent improvement on chest pathology classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' These results show that introducing multiple scales is more helpful to the segmentation of small regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Moreover, by increasing the number of scales from 3 to 5, we can improve the accuracy of all three tasks consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Not surprisingly, ET benefits the most from the introduction of multiple scales, indicating the necessity of utilizing multi- scale representations in medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Last but not the least, we investigate the significance of multi-crop (row 4) and sub-crop (row 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We empirically found that directly applying multi-crop to 3D medical vol- umes leads to the failure of model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The underlying reason might be that it is difficult for cropped global and local views to maintain clear spatial relations in the 3D space as in the 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In contrast, sub-crop can provide consistent performance gains on both types of tumor regions by successfully preserving the spatial relations in latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' When applying sub-crop to 2D X-rays, we observe a marginal improvement over multi-crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The underlying reason is that sub-crop is proposed to handle dispersed sampled views in a 3D space to guarantee the model captures local-global relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, in a 2D space, the sampled views usually (partly) overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 Semi-supervised chest pathology identification Table 4 presents the experimental results of applying semi- supervised learning on NIH ChestX-ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specifically, we use a specific amount of the training set (denoted as the labeling ratio in Table 4) as labeled data while the remaining training data is used for self-supervised pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' From Table 4, we see that self-supervised pre-training can dramatically boost the performance compared to train from scratch (TS), which verify the necessity of conduct- ing pre-training in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Comparing MG with TransVW, they show similar performance in different label- ing ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Such comparison is easy to explain as TransVW is built upon MG, and both are based on context restora- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' TransVW performs slightly better than MG, as it incorporates an additional classification head to encode more semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Compared to context restoration based methods, comparative methodologies (C2L and SimSiam) display better overall and class-specific results, especially in small labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The underlying reason might be that semantic information is more critical than pixel-level information in chest pathology detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' As for C2L and SimSiam, C2L performs better when the amount of labeled data is quite limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, SimSiam gradually produces better diagnosis results as the labeling ratio increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' After incorporating the semantic, pixel-level, and scale information into a unified framework, PCRLv2 outperforms various SSL baselines in different labeling ratios signifi- cantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' It surpasses the previous conference version by clear margins, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', PCRLv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Particularly, PCRLv2 seems to have more advantages in small labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For instance, when the labeling ratio is 5%, PCRLv2 outperforms PCRLv1 by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 percents on average, which verifies the significance of multi-scale latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 Semi-supervised pulmonary nodule detection In Table 5, we report the experimental results of semi- supervised pulmonary nodule detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Interestingly, we observe narrowed performance gaps between TS and SSL baselines than those reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' One possible ex- planation is that the task of detecting pulmonary nodules is less sensitive to the amount of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Among all SSL baselines, Cube++ gives better performance when utilizing small amounts of labeled data, while 3D-CPC is more advantageous in large labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In addition, we see TransVW quickly catching up with MG and Cube++ as the labeling ratio increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' PCRLv1 outperforms previous SSL approaches in dif- ferent labeling ratios by large margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' After incorporat- ing multi-scale latent representations, PCRLv2 consistently surpasses PCRLv1 in a range of labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' When the baseline SSL methods show similar performance as the labeling ratio increases, PCRLv2 can still provide impressive improvements over PCRLv1 and previous SSL approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 11 TABLE 4 Semi-supervised chest pathology identification (on NIH ChestX-ray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The labeling ratio denotes the amount of data with labels in the training set that is used for fine-tuning while the remaining data in the training set is used for self-supervised pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The best results are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Labeling ratio Methodology Mean Atelectasis Cardiomegaly Effusion Infiltration Mass Nodule Pneumonia Pneumothorax Consolidation Edema Emphysema Fibrosis Pleural Thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Hernia 5% TS 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 71.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 TransVW [16] 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 64.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 SimSiam [10] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 PCRLv1 [47] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 PCRLv2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 TABLE 5 Semi-supervised pulmonary nodule detection (on LUNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The labeling ratio indicates how much data from the training set with labels is utilized for fine-tuning while the rest of the data is used for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Best results are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Methodology Labeling ratio 10% 20% 30% 40% TS 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 MG [49] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 TransVW [16] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 Cube++ [35] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 3D-CPC [34] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 PCRLv1 [47] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 PCRLv2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 Transfer learning on chest pathology identification In Table 6, we validate the transferable ability of visual representations provided by different pre-training method- ologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Specifically, we compare PCRLv2 against train from scratch, ImageNet-based pre-training (IN), different SSL baselines, and PCRLv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Comparing MG/TransVW with IN, we see context restoration based SSL maintains the limited transferable ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' This phenomenon becomes more apparent when the target domain has quite limited annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The underlying reason is that semantic information plays a crucial role in transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In contrast, the significant performance gains brought by C2L and SimSiam again verify the effec- tiveness of comparative SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' C2L and SimSiam still cannot outperform IN by significant margins, especially when con- sidering that IN is more advantageous when the labeling ratio is 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' After integrating the benefits of context restoration based and comparative SSL, PCRLv1 is already capable of outperforming previous SSL methodologies by observable margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Furthermore, by exploiting multi-scale semantic and pixel-level information, PCRLv2 achieves consistent improvements over PCRLv1 in overall and class-specific results in different labeling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 Transfer learning on brain tumor segmentation We report the experimental results of applying transfer learning to brain tumor segmentation in Table 7, where we use LUNA dataset for self-supervised pre-training and fine-tune the pre-trained model with different amounts of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 12 TABLE 6 Transfer learning on chest pathology identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We pre-train the model using data from CheXpert (without labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Then, we fine-tune the pre-trained model on NIH ChestX-ray with different amounts of labeled data (denotes as different labeling ratios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The best results are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Labeling ratio Methodology Mean Atelectasis Cardiomegaly Effusion Infiltration Mass Nodule Pneumonia Pneumothorax Consolidation Edema Emphysema Fibrosis Pleural Thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Hernia 10% TS 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 TABLE 7 Transfer learning on brain tumor segmentation (on BraTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' WT, TC, and ET stand for the whole tumor, tumor core, and enhancing tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' For all SSL approaches, we use LUNA for pre-training, and then fine-tune the pre-trained model on BraTS with varying amounts of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Best results are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Methodology 10% 20% 30% 40% 100% Mean WT TC ET Mean WT TC ET Mean WT TC ET Mean WT TC ET Mean WT TC ET TS 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 66.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 PCRLv2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 13 TransVW PCRLv1 PCRLv2 GT TransVW PCRLv1 PCRLv2 GT 10% 10% 10% 20% 20% 20% TransVW PCRLv1 PCRLv2 GT 10% 20% 30% TransVW PCRLv1 PCRLv2 GT 10% 20% 30% b c Atelectasis TransVW PCRLv1 PCRLv2 Effusion Infiltration Mass Nodule Pneumonia TransVW PCRLv1 PCRLv2 a Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Visual interpretation of the transfer learning on chest pathology identification (a), and segmentation results of brain tumor (b) and liver (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We mainly compare PCRLv2 against PCRLv1 and TransVW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Red boxes in the top figure a denote the ground-truth (GT) annotations from radiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In figure b, we present the segmentation results of the enhancing tumor (ET) from BraTS when the labeling ratios are 10% and 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Similarly in the bottom figure, we display the liver segmentation results in three different labeling ratios (10%, 20%, and 30%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' PORTABLEPORTABLEPORTABLE14 TABLE 8 Transfer learning on abdominal organ segmentation (on LiTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We use LUNA for pre-training, and fine-tune the pre-trained model on LiTS with different amounts of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Best results are bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Methodology Labeling ratio 10% 20% 30% 40% 100% TS 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 MG [49] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 TransVW [16] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='4 Cube++ [35] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 3D-CPC [34] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 PCRLv1 [47] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='7 PCRLv2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='5 Nodule Infiltrate Atelectasis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Failure case analysis on chest pathology identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Red boxes stand for the lesion areas delineated by radiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Images are from NIH ChestX-ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Somewhat surprisingly, we find 3D-CPC does not out- perform context restoration based SSL (MG, TransVW, and Cube++) as obviously as those in Tables 4, 5, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' This comparison is consistent with our intuition: pixel- level information matters a lot in medical image segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Again, PCRLv1 and PCRLv2 outperform previous SSL methodologies in all three classes by large margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Compared to PCRLv1, PCRLv2 is more advantageous in segmenting the enhancing tumor (ET) regions, which are often smaller than WT and TC, and thus harder to segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The performance gains on ET again verify the effectiveness of multi-scale latent representations, which advances the segmentation of small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='9 Transfer learning on liver segmentation In Table 8, we present the results of liver segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' There exist three observable phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' First, we see that all SSL approaches provide substantial performance gains over train from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Second, we find the comparative methodology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=', 3D-CPC, achieves comparable segmen- tation performance to traditional context restoration based SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' This phenomenon verifies the necessity of utilizing pixel-level information in medical image segmentation (sim- ilar results also appear in Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Last but not the least, PCRLv2 consistently outperforms PCRLv1 in all labeling ratios, which again validates the effectiveness of introducing multiple scales into SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content='10 Visual analysis In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 8, we visually analyze the experimental results of transfer learning with limited annotations on chest pathol- ogy identification (10%), brain tumor segmentation (10% and 20%), and liver segmentation (10%, 20%, and 30%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Here, we compare PCRLv2 against generic SSL methodolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Considering TransVW was developed on top of MG, we exclude MG and compare PCRLv2 against PCRLv1 and TransVW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 8a presents the visual interpretation of chest pathol- ogy diagnoses using CAM [45] on six different pathologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We find that TransVW fails to capture the correct location of lesions on atelectasis, infiltration, nodule, and pneumonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In comparison, PCRLv1 can generate more interpretable diagnosis results but still yields inconsistent predictions on infiltration and nodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' By integrating multi-scale latent rep- resentations, PCRLv2 can capture the small lesion areas on infiltration and nodule, resulting in centralized yet accurate diagnosis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 8b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 8c, we visualize the segmentation results of the enhancing tumor (ET) on BraTS and liver on LiTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Compared to TransVW and PCRLv1, PCRLv2 reduces the false positive predictions and contains richer fine-grained details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We believe such superiority of PCRLv2 can be attributed to the integration of multi-scale pixel-level and semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We also provide some failure examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' One common characteristic of these detection results is that they include high-confidence predictions outside the lung area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' However, in daily clinical practice, such anomalies should not be located outside the lung area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Similar phenomena have been reported in [13], where the authors summarized them as “shortcuts” that are common in learning systems based on neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To mitigate this problem in self- supervised learning, we can add commonsense knowledge to pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Besides, it is also necessary to de- velop more powerful machine learning tools for model interpretation in various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' 5 CONCLUSION We present a unified visual information preservation frame- work for self-supervised learning in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' This framework aims to encode the pixel-level, semantic, and scale information into latent representations simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' To achieve this goal, we conduct multi-scale pixel restora- tion and feature comparison on the feature pyramid, which non-skip U-Net supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' The proposed PCRLv2 outper- forms previous self-supervised pre-training approaches by large margins and yields consistent improvements over its conference version (PCRLv1) on four well-established datasets in both quantitative and qualitative validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' We will continue to explore how to optimally integrate different types of information into SSL in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' REFERENCES [1] Samuel G Armato III, Geoffrey McLennan, Luc Bidaut, Michael F McNitt-Gray, Charles R Meyer, Anthony P Reeves, Binsheng Zhao, Denise R Aberle, 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+page_content=' [49] Zongwei Zhou, Vatsal Sodha, Jiaxuan Pang, Michael B Gotway, and Jianming Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Models genesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} +page_content=' Medical Image Analysis, 67:101840, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AyT4oBgHgl3EQf3vkG/content/2301.00772v1.pdf'} diff --git a/-9AzT4oBgHgl3EQfFvoN/content/tmp_files/2301.01014v1.pdf.txt b/-9AzT4oBgHgl3EQfFvoN/content/tmp_files/2301.01014v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..af2f4960de6a0caf1e9b1edde62d40982230a546 --- /dev/null +++ b/-9AzT4oBgHgl3EQfFvoN/content/tmp_files/2301.01014v1.pdf.txt @@ -0,0 +1,2032 @@ +arXiv:2301.01014v1 [math.DG] 3 Jan 2023 +TRICHOTOMY THEOREM FOR PRESCRIBED SCALAR AND MEAN +CURVATURES ON COMPACT MANIFOLDS WITH BOUNDARIES +JIE XU +Abstract. In this article, we give results of prescribing scalar and mean curvature functions +for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is +equipped on a compact manifold with boundary, with dimensions at least 3. The results are clas- +sified by the sign of the first eigenvalue of the conformal Laplacian. This leads to a “Trichotomy +Theorem” in terms of both scalar and mean curvature functions, which is a full extension of the +“Trichotomy Theorem” given by Kazdan and Warner. We also discuss prescribing Gauss and geo- +desic curvature problems on compact Riemann surfaces with boundary for metrics either pointwise +conformal or conformally equivalent to the original metric, provided that the Euler characteristic is +negative. The key step is a general version of monotone iteration scheme which handle the zeroth +order nonlinear term on the boundary conditions. +1. Introduction +In this article, we give a “Trichotomy Theorem” on compact manifolds ( ¯ +M, g) with non-empty +smooth boundaries ∂M, n := dim M ⩾ 3, involving both the scalar and mean curvatures. This +is a full generalization of the “Trichotomy Theorem” on closed manifolds, given by Kazdan and +Warner [9]. Precisely speaking, this “Trichotomy Theorem” concerns whether the given functions +S, H can be realized as scalar and mean curvatures, respectively, of a metric ˜g either within a +conformal class [g] or conformally equivalent to the metric g. Throughout this article, we assume +that ¯ +M is connected since otherwise we can easily apply arguments below equally to each connected +component. It is well-known that this problem is reduced to the existence of the positive solutions +of the nonlinear second order elliptic PDE +− a∆gu + Rgu = (S ◦ φ) up−1 in M, ∂u +∂ν + +2 +p − 2hgu = +2 +p − 2 (H ◦ φ) u +p +2 on ∂M. +(1) +Here Rg is the scalar curvature of the metric g, hg is the mean curvature. φ : ¯ +M → ¯ +M is some +diffeomorphism on +¯ +M. +When φ = Id, the PDE (1) is for prescribing functions S, H within a +conformal class [g]. The constants a, p are defined as +a = 4(n − 1) +n − 2 , p = +2n +n − 2. +∆g is the Laplace-Beltrami operator and ν is the unique outward unit normal vector field along +∂M. The functions S : C∞( ¯ +M) → R, and H : C∞(∂M) → R are given. We denote η1 to be the +first eigenvalue of the conformal Laplacian □g := −a∆g + Rg with associated eigenfunction ϕ, i.e. +ϕ is a positive, smooth function that solves the following PDE: +−a∆gϕ + Rgϕ = η1ϕ in M, ∂ϕ +∂ν + +2 +p − 2hgϕ = 0 on ∂M. +When the dimension of the manifold n = 2, we also discuss the pair of functions K, σ that can be +realized as Gaussian and geodesic curvatures, respectively, either for a pointwise conformal metric +1 + +2 +J. XU +or a conformally equivalent metric. The two dimensional case is reduced to the existence of the +solutions of the following elliptic PDE +− a∆gu + Kg = (K ◦ φ) e2u in M, ∂u +∂ν + σg = (σ ◦ φ) eu on ∂M. +(2) +Here Kg and σg are Gaussian and geodesic curvatures of g, respectively. +The functions K : +C∞( ¯ +M) → R and σ : C∞(∂M) → R are given. Again when the diffeomorphism φ : +¯ +M → +¯ +M +is the identity map, K, σ are prescribing Gauss and geodesic curvatures for some metric within the +conformal class [g]. +The main results of this article are given as follows: +Theorem 1.1. Let ( ¯ +M, g) be a connected, compact manifold with non-empty smooth boundary ∂M, +n = dim ¯ +M ⩾ 3. Let S, H ∈ C∞( ¯ +M) be given functions. +(i). If η1 < 0, then any function S < 0 somewhere in M can be realized as a scalar curvature +function of some metric conformally equivalent to g, with mean curvature cH for some small +enough constant c > 0 and any function H; +(ii). If η1 < 0, then any function S < 0 that changes sign in M can be realized as a scalar curvature +function of some metric conformally equivalent to g, with mean curvature cH for some small +enough constant c > 0 and any function H; +(iii). If η1 < 0, then any function S > 0 somewhere in M can be realized as a scalar curvature +function of some metric pointwise conformal to g, with mean curvature cH for some small +enough constant c > 0 and any function H. +Case (i) is given in §3 and §4; when S < 0 everywhere on ¯ +M, we can improve the result within +a pointwise conformal class [g] in Theorem 3.1 and Theorem 3.2; Case (ii) is given in §6; when S +satisfies +´ +M SdVolg < 0 in addition, we can improve the result within a pointwise conformal class +[g], see [18, Thm .1.2]; and Case (iii) is given in §7. The significance of this is that we can choose +arbitrary function with small enough sup-norm as our mean curvature function, provided that the +scalar curvature function is nontrivial. +Based on our best understanding, known results in this topic are mainly for the non-positive first +eigenvalue cases or non-positive Euler characteristic cases. In [5], Cruz-Bl´azquez, Malchiodi and +Ruiz discussed prescribing negative scalar functions and mean curvature functions with arbitrary +signs by variational method, for compact manifolds with dimensions at least 2. Some of our results +overlap their results, but with a different method and different hypotheses on prescribed functions. +However, our results are classified by the sign of the first eigenvalue of the conformal Laplacian. For +zero first eigenvalue case or zero Euler characteristic case, we follow the results of [18]. We point +out that Brezis and Merle discussed the PDE −∆eu = V eu on Ω ⊂ R2 with Dirichlet boundaray +condition in [3]. Other results for the local Yamabe equation with Dirichlet condition in higher +dimensions could be found in [12]. For more discussions with respect to (2) in 2-dimensional case, +we refer to [15, Ch. 13, Ch. 14]. When the first eigenvalue of conformal Laplacian is positive, a lot +of non-existence results are given, e.g. [2] and [11], etc. +We also give results on compact Riemann surfaces with boundary, provided that χ( ¯ +M) < 0. +Theorem 1.2. Let ( ¯ +M, g) be a compact Riemann surface with non-empty smooth boundary ∂M. +Let K, σ ∈ C∞( ¯ +M) be given functions. +(i). If K < 0 everywhere on ¯ +M, then there exists a metric pointwise conformal to g with Gauss +curvature K and geodesic curvature cσ for some small enough constant c > 0 and arbitrary +function σ; +(ii). If K < 0 somewhere on ¯ +M, then there exists a metric conformally equivalent to g with Gauss +curvature K and geodesic curvature cσ for some small enough constant c > 0 and arbitrary +function σ. + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +3 +Both results above are given in §5. Other than the related works on compact Riemann surfaces +with boundary we introduced above, many work has been done on closed Riemann surface, a +comprehensive study was given by Kazdan and Warner [8], including results for all signs of χ( ¯ +M). +For Nirenberg problem, we refer to Chang and Yang [4] and Struwe [13], etc.. +The most common method in analyzing this type of Kazdan-Warner problem is by calculus +of variations since we can consider the PDE as Euler-Lagrange equation with respect to some +functional; recently Morse theory is also involved. However, a new method, inspired by Kazdan +and Warner [10], has been developed recently. This new method applies monotone iteration scheme, +a local version of calculus of variation to classify the existence results by sign of the first eigenvalue +η1 of conformal Laplacian. This method has been applied to completely solve the Escobar problem +[16], the Han-Li conjecture [17], the prescribed scalar curvature problem on compact manifolds [19], +a trichotomy theorem in terms of prescribed scalar curvature with Dirichlet condition at boundary +[20], and a comprehensive study of zero first eigenvalue case on compact manifolds, possibly with +boundary, with dimensions at least 3 [18]. In this article, we apply a variation of the combination +of monotone iteration scheme and local analysis to show the results of prescribed scalar and mean +curvatures for the cases η1 > 0 and η1 < 0. We also develop a general monotone iteration scheme, +which can handle nonlinear terms both in the PDE and on the boundary condition; this new +monotone iteration scheme, see Theorem 2.3, allows us to work on 2-dimensional case without +using the calculus of variation. +This systematic procedure is powerful, but unfortunately this +direct method cannot be used to the classical manifold, the unit ball with spherical boundary. We +will explain why this direct method does not work in this case. Note that Escobar [7] has proved a +nontrivial Kazdan-Warner type obstruction of prescribed mean curvature functions for this case. +This paper is organized as follows: +In §2, we introduce the essential definitions and results that will be used throughout this article. +We assume the backgrounds of standard elliptic theory. We also introduced two versions of mono- +tone iteration schemes. Theorem 2.2 is for the PDE (1); Theorem 2.3 is more general, works for all +second order semi-linear elliptic PDE with Robin boundary conditions, possibly with zeroth order +nonlinear term on the boundary condition. Theorem 2.3 works well for the PDE like (2). +In §3, we give results for prescribing scalar curvature function S and mean curvature function +H within a conformal class [g] on ( ¯ +M, g), n = dim ¯ +M ⩾ 3, provided that η1 < 0. When S < 0 +everywhere and arbitrary H, the results are given in Theorem 3.1 and Theorem 3.2. When S < 0 +somewhere and arbitrary H, the results are given in Theorem 3.3 and Corollary 3.1 with some +restriction on S. The monotone iteration scheme plays a central role. +In §4, we give results for prescribing scalar curvature function S and mean curvature function +H for some metric conformally equivalent to g on ( ¯ +M, g), n = dim ¯ +M ⩾ 3, provided that η1 < 0. +It follows from Corollary 3.1. We conclude in Theorem 4.1 that any S that is negative somewhere +can be realized as a scalar curvature function of some metric conformally equivalent to g, with the +mean curvature cH for small enough constant c > 0 and arbitrary H. +In §5, we discuss prescribing Gauss and geodesic curvature functions K, σ on compact Riemann +surfaces with boundary for metrics conformally equivalent to the original metric g. We show in +Theorem 5.1 that any function K that is negative somewhere and satisfies some analytic condition +can be realized as Gaussian curvature function for some metric conformally equivalent to g, the +metric also has geodesic curvature cσ for some small enough constant c > 0 and arbitrary σ. The +result in Corollary 5.1 says that when K < 0 everywhere on ¯ +M, the metric can be chosen within a +conformal class [g]. +In §6, we give results for prescribing scalar function S and mean curvature function H for some +metric conformally equivalent to g on ( ¯ +M, g), n = dim ¯ +M ⩾ 3, provided that η1 = 0. We show +that any function S that changes sign can be realized as a scalar curvature function some metric + +4 +J. XU +conformally equivalent to g, with the mean curvature cH for small enough constant c > 0 and +arbitrary H in Corollary 6.2. Obviously there is a trivial case S ≡ H ≡ 0. +In §7, we consider the prescribing scalar and mean curvature problem for η1 > 0. The results in +Theorem 7.1 and Theorem 7.2 are for the case S > 0 somewhere and arbitrary H. We also explain +why our method cannot work on closed Euclidean ball with some nontrivial mean curvature on the +boundary Sn. +2. The Preliminaries and The Monotone Iteration Scheme +In this section, we first introduce the necessary definitions and essential results we need for the +later sections, then introduce a general version of the monotone iteration scheme given in [18], other +than many variations we have used in [16, 17, 19, 20, 21], with respect to the following Yamabe +equation with Robin boundary condition +− a∆gu + Rgu = Sup−1 in M, ∂u +∂ν + +2 +p − 2hgu = +2 +p − 2Hu +p +2 on ∂M. +(3) +for given functions S, H ∈ C∞( ¯ +M), and n = dim ¯ +M ⩾ 3. Lastly we introduce a W s,q-type regularity +for elliptic PDE with Robin boundary conditions. +First of all, we give definitions of Sobolev spaces, a local version and a global version. +Let +Ω be a connected, bounded, open subset of Rn with smooth boundary ∂Ω equipped with some +Riemannian metric g that can be extended smoothly to ¯Ω. We call (Ω, g) a Riemannian domain. +Throughout this article, we denote the space of smooth functions with compact support by C∞ +c , +smooth functions by C∞, and continuous functions by C0. +Definition 2.1. Let (Ω, g) be a Riemannian domain. +Let (M, g) be a closed Riemannian n- +manifold, and ( ¯ +M, g) be a compact Riemannian n-manifold with non-empty smooth boundary, with +volume density dVolg. Let u be a real valued function. Let ⟨v, w⟩g and |v|g = ⟨v, v⟩1/2 +g +denote the +inner product and norm with respect to g. +(i) For 1 ⩽ p < ∞, we define the Lebesgue spaces on Ω and ¯ +M to be +Lp(Ω) is the completion of +� +u ∈ C∞ +c (Ω) : ∥u∥p +p := +ˆ +Ω +|u|pdx < ∞ +� +, +Lp(Ω, g) is the completion of +� +u ∈ C∞ +c (Ω) : ∥u∥p +p,g := +ˆ +Ω +|u|p dVolg < ∞ +� +, +Lp(M, g) is the completion of +� +u ∈ C∞(M) : ∥u∥p +p,g := +ˆ +M +|u|p dVolg < ∞ +� +. +(ii) For ∇u the Levi-Civita connection of g, and for u ∈ C∞(Ω) or u ∈ C∞( ¯ +M), +|∇ku|2 +g := (∇α1 . . . ∇αku)(∇α1 . . . ∇αku). +(4) +In particular, |∇0u|2 +g = |u|2 and |∇1u|2 +g = |∇u|2 +g. + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +5 +(iii) For s ∈ N, 1 ⩽ p < ∞, we define the (s, p)-type Sobolev spaces on Ω and ¯ +M to be +W s,p(Ω) = + + +u ∈ Lp(Ω) : ∥u∥p +W s,p(Ω) := +ˆ +Ω +s +� +j=0 +��Dju +��p dx < ∞ + + + , +(5) +W s,p(Ω, g) = + + +u ∈ Lp(Ω, g) : ∥u∥p +W s,p(Ω,g) = +s +� +j=0 +ˆ +Ω +��∇ju +��p +g dVolg < ∞ + + + , +W s,p(M, g) = + + +u ∈ Lp(M, g) : ∥u∥p +W s,p(M,g) = +s +� +j=0 +ˆ +M +��∇ju +��p +g dVolg < ∞ + + + . +Here |Dju|p := � +|α|=j|∂αu|p in the weak sense. Similarly, W s,p +0 (Ω) is the completion of C∞ +c (Ω) +with respect to the W s,p-norm. +In particular, Hs(Ω) := W s,2(Ω) and Hs(Ω, g) := W s,2(Ω, g), +Hs(M, g) := W s,2(M, g) are the usual Sobolev spaces. We similarly define Hs +0(Ω), Hs +0(Ω, g) and +Hs +0(M, g). +(iv) On closed manifolds (M, g), we say that a function u ∈ Hs(M, g) if u ∈ L2(M, g) , and for +any coordinate chart U ⊂ M, any ψ ∈ C∞ +c (U), the function ψu ∈ Hs(U, g). +We assume the background of the standard elliptic theory, including the solvability of standard +linear elliptic PDEs, elliptic regularity of Hs-type, trace theorem, Sobolev embedding, Schauder +estimates, etc. We introduce a W s,q-type elliptic regularity for later use. +Theorem 2.1. [17, Thm. 2.2] Let ( ¯ +M, g) be a compact manifold with smooth boundary ∂M. Let +ν be the unit outward normal vector along ∂M and q > n = dim ¯ +M. Let L : C∞( ¯ +M) → C∞( ¯ +M) +be a uniform second order elliptic operator on M with smooth coefficients up to ∂M and can be +extended to L : W 2,q(M, g) → Lq(M, g). Let f ∈ Lq(M, g), ˜f ∈ W 1,q(M, g). Let u ∈ H1(M, g) be a +weak solution of the following boundary value problem +Lu = f in M, Bu = ∂u +∂ν + c(x)u = ˜f on ∂M. +(6) +Here c ∈ C∞(M). Assume also that Ker(L) = {0} associated with the homogeneous Robin boundary +condition. If, in addition, u ∈ Lq(M, g), then u ∈ W 2,q(M, g) with the following estimates +∥u∥W 2,q(M,g) ⩽ γ′ � +∥Lu∥Lq(M,g) + ∥Bu∥W 1,q(M,g) +� +(7) +Here γ′ depends on L, q, c and the manifold ( ¯ +M, g) and is independent of u. +We then introduce the first eigenvalue of conformal Laplacian. Note that a = 4(n−1) +n−2 +and p = +2n +n−2, +hence it only makes sense when n ⩾ 3. +Definition 2.2. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M. We +denote η1 be the first eigenvalue of conformal Laplacian with its corresponding eigenfunction ϕ > 0 +if and only if the following PDE holds. +− a∆gϕ + Rgϕ = η1ϕ in M, ∂ϕ +∂ν + +2 +p − 2hgϕ = 0 on ∂M. +(8) +We now introduce a variation of the monotone iteration scheme we used in [16], [17] and [19]. +In particular, we do require hg = h > 0 to be some positive constant on ∂M, this can be done due +to the proof of the Han-Li conjecture in [17]. We will also use other versions of monotone iteration +schemes introduced in eariler work [16, 17, 19, 20, 21]. +Theorem 2.2. [18, Thm. 2.4] Let ( ¯ +M, g) be a compact manifold with smooth boundary ∂M. Let +ν be the unit outward normal vector along ∂M and q > dim ¯ +M. Let S ∈ C∞( ¯ +M) and H ∈ C∞( ¯ +M) + +6 +J. XU +be given functions. Let the mean curvature hg = h > 0 be some positive constant. In addition, +we assume that sup ¯ +M|H| is small enough. Suppose that there exist u− ∈ C0( ¯ +M) ∩ H1(M, g) and +u+ ∈ W 2,q(M, g) ∩ C0( ¯ +M), 0 ⩽ u− ⩽ u+, u− ̸≡ 0 on ¯ +M, some constants θ1 ⩽ 0, θ2 ⩾ 0 such that +−a∆gu− + Rgu− − Sup−1 +− +⩽ 0 in M, ∂u− +∂ν + +2 +p − 2hgu− ⩽ θ1u− ⩽ +2 +p − 2Hu +p +2 +− on ∂M +−a∆gu+ + Rgu+ − Sup−1 ++ +⩾ 0 in M, ∂u+ +∂ν + +2 +p − 2hgu+ ⩾ θ2u+ ⩾ +2 +p − 2Hu +p +2 ++ on ∂M +(9) +holds weakly. In particular, θ1 can be zero if H ⩾ 0 on ∂M, and θ1 must be negative if H < 0 +somewhere on ∂M; similarly, θ2 can be zero if H ⩽ 0 on ∂M, and θ2 must be positive if H > 0 +somewhere on ∂M. Then there exists a real, positive solution u ∈ C∞(M) ∩ C1,α( ¯ +M) of +□gu = −a∆gu + Rgu = Sup−1 in M, Bgu = ∂u +∂ν + +2 +p − 2hgu = +2 +p − 2Hu +p +2 on ∂M. +(10) +The following two results are necessary, which shows the existence of the solution of some local +Yamabe-type problem. When the manifold is not locally conformally flat, we need +Proposition 2.1. [18, Prop. 3.2] Let (Ω, g) be a Riemannian domain in Rn, n ⩾ 3, not locally +conformally flat, with C∞ boundary, with Volg(Ω) and the Euclidean diameter of Ω sufficiently +small. Let f ∈ Ω′ ⊃ Ω be a positive, smooth function in some open region Ω′. In addition, we +assume that the first eigenvalue of Laplace-Beltrami operator −∆g on Ω with Dirichlet condition +satisfies λ1 → ∞ as Ω shrinks. Assume Rg < 0 within the small enough closed domain ¯Ω. Then +the Dirichlet problem +− a∆gu + Rgu = fup−1 in Ω, u ≡ 0 on ∂Ω +(11) +has a real, positive, smooth solution u ∈ C∞(Ω) ∩ H1 +0(Ω, g) ∩ C0(¯Ω). The size of Ω is depending on +the function f. +When the manifold is locally conformally flat, we give the local solution of (11) provided that Ω +is not topologically trivial. +Proposition 2.2. [19, Prop. 2.5] Let (Ω, g) be a Riemannian domain in Rn, n ⩾ 3, with C∞ +boundary. Let the metric g be locally conformally flat on some open subset Ω′ ⊃ ¯Ω. For any point +ρ ∈ Ω and any positive constant ǫ, denote the region Ωǫ to be +Ωǫ = {x ∈ Ω||x − ρ| > ǫ}. +Assume that Q ∈ C2(¯Ω), minx∈¯Ω Q(x) > 0 and ∇Q(ρ) ̸= 0. Then there exists some ǫ0 such that for +every ǫ ∈ (0, ǫ0) the Dirichlet problem +− a∆gu + Rgu = Qup−1 in Ωǫ, u = 0 on ∂Ωǫ +(12) +has a real, positive, smooth solution u ∈ C∞(Ωǫ) ∩ H1 +0(Ωǫ, g) ∩ C0( ¯Ωǫ). +Remark 2.1. It is straightforward to see that under conformal change ˜g = φp−2g, we have +˜g = φp−2g ⇒ −a∆˜g + R˜g = φ− n+2 +n−2 (−a∆g + Rg) φ ⇔ □˜g = φ1−p□gφ. +(13) +We call (13) the conformal invariance of the conformal Laplacian. It follows from Proposition 2.2 +and (13) that if the manifold ( ¯ +M, g) is locally conformally flat in the interior, the equation (12) is +equivalent to +− a∆geu = Qup−1 in Ωǫ, u = 0 on ∂Ωǫ +(14) +which admits a positive solution u ∈ C∞(Ωǫ) ∩ H1 +0(Ωǫ, g) ∩ C0( ¯Ωǫ). +As a prerequisite, we also need a result in terms of the perturbation of negative first eigenvalue +of conformal Laplacian. + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +7 +Proposition 2.3. Let ( ¯ +M, g) be a compact Riemannian manifold with non-empty smooth boundary +∂M, n = dim ¯ +M ⩾ 3. Let β > 0 be a small enough constant. If η′ +1 < 0, then the quantity +η′ +1,β = inf +u̸=0 +a +´ +M|∇gu|2dVolg + +´ +M Rgu2dVolg + +2a +p−2 +´ +∂M(hg + β)u2dS +´ +M u2dVolg +< 0. +In particular, η′ +1,β satisfies +− a∆gϕ + Rgϕ = η′ +1,βϕ in M, ∂ϕ +∂ν + +2 +p − 2(hg + β)ϕ = 0 on ∂M +(15) +with some positive function ϕ ∈ C∞( ¯ +M). +Proof. Since η′ +1 < 0, the normalized first eigenfunction ϕ1, i.e. +´ +M ϕ2 +1dVolg = 1, satisfies +η′ +1 = a +ˆ +M +|∇gϕ1|2dVolg + +ˆ +M +Rgϕ2 +1dVolg + +2a +p − 2 +ˆ +∂M +hgϕ2 +1dS +By characterization of η′ +1,β, we have +η′ +1,β ⩽ a +ˆ +M +|∇gϕ1|2dVolg + +ˆ +M +Rgϕ2 +1dVolg + +2a +p − 2 +ˆ +∂M +(hg + β)ϕ2 +1dS = η′ +1 + β +ˆ +∂M +ϕ2 +1dS. +Since ϕ1 is fixed, it follows that η′ +1,β < 0 if β > 0 is small enough. +□ +When n = 2, i.e. M or ¯ +M is a compact Riemann surface (possibly with boundary), all tools +above are not available. We thus need a new version of the monotone iteration scheme for compact +Riemann surfaces with non-empty smooth boundary. We point out that the monotone iteration +scheme below works for all compact manifolds with non-empty boundary, with dimensions at least +2. +Theorem 2.3. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim M ⩾ 2. Let q > n be a positive integer. Let F(·, ·), G(·, ·) : ¯ +M × R → R be smooth functions. +Let ν be the unit outward normal vector along ∂M. Let σ be some nonnegative, small enough +constant. If +(i) there exists two functions u+ ∈ C∞( ¯ +M) and u− ∈ C0( ¯ +M) ∩ H1(M, g) such that +−∆gu+ ⩾ F(·, u) in M, ∂u +∂ν + σu ⩾ G(·, u+) on ∂M; +−∆gu− ⩽ F(·, u) in M, ∂u +∂ν + σu ⩽ G(·, u−) on ∂M, +(16) +where the sub-solution may hold in the weak sense; and +(ii) in addition, sup ¯ +M|G(·, u+)|, sup ¯ +M|∇G(·, u+)| are small enough; +(iii) furthermore, u+ ⩾ u− pointwise on ¯ +M; +then there exists a smooth function u ∈ C∞( ¯ +M) with u− ⩽ u ⩽ u+ such that +− ∆gu = F(·, u) in M, ∂u +∂ν + σu = G(·, u) on ∂M. +(17) +Remark 2.2. The proof of Theorem 2.3 is essentially the same as the proof of [18, Thm. 2.4], +except some minor change, e.g. here we use general smooth functions F and G but not specific +Yamabe equations. We therefore will give a relatively concise proof for Theorem 2.3. + +8 +J. XU +Proof. +¯ +M is compact, so extremal values of continuous functions u+, u− can be achieved. Choose +positive constant A and nonnegative constant B such that +A ⩾ −∂F +∂u (x, u(x)), ∀x ∈ ¯ +M, u(x) ∈ [min +¯ +M u−, max +¯ +M u+]; +B ⩾ σ − ∂G +∂u (x, u(x)), ∀x ∈ ¯ +M, u(x) ∈ [min +¯ +M u−, max +¯ +M u+]. +(18) +Denote u0 = u+ ∈ C∞( ¯ +M), and consider the iteration scheme +−∆guk + Auk = Auk−1 + F(·, uk−1) in M, k ∈ N, +∂uk +∂ν + Buk = Buk−1 − σuk−1 + G(·, uk−1) on ∂M, k ∈ N. +(19) +Since A > 0, B ⩾ 0, the operator +� +−∆g + A, ∂ +∂ν + B +� +is invertible due to the standard argument. Clearly when k = 1, the first iteration step in (19) +gives a unique smooth solution u1 ∈ C∞( ¯ +M). The regularity argument is also standard. +We show that u− ⩽ u ⩽ u+. For u− ⩽ u, we have to use the sub-solution in the weak sense, +since u0 = u+ ⩾ u−, we pair (19) for k = 1 with arbitrary non-negative function v ∈ C∞( ¯ +M), and +subtract this with the sub-solution (adding Au− and Bu− on both sides of the PDE and boundary +conditions respectively) in the weak sense, we have +ˆ +M +(A (u0 − u−) + F (x, u0) − F (x, u−)) vdVolg ⩽ +ˆ +M +(−∆g (u1 − u−) + A (u1 − u−)) vdVolg +⩽ +ˆ +∂M +B (u1 − u−) vdS − +ˆ +∂M +(B (u0 − u−) − σ (u0 − u−) + G (x, u0) − G (x, u−)) vdS ++ +ˆ +M +A (u1 − u−) vdVolg + +ˆ +M +∇g (u1 − u−) · ∇gvdVolg. +Taking v = w := max (u− − u1, 0), and applying the mean value theorem for F, G, due to the +definitions of A, B in (18), we observe that +ˆ +M +|∇gw|2 + +ˆ +∂M +Bw2 + +ˆ +M +Aw2 ⩽ 0. +It follows that w = 0, therefore u− ⩽ u1. By a very similar argument in terms of the subtraction +between (19) and the super-solution, we conclude that u+ ⩾ u1. +Inductively, we may assume the existence of the solutions u1, . . . , uk with +u− ⩽ uk ⩽ uk−1 ⩽ . . . ⩽ u1 ⩽ u0. +By the same argument in the first iteration step, we conclude the existence of uk+1 ∈ C∞( ¯ +M); in +addition, uk+1 satisfies +u− ⩽ uk+1 ⩽ uk ⩽ uk−1 ⩽ . . . ⩽ u1 ⩽ u0. +Therefore we show the existence of the sequence of solutions of (19) with the monotonicity +u− ⩽ . . . ⩽ uk+1 ⩽ uk ⩽ uk−1 ⩽ . . . ⩽ u0, k ∈ N. +(20) +We now show the uniform boundedness of ∥uk∥C1,α( ¯ +M). +Since q > n, showing the uniform +boundedness of ∥uk∥C1,α( ¯ +M) is equivalent to show the uniform boundedness of ∥uk∥W 2,q(M,g). We +have mentioned that the operator is invertible and thus the W s,q-type estimates (7) applies. We L +and the boundary condition c to be the operators here with associated constant γ′. Mimicking the + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +9 +boundedness proof in [18, Thm. 2.4], we should require σ and sup ¯ +M|G(·, u)|, and sup ¯ +M|∇G(·, u)| +to be small enough. Denote +C = +sup +x∈ ¯ +M,u(x)∈[min ¯ +M u−,max ¯ +M u+] +|F(x, u(x))|; +D1 = +sup +x∈ ¯ +M,u(x)∈[min ¯ +M u−,max ¯ +M u+] +|G(x, u(x))| ; +D2 = +sup +x∈ ¯ +M,u(x)∈[min ¯ +M u−,max ¯ +M u+] +|∇G(x, u(x))| ; +(21) +We require that G(·, u+), D1, D2 satisfies +∥(B − σ) u+ + G (·, u+)∥W 1,q(M,g) ⩽ 1; +(B − σ) · γ′ +�� +A max +¯ +M (|u+|, |u−|) + C +� +· Volg(M) +1 +q + 1 +� ++ D1 · Volg(M) +1 +q ++ D2 · γ′ +�� +A max +¯ +M (|u+|, |u−|) + C +� +· Volg(M) +1 +q + 1 +� +⩽ 1. +(22) +By (7) and the first inequality in (22), we observe from the PDE (19) with k = 1 that +∥u1∥W 2,q(M,g) ⩽ γ′ � +∥Au+ + F(·, u+)∥Lq(M,g) + ∥(B − σ) u+ + G (·, u+)∥W 1,q(M,g) +� +⩽ γ′ +�� +A max +¯ +M |u+| + C +� +· Volg(M) +1 +q + 1 +� +⩽ γ′ +�� +A max +¯ +M (|u+|, |u−|) + C +� +· Volg(M) +1 +q + 1 +� +. +Inductively, assume that +∥uk∥W 2,q(M,g) ⩽ γ′ +�� +A max +¯ +M (|u+|, |u−|) + C +� +· Volg(M) +1 +q + 1 +� +. +(23) +To check ∥uk+1∥W 2,q(M,g), we apply the W s,q-type elliptic estimate with the solution of (19) again, +∥uk+1∥W 2,q(M,g) ⩽ γ′ � +∥Auk + F(·, uk)∥Lq(M,g) + ∥(B − σ) uk + G (·, uk)∥W 1,q(M,g) +� +⩽ γ′ +�� +A max +¯ +M (|u+|, |u−|) + C +� +· Volg(M) +1 +q +� ++ γ′ � +(B − σ)∥uk∥W 1,q(M,g) + ∥G(·, uk)∥Lq(M,g) + ∥∇G(·, uk)∥Lq(M,g) +� +⩽ γ′ +�� +A max +¯ +M (|u+|, |u−|) + C +� +· Volg(M) +1 +q +� ++ +� +γ′�2 (B − σ) +� +A max +¯ +M ((|u+|, |u−|) + C) · Volg(M) +1 +q + 1 +� ++ γ′D1 · Volg(M) +1 +q + +� +γ′�2 D2 +� +A max +¯ +M ((|u+|, |u−|) + C) · Volg(M) +1 +q + 1 +� +⩽ γ′ +�� +A max +¯ +M (|u+|, |u−|) + C +� +· Volg(M) +1 +q + 1 +� +. +It turns that ∥uk∥W 2,q(M,g) is uniformly bounded. The rest of the argument, in applying Arzela- +Ascoli, the monotonicity of the sequence, and the elliptic regularity, is essentially the same as in +[18, Thm. 2.4]. We omit the details here. +In conclusion, the sequence uk converges classically to a smooth function u which solves (17). In +addition, u− ⩽ u ⩽ u+ pointwise on ¯ +M. +□ + +10 +J. XU +Remark 2.3. Theorem 2.2 is a special case of Theorem 2.3 by taking F(·, u) = −Rgu+Sup−1 and +G(·, u) = − +2 +p−2hgu + +2 +p−2Hu +p +2 . +3. Prescribed Scalar and Mean Curvature Functions under Pointwise Conformal +Deformation When η1 < 0 +Recall the Yamabe equation with Robin condition +− a∆gu + Rgu = Sup−1 in M, ∂u +∂ν + +2 +p − 2hgu = +2 +p − 2Hu +p +2 on ∂M. +(24) +In this section, we consider the existence of the solution of (24) for given functions S, H ∈ C∞( ¯ +M), +provided that η1 < 0. In particular, we will discuss the following cases: +(i). S < 0 in M, and H ⩽ 0 everywhere on ∂M, H ̸≡ 0, with η1 < 0; +(ii). S < 0 in M, and H > 0 somewhere on ∂M, with η1 < 0; +(iii). S changes sign in M, and H is arbitrary on ∂M, with η1 < 0. +Note that the Case (ii) above covers the possibilities when H > 0 everywhere on ∂M, or +´ +∂M HdS > +0. Note also that the case S < 0 everywhere in M and H = 0 on ∂M has been discussed in [19]. +For Case (iii), obviously we have to impose some restrictions on S and H, as we shall see later; +there is no free choice of S especially, due to Kazdan and Warner [10]. The first result concerns +the Case (i). +Theorem 3.1. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M ⩾ 3. Let S1 < 0 be any smooth function on +¯ +M. Let H1 ∈ C∞( ¯ +M) such that H1 < 0 +everywhere on ∂M. If η1 < 0, then there exists a small enough constant c > 0 such that (24) +admits a positive solution u ∈ C∞( ¯ +M) with S = S1 and H = cH1. Equivalently, there exists a +Yamabe metric ˜g = up−2g such that R˜g = S1 and h˜g = cH1 +���� +∂M +. +Proof. Due to the proof of Han-Li conjecture [17, Theorem], we may assume that hg = h > 0 and +Rg < 0. Since η1 < 0, it follows that η1,β < 0 with small enough positive constant β > 0, due to +Proposition 2.3. Any constant multiple of ϕ solves (15). Denote φ = δϕ, we choose the constant +δ > 0 small enough so that +η1,β inf +¯ +M ϕ ⩾ δp−2 · inf +¯ +M S1 · sup +¯ +M +ϕp−1. +This can be done since both η1,β and S1 are negative functions. It follows that +−a∆gφ + Rgφ = η1,βφ ⩽ S1φp−1 in M. +Fix this δ. We check the boundary condition +−∂φ +∂ν + +2 +p − 2hgφ = −β · +2 +p − 2φ ⩽ +2 +p − 2 · (cH1) φ +p +2 +for small enough positive constant c > 0. Again it works since both −β and H1 are negative. We +set +u− := φ. +(25) +The argument above shows that u− is a sub-solution of (24) with S = S1 and H = cH1 for small +enough c. For super-solution, we set +u+ := C ≫ 1. +(26) +When C large enough, we have +−a∆gu+ + Rgu+ = RgC ⩾ S1Cp−1 in M. + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +11 +Since H1 < 0, it is straightforward to check that for any c > 0, we have +−∂u+ +∂ν + +2 +p − 2hgu+ ⩾ 0 > +2 +p − 2 (cH1) u +p +2 ++. +We can enlarge C so that C ⩾ sup ¯ +M u−. Lastly we shrink c if necessary since we require the +smallness of the sup-norm of the prescribing mean curvature function in the proof of Theorem 2.2. +Since 0 < u− ⩽ u+ and both u+ and u− are smooth functions, we conclude by Theorem 2.2 that +(24) has a positive solution u ∈ C∞( ¯ +M) with S = S1 and H = cH1 for small enough c > 0. +□ +We now consider the Case (ii) at the beginning of this section. Actually the proof is very similar +to Theorem 3.1 above. +Theorem 3.2. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M ⩾ 3. Let S2 < 0 be any smooth function on +¯ +M. Let H2 ∈ C∞( ¯ +M) such that H2 > 0 +somewhere on ∂M. If η1 < 0, then there exists a small enough constant c > 0 such that (24) +admits a positive solution u ∈ C∞( ¯ +M) with S = S2 and H = cH2. Equivalently, there exists a +Yamabe metric ˜g = up−2g such that R˜g = S2 and h˜g = cH2 +���� +∂M +. +Proof. The choice of the sub-solution is exactly the same as in Theorem 3.1. When we fix the sub- +solution u−, we choose u+ = C ≫ 1 with C ⩾ u−, also large enough so that the same argument in +Theorem 3.1 holds. Fix this C from now on. The only difference is that since H2 > 0 somewhere, +we may need to shrink c, if necessary, so that +∂C +∂ν + +2 +p − 2hgC ⩾ +2 +p − 2 · sup +∂M +(cH2)C +p +2 +The rest of the argument is exactly the same as in Theorem 3.1. +□ +Remark 3.1. The method of monotone iteration scheme has its limits, as we cannot obtain the +prescribed mean curvature to be H, due to the technical issue, see [18, Thm. 2.4]. +We now discuss the Case (iii). The following argument is inspired by Kazdan and Warner [10]. +When η1 < 0, Kazdan and Warner showed that the key is to get the super-solution of (24), if we +are not using the variational method but instead the monotone iteration scheme. Next result shows +that a super-solution of (24) can be converted to another relation. We point out that the following +result is not specific for η1 < 0 case only. +Lemma 3.1. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M ⩾ 3. +Let S, H ∈ C∞( ¯ +M) be given functions. +Then there exists some positive function +u ∈ C∞( ¯ +M) satisfying +− a∆gu + Rgu ⩾ Sup−1 in M, ∂u +∂ν + +2 +p − 2hgu ⩾ +2 +p − 2Hu +p +2 on ∂M +(27) +if and only if there exists some positive function w ∈ C∞( ¯ +M) satisfying +− a∆gw + (2 − p)Rgw + (p − 1)a +p − 2 +· |∇gw|2 +w +⩽ (2 − p)S in M, ∂w +∂ν − 2hgw ⩽ −2Hw +1 +2. +(28) +Moreover, the equality in (27) holds if and only if the equality in (28) holds. +Proof. Assume that there is a positive function u ∈ C∞(M) that satisfies (27). Define +w = u2−p. +Note that 2 − p = − +4 +n−2 < 0 since n ⩾ 3 by hypothesis. We compute that +∇w = (2 − p)u1−p∇u ⇔ ∇u = up−1(2 − p)−1∇w, + +12 +J. XU +and +∆gw = (2 − p)u1−p∆gu + (2 − p)(1 − p)u−p|∇gu|2. +By the inequality (27), we have +a∆gw = (2 − p)u1−p (a∆gu) + a(2 − p)(1 − p)u−p|∇gu|2 +⩾ (p − 2)u1−p � +−Rgu + Sup−1� ++ a(2 − p)(1 − p)(2 − p)−2u2p−2u−p|∇gv|2 += (p − 2)S + (2 − p)Rgu1−p + a(p − 1) +p − 2 up−2|∇gv|2 += (p − 2)S + (2 − p)Rgw + a(p − 1) +p − 2 +|∇gw|2 +w +. +Shifting (p − 2)S to the left side and a∆gw to the right side, we get the first part of the inequality +(28). For the boundary condition, recall that u = w +1 +2−p and p = +2n +n−2, it follows that +∂u +∂ν + +2 +p − 2Hu +p +2 ⩾ +2 +p − 2hgu ⇔ +1 +2 − pw +1 +2−p −1 ∂w +∂ν + +2 +p − 2hgw +1 +2−p ⩾ +2 +p − 2Hw +p +2(2−p) +⇔ − n − 2 +4 +w− n +4 − 1 +2 ∂w +∂ν + n − 2 +2 +hgw− n +4 + 1 +2 ⩾ n − 2 +2 +Hw− n +4 +⇔∂w +∂ν − 2hgw ⩽ −2Hw +1 +2 . +Hence the second part of (28) holds. It is clear that the equality holds if an only if all inequalities +above are equalities. +If we assume (28) for some w, we just define u = w +1 +2−p . The argument is very similar and we +omit the details. +□ +We now introduce the result of prescribing scalar and mean curvature functions for Case (iii), +with a technical restriction very similar to the condition given by Kazdan and Warner. +This +technical condition, in principle, is to show the positivity of the function that satisfies (28). Due +to the Han-Li conjecture [17, Theorem], we may assume that the initial metric g has Rg = λ < 0 +and hg = ζ > 0, since η1 < 0. Before we start with the special case, recall that if there exists a +constant q > n, and some function u ∈ C∞( ¯ +M) satisfies +∥u∥W 2,q(M,g) ⩽ γ′ � +∥F1∥Lq(M,g) + ∥F2∥W 1,q(M,g) +� +for some functions F1 ∈ Lq(M, g) and F2 ∈ W 1,q(M, g), the H¨older estimates implies that +∥u∥L∞( ¯ +M) + ∥∇u∥L∞( ¯ +M) ⩽ γ +� +∥F1∥Lq(M,g) + ∥F2∥W 1,q(M,g) +� +. +(29) +This inequality is due to the Sobolev embedding in [1, §2]. +Theorem 3.3. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M ⩾ 3. Assume that η1 < 0, Rg = λ < 0 and hg = ζ > 0 for some constants λ and ζ. Let +S3, H3 ∈ C∞( ¯ +M) and q > n be a positive integer. Let γ be the constant in the estimate (29). Set +D = (p−1)a +p−2 . If there exists a function F ∈ C∞( ¯ +M) and a positive constant A > 0, such that +(2 − p)S3 ⩾ F on ∂M, ∥F − A∥Lq(M,g) ⩽ +A +2γ (1 + (D + 1) (2 − p)λ), +(30) +then there exists a small enough constant c > 0 such that (24) admits a positive solution u ∈ C∞( ¯ +M) +with S = S3 and H = cH3. Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S3 +and h˜g = cH3 +���� +∂M +. + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +13 +Proof. In this proof, we always denote Rg = λ and hg = ζ. We construct the super-solution of +(24) first. +Due to Lemma 3.1, it is equivalent to show the existence of some positive function +w ∈ C∞( ¯ +M) such that (28) holds for S = S3 and H = cH3 for some constant c. Take +δ := +A +(1 + (D + 1)(2 − p)λ) > 0. +We also choose some negative constant +δ′ = − +δ +2γVolg(M) +1 +q +< 0. +By standard elliptic theory, there exists a unique solution w of the following PDE +−a∆gw + (2 − p)λw = F − δ in M, ∂w +∂ν = δ′ on ∂M. +The uniqueness comes from the fact that (2−p)λ > 0, which implies the invertibility of the operator +� +−a∆g + (2 − p)λ, ∂ +∂ν +� +. Clearly the constant (D + 1)δ solves the PDE +−a∆g((D + 1)δ) + (2 − p)λ · ((D + 1)δ) = (2 − p)λ · ((D + 1)δ) in ∂M, ∂((D + 1)δ) +∂ν += 0 on ∂M. +Denote +w0 := w − (D + 1)δ. +The function w0 satisfies +−a∆gw0 + (2 − p)λw0 = F − δ − (D + 1)(2 − p)λδ = F − A in M, +∂w +∂ν = δ′ on ∂M. +(31) +The first line in (31) is due to the definition of δ. Since the differential operator with the boundary +operator is invertible, we apply W s,q-type elliptic estimates (7) as well as the estimates of (29), +∥w0∥L∞( ¯ +M) + ∥∇w0∥L∞( ¯ +M) ⩽ γ +� +∥F − A∥Lq(M,g) + ∥δ′∥W 1,q(M,g) +� +⩽ γ +� +A +2γ (1 + (D + 1)(2 − p)λ) + |δ′| · Volg(M) +1 +q +� +⩽ δ. +(32) +The last inequality is due to the definitions of δ and δ′. By definition of w0, the inequality (32) +implies +∥w − (D + 1)δ∥L∞( ¯ +M) ⩽ δ, ∥∇w∥L∞(¯Ω) ⩽ δ. +It follows that +0 < Dδ ⩽ w ⩽ (D + 2)w on ¯ +M, sup +¯ +M +|∇w| ⩽ δ ⇒ (p − 1)a +p − 2 +· |∇w|2 +w +⩽ δ on ¯ +M. +(33) +With (36), (33), we have +− a∆gw + (2 − p)λw + (p − 1)a +p − 2 +· |∇w|2 +w += F − δ + (p − 1)a +p − 2 +· |∇w|2 +w +⩽(2 − p)S3; +∂w +∂ν − 2hgw = δ′ − 2hgw ⩽ −2 · (cH3) w +1 +2 . +The last inequality holds for small enough constant c > 0, regardless of the sign of H3 since +δ′ − 2hgw < 0 by set-up. By (33) again, we conclude that w > 0 on +¯ +M. By Lemma 3.1, the +positive, smooth function +u = w +1 +2−p + +14 +J. XU +is a super-solution of (24) with S = S3 and H = cH3. Note that u is still a super-solution if we +make c smaller. +For sub-solution, we apply the perturbed eigenvalue problem in Proposition 2.3 again. There +exists a small enough constant β > 0 such that +− a∆gϕ + λϕ = η1,βϕ in M, ∂ϕ +∂ν + +2 +p − 2 (ζ + β) ϕ = 0 on ∂M. +(34) +Any scaling of ϕ solves (35). Set the positive constant ξ ≪ 1 such that +φ := ξϕ ⩽ u on ¯ +M +for the fixed super-solution u defined just above. We shrink ξ further, if necessary, such that +η1,β (ξϕ) ⩽ S3 (ξϕ)p−1 in M, − +2 +p − 2 · β (ξϕ) ⩽ +2 +p − 2 · (cH3) · (ξϕ) +p +2 on ∂M. +(35) +Note that the boundary condition holds for every c, as long as we take ξ small enough. We point +out that the choice of the constant c depends on the construction of the super-solution as well as +the technical condition of the monotone iteration scheme, which only depends on the super-solution +but not the sub-solution, see Equation (19) in [18]. Thus we can choose c first, then determine ξ. +It follows that φ is a sub-solution of (24) with S = S3 and H = cH3. Furthermore, 0 < φ ⩽ u +on ¯ +M. Applying Theorem 2.2, we conclude that there exists a positive function u ∈ C∞( ¯ +M) as +desired. +□ +The general case when η1 < 0 is a straightforward consequence of the result above. +Corollary 3.1. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M ⩾ 3. Let S4, H4 ∈ C∞( ¯ +M) and q > n be a positive integer. Let γ be the constant in the +estimate (29) and λ be some negative constant. Set D = (p−1)a +p−2 . Assume that η1 < 0. If there +exists a function F ∈ C∞( ¯ +M) and a positive constant A > 0, such that +(2 − p)S4 ⩾ F on ∂M, ∥F − A∥Lq(M,g) ⩽ +A +2γ (1 + (D + 1) (2 − p)λ), +(36) +then there exists a small enough constant c > 0 such that (24) admits a positive solution u ∈ C∞( ¯ +M) +with S = S4 and H = cH4. Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S4 +and h˜g = cH4 +���� +∂M +. +Proof. By the result of the Han-Li conjecture [17, Theorem], there exists a conformal metric g1 = +vp−2g such that Rg1 = λ and hg1 = ζ. We then apply Theorem 3.3 for the metric g1, i.e. there +exists ˜g = up−2g1 with R˜g = S4 and h˜g = cH4 with small enough c > 0. The conformal change +˜g = (uv)p−2 g +is the desired metric. +□ +4. Prescribed Scalar and Mean Curvature Functions for Conformal Equivalent +Metrics When η1 < 0 +Inspired by the “Trichotomy Theorem” on closed manifolds, we would like to discuss the pre- +scribing scalar and mean curvature problem on ( ¯ +M, g), n = dim ¯ +M ⩾ 3, but not restricted in a +conformal class [g] only. Instead, we are interested in the conformally equivalent metrics. + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +15 +Definition 4.1. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, we say +that a metric ˜g is conformally equivalent to the metric g if there exists a positive, smooth function +u ∈ C∞( ¯ +M) and a diffeomorphism φ : ¯ +M → ¯ +M such that +φ∗˜g = up−2g. +Within in a conformal class, the prescribing scalar and mean curvature problem for given func- +tions S, H ∈ C∞( ¯ +M) is reduced to the PDE (1). For conformlaly equivalent metrics, the prescribing +scalar and mean curvature problem is reduced to the existence of a positive, smooth solution of the +following PDE +− a∆gu + Rgu = (S ◦ φ) up−1 in M, ∂u +∂ν + +2 +p − 2hgu = +2 +p − 2 · (H ◦ φ) · u +p +2 on ∂M. +(37) +Our next result extends the result of prescribing scalar curvature problem on closed manifolds with +dimensions at least 3 [10, Thm. 3.3] to compact manifolds with non-empty smooth boundaries, +provided that the first eigenvalue η1 of the conformal Laplacian with Robin boundary condition is +negative. The method is essentially due to Kazdan and Warner [8, 10]. +Theorem 4.1. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M ⩾ 3. Let S5 be any smooth function on +¯ +M that is negative somewhere in M. Let H5 ∈ +C∞( ¯ +M) and q > n be a positive integer. If η1 < 0, then there exists a small enough constant c > 0 +and a diffeomorphism φ : ¯ +M → ¯ +M such that (37) admits a positive solution u ∈ C∞( ¯ +M) with S = S5 +and H = cH5. Equivalently, there exists a conformally equivalent metric ˜g = +� +φ−1�∗ � +up−2g +� +such +that R˜g = S5 and h˜g = cH5 +���� +∂M +. +Proof. By Han-Li conjecture [17, Theorem], we may assume that Rg = λ < 0 and hg = ζ > 0 +for some constants λ, ζ. +Fix some constant q > n. +Due to Theorem 3.3, it suffices to find a +diffeomorphism φ : ¯ +M → ¯ +M, a smooth function F ∈ C∞( ¯ +M), a positive constant A > 0 and a small +enough positive constant c such that +(2 − p)S5 ◦ φ ⩾ F on ¯ +M, ∥F − A∥Lq(M,g) ⩽ +A +2γ (1 + (D + 1)(2 − p)λ); +(38) +in addition, sup ¯ +M c (H ◦ φ) is small enough. Here γ is the constant in the estimate (29), the constant +D is defined to be +D = (p − 1)a +p − 2 . +We determine φ, F and A first. If S5 < 0 everywhere on ¯ +M, we just choose φ to be the identity +map and set +F = A = (2 − p) max +¯ +M S5. +It is straightforward to check that (38) holds. +If S5 ⩾ 0 somewhere and changes sign, we choose A first to be any positive constant such that +0 < A < (2 − p) min +¯ +M S5. +(39) +Just note that (2 − p) < 0. We pick interior open submanifolds U, V ⊂ M such that +V ⊂ ¯V ⊂ U ⊂ M ⊂ ¯ +M. +In particular, we require that +Volg(U − V ) ⩽ + + +A +2γ (1 + (D + 1)(2 − p)λ) · +� +(2 − p)∥S3∥L∞( ¯ +M) − A +� + + +q +. +(40) + +16 +J. XU +We select the diffeomorphism φ such that +(2 − p)S3 ◦ φ > A in U. +(41) +We then take the function F to be +F = A in V ; +(2 − p) max +¯ +M S3 ◦ φ ⩽ F ⩽ A in U − V ; +F = (2 − p) max +¯ +M S3 ◦ φ in ¯ +M − U. +(42) +Clearly F ⩽ (2 − p)S3 ◦ φ on ¯ +M by (42). The function F only differs with A in U − V , by (40), it +is immediate to check that the second inequality in (38) holds. +Lastly we choose c so that the condition in Theorem 2.2 holds for the function S3 ◦ φ, i.e. +c sup ¯ +M|H5| is small enough. +The same c applies for the smallness of c sup ¯ +M|H5 ◦ φ| since the +diffeomorphism does not change the extremal values of a function. Therefore the function S3 ◦ φ +and cH5 ◦ φ can be realized as prescribed scalar and mean curvature functions, respectively, for +some metric φ∗˜g = up−2g where u is positive and smooth on ¯ +M. Equivalently, S5 and cH5 can +be realized as prescribed scalar and mean curvature functions, respectively, for some metric ˜g = +� +φ−1�∗ up−2g. +□ +Remark 4.1. The result of Theorem 4.1 indicates that on ( ¯ +M, g) with n = dim ¯ +M ⩾ 3, any +function that is negative somewhere can be realized as a scalar curvature function of some metric +g, meanwhile the mean curvature function of g can be some small enough scaling of any smooth +function, provided that the manifold admits a metric with negative first eigenvalue of the conformal +Laplacian, or equivalently, negative Yamabe invariant [6, §1]. +5. Prescribed Gauss and Geodesic Curvature Functions When χ( ¯ +M) < 0 +In this section, we discuss the prescribing Gauss and geodesic curvatures problem within a +conformal class [g] of compact manifolds ( ¯ +M, g) with non-empty smooth boundary ∂M, provided +that χ( ¯ +M) < 0 and n = dim ¯ +M = 2. This is a 2-dimensional analogy of prescribing scalar and +mean curvatures problem with η1 < 0, provided that the dimension is at least 3. +Let K, σ ∈ C∞( ¯ +M) be given functions. This type of Kazdan-Warner problem is reduced to the +existence of a smooth solution u of the following PDE +− a∆gu + Kg = Ke2u in M, ∂u +∂ν + σg = σeu on ∂M. +(43) +Here Kg and σg are Gaussian and geodesic curvatures of g, respectively. The solvability of this +PDE implies that the metric ˜g = e2ug has Gauss curvature K˜g = K and geodesic curvature σ˜g = σ. +We mainly discuss to cases: +(i). K ⩽ 0 everywhere in ¯ +M, and arbitrary σ, with χ( ¯ +M) < 0; +(ii). K > 0 somewhere in ¯ +M and changes sign, σ is an arbitrary function, with χ( ¯ +M) < 0. +We would like to apply the monotone iteration scheme to solve (43), it is equivalent to construct the +sub- and super-solutions of (43). The key is to construct the super-solution. As in §3, we convert +the super-solution of (43) into another inequality involving derivatives. +Lemma 5.1. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M = 2. Let K, σ ∈ C∞( ¯ +M) be given functions. Then there exists some function u ∈ C∞( ¯ +M) +satisfying +− ∆gu + Kg ⩾ Ke2u in M, ∂u +∂ν + σg ⩾ σeu on ∂M +(44) + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +17 +if and only if there exists some positive function w ∈ C∞( ¯ +M) satisfying +− ∆gw − 2wKg + |∇gw|2 +w +⩽ −2K in M, ∂w +∂ν − 2wσg ⩽ −2σw +1 +2 on ∂M. +(45) +Moreover, the equality in (44) holds if and only if the equality in (45) holds; and the inequality in +(44) is in the reverse direction if and only if the inequality in (45) is in the reverse direction. +Proof. Assume (44) for some function u first. Define +w := e−2u +We observe that +∇gw = −2e−2u∇gu ⇒ ∇gu = −1 +2e2u∇gw, +∆gw = −2e−2u∆gu + 4e−2u|∇gu|2 = −2e−2u∆gu + e2u|∇gw|2. +Thus we have +−∆gw = 2e−2u∆gu − e2u|∇gw|2 ⩽ 2e−2u � +Kg − Ke2u� +− |∇gw|2 +w += 2wKg − 2K − |∇gw|2 +w +⇒ − ∆gw − 2wKg + |∇gw|2 +w +⩽ −2K in M. +For the boundary condition, we have +∂w +∂ν = ∂e−2u +∂ν += −2e−2u ∂u +∂ν ⩽ −2e−2u (−σg + σeu) += 2wσg − 2σw +1 +2 +⇒∂w +∂ν − 2wσg ⩽ −2σw +1 +2 on ∂M. +Therefore (45) holds for w = e−2u > 0 on ¯ +M. It is clear that equality holds when all inequalities +above are equalities. It is also straightforward to see that the inequalities are in the reverse directions +if and only if the inequalities are in the reverse directions in each step above. +For the opposite direction, we assume (45) holds for some positive, smooth function w. Define +u = −1 +2 log w. +We can show that u satisfies (44). +The argument is quite similar to above and we omit the +details. +□ +Due to the uniformization theorem, we may assume Kg = −1 and σg = 0 in (43) from now on, as +our model case up to some pointwise conformal change, provided that χ( ¯ +M) < 0. In 2-dimensional +case, we also have the W s,q-type estimates from Theorem 2.1. We choose q = 3, the estimate in +(7) plus the Sobolev embedding into H¨older space, the inequality in (29) becomes +∥u∥L∞( ¯ +M) + ∥∇u∥L∞( ¯ +M) ⩽ γ +� +∥F1∥L3(M,g) + ∥F2∥W 1,3(M,g) +� +. +(46) +Here F1, F2 and u comes from the PDE (6) with the operators L = −∆g + 2 and B = +∂ +∂ν , so is the +constant γ. Our main result of this section is the following, which covers both Case (i) and Case +(ii) at the beginning of this section. + +18 +J. XU +Theorem 5.1. Let ( ¯ +M, g) be a compact Riemann surface with non-empty smooth boundary ∂M. +Let K1, σ1 ∈ C∞( ¯ +M) be given functions. Let γ be the constant in the estimate (46). Assume that +χ( ¯ +M) < 0. If there exists a function F ∈ C∞( ¯ +M) and a positive constant A > 0, such that +− 2K1 ⩾ F on ∂M, ∥F − A∥L3(M,g) ⩽ A +6γ , +(47) +then there exists a small enough constant c > 0 such that (43) admits a positive solution u ∈ C∞( ¯ +M) +with K = K1 and σ = cσ1. Equivalently, there exists a Yamabe metric ˜g = e2ug such that K˜g = K1 +and σ˜g = cσ1 +���� +∂M +. +Proof. The proof is essentially the same as in Theorem 4.1. By Lemma 43, the construction of the +super-solution is equivalent to the construction of a function w that satisfies (45) for K1, σ1 and +some small enough positive constant c. We set +δ = A +3 , δ′ = − +δ +2γVolg(M) +1 +3 +. +(48) +There is a unique solution for the PDE +−∆gw + 2w = F − δ in M, ∂w +∂ν = δ′ on ∂M. +Define +w0 = w − 2δ, +it follows that w0 satisfies the PDE +− ∆gw0 + 2w0 = F − 3δ = F − A in M, ∂w0 +∂ν = δ′ on ∂M. +(49) +Apply the estimate (46) for w0 in (49), it follows that +∥w0∥L∞( ¯ +M) + ∥∇w0∥L∞( ¯ +M) ⩽ γ +� +∥F − A∥L3(M,g) + ∥δ′∥W 1,3(M,g) +� +⩽ δ. +It follows from the definition of w0 that +0 < δ ⩽ w ⩽ 3δ on ¯ +M, ∥∇w∥L∞( ¯ +M) ⩽ δ. +Therefore we conclude that +−∆gw + 2w + |∇w|2 +w += F − δ + |∇w|2 +w +⩽ F ⩽ −2K1 in M. +In addition, we take c small enough so that +∂w +∂ν = δ′ ⩽ −2cσ1w +1 +2 on ∂M. +This can be done since δ′ < 0. It follows that the function +u+ := −1 +2 log w +is a super-solution of (43) with K = K1 and σ = cσ1. Clearly u+ ∈ C∞( ¯ +M). +We construct a sub-solution now. Consider the PDE +−∆gu0 = 1 +2 in M, ∂u0 +∂ν = C on ∂M. +By standard elliptic PDE theory, see e.g. [14, Prop. 7.7, Ch. 4], the above PDE is solvable by some +smooth function u0 ∈ C∞( ¯ +M) if − +´ +M +1 +2dVolg = +´ +∂M CdSg. We choose the constant C < 0 so that +the compatibility condition just mentioned holds. Clearly +u− := u0 + C1 + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +19 +solves the PDE above also for any constant C1. We just choose C1 to be very negative such that +u− ⩽ u+ on ∂M; +In addition, +−∆gu− − 1 = −1 +2 ⩽ K1e2u− = K1e2u0 · e2C1 in M, +∂u− +∂ν = C ⩽ cσ1eu− = cσ1eu0 · eC1 on ∂M. +These can be done since the constants on the left sides of the inequalities are both negative. Note +that both the super-solution and sub-solution holds for smaller constant c by adjusting the constant +C1 only. +Note that when F(·, u) = K1e2u + 1, G(·, u) = cσ1eu, the condition (22) is independent of the +sub-solution u− as we can see the very similar case in Theorem 2.2 for the Yamabe equation. Thus +we take c small enough so that the hypotheses in Theorem 2.3 holds. It follows that there exists +some smooth function u that solves (43) with K = K1 and σ = cσ1. +□ +We can partially answer the two cases we are interested in. For Case (ii), not every function that +changes sign can be a prescribed scalar curvature function unless it is not too positive too often. We +show that every function that is negative everywhere can be realized as a scalar curvature function, +meanwhile, a small enough scaling of any function can be realized as prescribed mean curvature +function, under pointwise conformal deformation. This is Case (i). +Corollary 5.1. Let ( ¯ +M, g) be a compact Riemann surface with non-empty smooth boundary ∂M. +Let K2, σ2 ∈ C∞( ¯ +M) be given functions. Assume that K2 < 0 everywhere on +¯ +M. If χ( ¯ +M) < 0, +then there exists a small enough constant c, a smooth function u ∈ C∞( ¯ +M) such that u solves (43) +with K = K2 and σ = cσ2. It is equivalent to say that the metric ˜g = e2ug has Gauss curvature +K˜g = K2 and geodesic curvature σ˜g = cσ2. +Proof. We show that the condition (47) holds. Since K2 < 0 everywhere, we just choose +F = A = −2 max +¯ +M K2 ⇒ −2K2 ⩾ F, ∥F − A∥L3(M,g) = 0. +We just need to choose a small enough c such that the hypotheses in Theorem 5.1 and Theorem +2.3 hold. +□ +For Case (ii), we can get a more comprehensive answer by considering the class of conformally +equivalent metrics. +Analogous to §4, we are looking for a metric ˜g = +� +φ−1�∗ e2ug with some +diffeomorphism φ : +¯ +M → +¯ +M and smooth function u ∈ C∞( ¯ +M) such that the scalar and mean +curvatures of ˜g are given functions K, σ ∈ C∞( ¯ +M), respectively. This problem is reduced to the +PDE +− ∆gu + Kg = (K ◦ φ) e2u in M, ∂u +∂ν + σgu = (σ ◦ φ) eu on ∂M. +(50) +Similar to Theorem 4.1 for dimensions at least 3, we introduce the following result for compact +Riemann surfaces. +Corollary 5.2. Let ( ¯ +M, g) be a compact Riemann surface with non-empty smooth boundary ∂M. +Let σ3 ∈ C∞( ¯ +M) be any function and K3 ∈ C∞( ¯ +M) be a function that is negative somewhere in M. +If χ( ¯ +M) < 0, then there exists a small enough constant c, a smooth function u ∈ C∞( ¯ +M) and a +diffeomorphism φ : ¯ +M → ¯ +M such that u solves (50) with K = K3 and σ = cσ3. It is equivalent to +say that the metric ˜g = +� +φ−1�∗ e2ug has Gauss curvature K˜g = K3 and geodesic curvature σ˜g = cσ3. + +20 +J. XU +Proof. The proof is essentially the same as in Theorem 4.1. We determine φ, F, A first so that (47) +holds; then determine the constant c. We may assume that K3 is negative somewhere but not +everywhere since otherwise it is reduced to the result of Corollary 5.1. +We choose A first to be any positive constant such that +0 < A < −2 min +¯ +M K3. +(51) +We pick interior open submanifolds U, V ⊂ M such that +V ⊂ ¯V ⊂ U ⊂ M ⊂ ¯ +M. +In particular, we require that +Volg(U − V ) ⩽ + + +A +6γ · +� +2∥K3∥L∞( ¯ +M) − A +� + + +3 +. +(52) +We select the diffeomorphism φ such that +− 2K3 ◦ φ > A in U. +(53) +We then take the function F to be +F = A in V ; +− 2 max +¯ +M K3 ◦ φ ⩽ F ⩽ A in U − V ; +F = −2 max +¯ +M K3 ◦ φ in ¯ +M − U. +(54) +Clearly F ⩽ −2K3 ◦ φ on ¯ +M by (54). The function F only differs with A in U − V , by (52), it is +immediate to check that the second inequality in (47) holds. +Lastly we choose c so that the condition in Theorem 2.3 holds for the function K3 ◦ φ, i.e. +c sup ¯ +M|σ3| is small enough. +The same c applies for the smallness of c sup ¯ +M|σ3 ◦ φ| since the +diffeomorphism does not change the extremal values of a function. Therefore the function K3 ◦ φ +and cσ3 ◦ φ can be realized as prescribed scalar and mean curvature functions, respectively, for +some metric φ∗˜g = up−2g where u is positive and smooth on ¯ +M. Equivalently, K3 and cσ3 can +be realized as prescribed scalar and mean curvature functions, respectively, for some metric ˜g = +� +φ−1�∗ up−2g. +□ +Remark 5.1. The result of Corollary 5.2, combining Theorem 4.1 indicate that on ( ¯ +M, g) with +n = dim ¯ +M ⩾ 2, any function that is negative somewhere can be realized as a scalar/Gauss +curvature function of some metric g, meanwhile the mean/geodesic curvature function of g can be +some small enough scaling of any smooth function, provided that the manifold admits a metric with +negative first eigenvalue of the conformal Laplacian, or negative Euler characteristics, respectively, +depending on the dimension of the manifold. This improve the result mentioned in Remark 4.1. +6. Prescribed Scalar and Mean Curvature Functions for Conformally Equivalent +Metrics When η1 = 0 +In this section, we discuss the prescribing scalar and mean curvatures problem for metrics con- +formally equivalent to the metric g on compact manifolds ( ¯ +M, g) with non-empty smooth boundary +∂M, provided that η1 = 0 and n = dim ¯ +M ⩾ 3. We gave a comprehensive study for manifolds +with dimensions at least 3 in [18] for pointwise conformal change. Here we consider whether there +exists some smooth function u ∈ C∞( ¯ +M) and some diffeomorphism φ : +¯ +M → +¯ +M such that the +metric ˜g = +� +φ−1�∗ up−2g has scalar curvature S and mean curvature H for some given functions + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +21 +S, H ∈ C∞( ¯ +M). Since the model case for zero first eigenvalue case is Rg = hg = 0, the problem +above is reduced to the existence of the solution of the following PDE +− a∆gu = (S ◦ φ) · up−1 in M, ∂u +∂ν = +2 +p − 2 · (H ◦ φ) · u +p +2 on ∂M. +(55) +Recall the result of prescribing scalar and mean curvature problems for conformal metrics on ( ¯ +M, g). +Theorem 6.1. [18, Thm. 1.4] Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary +∂M, n = dim ¯ +M ⩾ 3. Let S, H ∈ C∞( ¯ +M) be given nonzero functions. Assume that η1 = 0. If the +function S satisfies +S changes sign and +ˆ +M +SdVolg < 0, +then there exists a pointwise conformal metric ˜g ∈ [g] that has scalar curvature R˜g = S and h˜g = cH +for some small enough positive constant c. +The conformally equivalent case follows from the result of Theorem 6.1, we show it below. Note +that the case S = H = 0 is the trivial case. +Theorem 6.2. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M ⩾ 3. Let S6, H6 ∈ C∞( ¯ +M) be given nonzero functions. Assume that η1 = 0. If the function +S satisfies +S6 changes sign, +then there exists a diffeomorphism φ : ¯ +M → ¯ +M and a small enough constant c > 0 such that (55) +has a smooth solution u ∈ C∞( ¯ +M) for φ, S = S6 and H = cH6. It is equivalent to say that the +conformally equivalent metric ˜g = +� +φ−1�∗ up−2g has scalar curvature R˜g = S6 and mean curvature +h˜g = cH6. +Proof. Due to Theorem 6.1, it suffices to show that there exist a diffeomorphism φ : ¯ +M → ¯ +M such +that +ˆ +M +(S6 ◦ φ) dVolg < 0. +Due to the same reason in [9, 8], it is straightforward that such a diffeomorphism does exist since +S6 changes sign. The smallness of c is then determined by S6 ◦ φ, sup ¯ +M|H6| as well as the choice +of sub- and super-solutions in the proofs of [18, Thm. 5.1, Cor. 5.1, Cor. 5.2]. +Note that any +diffeomorphism φ will not change the supremum of |H6| on ¯ +M. +□ +Remark 6.1. The result of Theorem 6.2 indicates that on ( ¯ +M, g) with n = dim ¯ +M ⩾ 3, any +function that changes sign or identically zero can be realized as a scalar curvature function of some +metric g, meanwhile the mean curvature function of g can be some small enough scaling of any +smooth function or zero function, respectively, provided that the manifold admits a metric with +zero first eigenvalue of the conformal Laplacian, or equivalently, zero Yamabe invariant [6, §1]. +7. Prescribed Scalar and Mean Curvature Functions When η1 > 0 +In this section, we seek for a positive, smooth solution of the following PDE +− a∆gu + Rgu = Sup−1 in M, ∂u +∂ν + +2 +p − 2hgu = +2 +p − 2Hu +p +2 on ∂M. +(56) +on compact manifolds ( ¯ +M, g) with non-empty smooth boundary ∂M, n = dim ¯ +M ⩾ 3, for given +functions S, H ∈ C∞( ¯ +M), provided that η1 > 0. +As we have shown in [16], [17] and [19], we +need to use local analysis, gluing a super-solution, and then apply monotone iteration scheme here. +According to the “Trichotomy Theorem” in [20], we expect few restrictions on prescribed scalar +and mean curvature functions. We will discuss the following case: + +22 +J. XU +(i). S > 0 somewhere in M, and H > 0 somewhere on ∂M, with η1 > 0; +(ii). S > 0 somewhere in M, and H ⩽ 0 everywhere on ∂M but H ̸≡ 0, with η1 > 0. +Note that we have discussed the case S > 0 somewhere and H ≡ 0 in [19]. Currently we do not see +how to apply our method to the case mentioned in [7], +− ∆eu = 0 in Bn, ∂u +∂ν + +2 +p − 2hgu = +2 +p − 2Hu +p +2 on ∂Bn, u > 0 +(57) +for some given function H. Escobar showed that there is an obstruction for the choice of H +ˆ +∂Bn X · ∇gHdS = 0. +Here X is some conformal Killing field on ∂Bn. With standard Euclidean metric in Bn and the +induced metric on ∂Bn, the first eigenvalue of conformal Laplacian with Robin condition is positive. +However, since the right side is zero, we are not able to get a nontrivial local solution of the Dirichlet +problem +−∆eu = 0 in Ω, u = 0 on ∂Ω. +Therefore we may need some alternative method to resolve this issue. +However, we can get some interesting results provided that S ̸≡ 0. According to the detailed +analysis in [19, §5], we know that there will be obstructions for the choices of prescribed scalar +curvature functions on Sn/Γ for some Kleinian group Γ. The map Sn → Sn/Γ must be a covering +map since otherwise Sn/Γ cannnot be a manifold. It follows that Sn/Γ has empty boundary, which +follows that there will be no obstruction for the choice of prescribed scalar curvature functions on +( ¯ +M, g). +The first result concerns the Case (i) above: +Theorem 7.1. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M ⩾ 3. Let S7 > 0 somewhere be any smooth function on +¯ +M. Let H7 ∈ C∞( ¯ +M) such that +H7 > 0 somewhere on ∂M. If η1 > 0, then there exists a small enough constant c > 0 such that +(56) admits a positive solution u ∈ C∞( ¯ +M) with S = S7 and H = cH7. Equivalently, there exists a +Yamabe metric ˜g = up−2g such that R˜g = S7 and h˜g = cH7 +���� +∂M +. +Proof. Without loss of generality, we may assume that Sg > 0 and hg = h > 0 with positive +constant h, by Theorem 2.1. According to Proposition 2.3, we fix some β < 0 small enough so that +η1,β > 0 and satisfies +− a∆gϕ + Rgϕ = η1,βϕ in M, ∂ϕ +∂ν + +2 +p − 2hgϕ = 0 on ∂M. +(58) +Here ϕ > 0 on ¯ +M. Any scaling of ϕ solves (58). Denote φ = δϕ for some δ > 0. We choose δ > 0 +small enough so that +η1,β inf +¯ +M ϕ ⩾ δp−2 sup +¯ +M +S7 sup +¯ +M +ϕp−1. +It follows that +−a∆gφ + Rgφ ⩾ S7φp−1 in M. +Fix this δ. We then choose c > 0 small enough so that +βφ ⩾ (cH7) φ +p +2 on ∂M. +It follows that +∂φ +∂ν + +2 +p − 2hgφ ⩾ +2 +p − 2 (cH7) φ +p +2 on ∂M. +(59) +Note that (59) still holds for any smaller c. For the sub-solution, we apply Proposition 2.1 or +Proposition 2.2, depending on the vanishing of the Weyl tensor in the interior M, to construct local + +TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES +23 +solution u0 of the Yamabe equation with Dirichlet boundary condition on some domain Ω. Apply +Lemma 3.2 in [19], we can construct a local super-solution f of the Yamabe equation in Ω such +that f = φ near ∂Ω. We then define +u− = +� +u0 in Ω +0 in M\Ω +u+ := +� +f in Ω +φ in M\Ω . +Since u− ≡ 0 on ∂M, it follows from the same argument in Lemma 3.1 in [19] that u− is a sub- +solution of the (56) with S = S7 and H = cH7 for any constant c. According to the construction in +Lemma 3.2 of [19], we conclude that 0 ⩽ u− ⩽ u+, u− ̸≡ 0. In addition, u− ∈ H1(M, g) ∩ C0( ¯ +M), +and u+ ∈ C∞( ¯ +M). According to (59), we have seen that u+ is a super-solution of the (56) with +S = S7 and H = cH7 for small enough c. Shrinking c, if necessary, so that the hypotheses of +smallness of c sup ¯ +M|H7| holds. A direct application of Theorem 2.2 indicates the existence of a +positive solution u ∈ C∞( ¯ +M) with S = S7 and H = cH7. +□ +The proof of the Case (ii) is very similar as in Theorem 7.1. +Theorem 7.2. Let ( ¯ +M, g) be a compact manifold with non-empty smooth boundary ∂M, n = +dim ¯ +M ⩾ 3. Let S8 > 0 somewhere be any smooth function on +¯ +M. Let H8 ∈ C∞( ¯ +M) such that +H8 ⩽ 0 everywhere on ∂M. If η1 > 0, then there exists a small enough constant c > 0 such that +(56) admits a positive solution u ∈ C∞( ¯ +M) with S = S8 and H = cH8. Equivalently, there exists a +Yamabe metric ˜g = up−2g such that R˜g = S8 and h˜g = cH8 +���� +∂M +. +Proof. Everything is exactly the same as in Theorem 7.1, except at (59), there is no restriction for +the choice of the constant c. However, c should be small enough so that the hypotheses in Theorem +2.2 holds. +□ +Remark 7.1. The result of Theorem 7.1 and Theorem 7.2 indicate that on ( ¯ +M, g) with n = +dim ¯ +M ⩾ 3, any function that is positive somewhere can be realized as a scalar curvature function +of some metric g, meanwhile the mean curvature function of g can be some small enough scaling +of any smooth function, provided that the manifold admits a metric with positive first eigenvalue +of the conformal Laplacian, or equivalently, positive Yamabe invariant [6, §1]. +References +[1] T. Aubin. Nonlinear Analysis on Manifolds. Monge-Amp´ere Equations. Grundlehren der mathematischen Wis- +senschaften. Springer, Berlin, Heidelberg, New York, 1982. +[2] S. Brendle and F. Marques. Recent progress on the Yamabe problem. arXiv:1040.4960. +[3] H. Brezis and F. Merle. Uniform esitmates and blow-up behavior for solutions of −δu = v(x)eu in two dimensions. +Commun. Partial. Differ., 16(8-9):1223–1253, 1991. +[4] A. Chang and P. Yang. Prescribing Gaussian curvature on S2. Acta Math., 159:215–259, 1987. +[5] S. Cruz-Bl´azquez, A. Malchiodi, and D. Ruiz. Conformal metrics with prscribed scalar and mean curvature. +arXiv:2105.04185. +[6] J. Escobar. The Yamabe problem on manifolds with boundary. J. Differential Geom., 35:21–84, 1992. +[7] J. Escobar. Conformal metrices with prescribed mean curvature on the boundary. Calc. Var. Partial Differential +Equations, 4:559–592, 1996. +[8] J. Kazdan and F. Warner. Curvature functions for compact 2−manifolds. Ann. of Math., 99:14–47, 1974. +[9] J. Kazdan and F. Warner. Existence and conformal deformations of metrices with prescribed Gaussian and scalar +curvatures. Ann. of Math. (2), 101(2):317–331, 1975. +[10] J. Kazdan and F. Warner. Scalar curvature and conformal deformation of Riemannian structure. J. Differential +Geom., 10:113–134, 1975. + +24 +J. XU +[11] A. Malchiodi and M. Mayer. Prescribing Morse scalar curvatures: Pinching and Morse theory. Commun. Pure +Appl. Math, 2021. +[12] S. Rosenberg and J. Xu. Solving the Yamabe problem by an iterative method on a small Riemannian domain. +arXiv:2110.14543. +[13] M. Struwe. A flow approach to Nirenberg’s problem. Duke Math. J., 128(19-64), 2005. +[14] M. Taylor. Partial Differential Equations I. Springer-Verlag, New York, New York, 2011. +[15] M. Taylor. Partial Differential Equations III. Springer-Verlag, New York, New York, 2011. +[16] J. Xu. The boundary Yamabe problem, I: Minimal boundaray case. arXiv:2111:03219. +[17] J. Xu. The boundary Yamabe problem, II: General constant mean curvature case. arXiv:2112.05674. +[18] J. Xu. The conformal Laplacian and the Kazdan-Warner problem: Zero first eigenvalue case. arXiv:2211.15024. +[19] J. Xu. Prescribed scalar curvature on compact manifolds under conformal deformation. arXiv:2205.15453. +[20] J. Xu. Prescribed scalar curvature problem under conformal deformation of a Riemannian metric with Dirichlet +boundary condition. arXiv:2208.11318. +[21] J. Xu. Solving the Yamabe-type equations on closed manifolds by iteration schemes. arXiv: 2110.15436. +Department of Mathematics and Statistics, Boston University, Boston, MA, U.S.A. +Email address: xujie@bu.edu +Institute for Theoretical Sciences, Westlake University, Hangzhou, Zhejiang Province, China +Email address: xujie67@westlake.edu.cn + diff --git a/-9AzT4oBgHgl3EQfFvoN/content/tmp_files/load_file.txt b/-9AzT4oBgHgl3EQfFvoN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0813712fadaca7e40d42bf27882ba42fffbd98e --- /dev/null +++ b/-9AzT4oBgHgl3EQfFvoN/content/tmp_files/load_file.txt @@ -0,0 +1,945 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf,len=944 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='01014v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='DG] 3 Jan 2023 TRICHOTOMY THEOREM FOR PRESCRIBED SCALAR AND MEAN CURVATURES ON COMPACT MANIFOLDS WITH BOUNDARIES JIE XU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with dimensions at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The results are clas- sified by the sign of the first eigenvalue of the conformal Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This leads to a “Trichotomy Theorem” in terms of both scalar and mean curvature functions, which is a full extension of the “Trichotomy Theorem” given by Kazdan and Warner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We also discuss prescribing Gauss and geo- desic curvature problems on compact Riemann surfaces with boundary for metrics either pointwise conformal or conformally equivalent to the original metric, provided that the Euler characteristic is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The key step is a general version of monotone iteration scheme which handle the zeroth order nonlinear term on the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Introduction In this article, we give a “Trichotomy Theorem” on compact manifolds ( ¯ M, g) with non-empty smooth boundaries ∂M, n := dim M ⩾ 3, involving both the scalar and mean curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This is a full generalization of the “Trichotomy Theorem” on closed manifolds, given by Kazdan and Warner [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Precisely speaking, this “Trichotomy Theorem” concerns whether the given functions S, H can be realized as scalar and mean curvatures, respectively, of a metric ˜g either within a conformal class [g] or conformally equivalent to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Throughout this article, we assume that ¯ M is connected since otherwise we can easily apply arguments below equally to each connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It is well-known that this problem is reduced to the existence of the positive solutions of the nonlinear second order elliptic PDE − a∆gu + Rgu = (S ◦ φ) up−1 in M, ∂u ∂ν + 2 p − 2hgu = 2 p − 2 (H ◦ φ) u p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (1) Here Rg is the scalar curvature of the metric g, hg is the mean curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' φ : ¯ M → ¯ M is some diffeomorphism on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' When φ = Id, the PDE (1) is for prescribing functions S, H within a conformal class [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The constants a, p are defined as a = 4(n − 1) n − 2 , p = 2n n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ∆g is the Laplace-Beltrami operator and ν is the unique outward unit normal vector field along ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The functions S : C∞( ¯ M) → R, and H : C∞(∂M) → R are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We denote η1 to be the first eigenvalue of the conformal Laplacian □g := −a∆g + Rg with associated eigenfunction ϕ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ϕ is a positive, smooth function that solves the following PDE: −a∆gϕ + Rgϕ = η1ϕ in M, ∂ϕ ∂ν + 2 p − 2hgϕ = 0 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' When the dimension of the manifold n = 2, we also discuss the pair of functions K, σ that can be realized as Gaussian and geodesic curvatures, respectively, either for a pointwise conformal metric 1 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU or a conformally equivalent metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The two dimensional case is reduced to the existence of the solutions of the following elliptic PDE − a∆gu + Kg = (K ◦ φ) e2u in M, ∂u ∂ν + σg = (σ ◦ φ) eu on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (2) Here Kg and σg are Gaussian and geodesic curvatures of g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The functions K : C∞( ¯ M) → R and σ : C∞(∂M) → R are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Again when the diffeomorphism φ : ¯ M → ¯ M is the identity map, K, σ are prescribing Gauss and geodesic curvatures for some metric within the conformal class [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The main results of this article are given as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a connected, compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S, H ∈ C∞( ¯ M) be given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If η1 < 0, then any function S < 0 somewhere in M can be realized as a scalar curvature function of some metric conformally equivalent to g, with mean curvature cH for some small enough constant c > 0 and any function H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If η1 < 0, then any function S < 0 that changes sign in M can be realized as a scalar curvature function of some metric conformally equivalent to g, with mean curvature cH for some small enough constant c > 0 and any function H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If η1 < 0, then any function S > 0 somewhere in M can be realized as a scalar curvature function of some metric pointwise conformal to g, with mean curvature cH for some small enough constant c > 0 and any function H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Case (i) is given in §3 and §4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' when S < 0 everywhere on ¯ M, we can improve the result within a pointwise conformal class [g] in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Case (ii) is given in §6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' when S satisfies ´ M SdVolg < 0 in addition, we can improve the result within a pointwise conformal class [g], see [18, Thm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' and Case (iii) is given in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The significance of this is that we can choose arbitrary function with small enough sup-norm as our mean curvature function, provided that the scalar curvature function is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Based on our best understanding, known results in this topic are mainly for the non-positive first eigenvalue cases or non-positive Euler characteristic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In [5], Cruz-Bl´azquez, Malchiodi and Ruiz discussed prescribing negative scalar functions and mean curvature functions with arbitrary signs by variational method, for compact manifolds with dimensions at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Some of our results overlap their results, but with a different method and different hypotheses on prescribed functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' However, our results are classified by the sign of the first eigenvalue of the conformal Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For zero first eigenvalue case or zero Euler characteristic case, we follow the results of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We point out that Brezis and Merle discussed the PDE −∆eu = V eu on Ω ⊂ R2 with Dirichlet boundaray condition in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Other results for the local Yamabe equation with Dirichlet condition in higher dimensions could be found in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For more discussions with respect to (2) in 2-dimensional case, we refer to [15, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 13, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' When the first eigenvalue of conformal Laplacian is positive, a lot of non-existence results are given, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' [2] and [11], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We also give results on compact Riemann surfaces with boundary, provided that χ( ¯ M) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact Riemann surface with non-empty smooth boundary ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let K, σ ∈ C∞( ¯ M) be given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If K < 0 everywhere on ¯ M, then there exists a metric pointwise conformal to g with Gauss curvature K and geodesic curvature cσ for some small enough constant c > 0 and arbitrary function σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If K < 0 somewhere on ¯ M, then there exists a metric conformally equivalent to g with Gauss curvature K and geodesic curvature cσ for some small enough constant c > 0 and arbitrary function σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 3 Both results above are given in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Other than the related works on compact Riemann surfaces with boundary we introduced above, many work has been done on closed Riemann surface, a comprehensive study was given by Kazdan and Warner [8], including results for all signs of χ( ¯ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For Nirenberg problem, we refer to Chang and Yang [4] and Struwe [13], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='. The most common method in analyzing this type of Kazdan-Warner problem is by calculus of variations since we can consider the PDE as Euler-Lagrange equation with respect to some functional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' recently Morse theory is also involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' However, a new method, inspired by Kazdan and Warner [10], has been developed recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This new method applies monotone iteration scheme, a local version of calculus of variation to classify the existence results by sign of the first eigenvalue η1 of conformal Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This method has been applied to completely solve the Escobar problem [16], the Han-Li conjecture [17], the prescribed scalar curvature problem on compact manifolds [19], a trichotomy theorem in terms of prescribed scalar curvature with Dirichlet condition at boundary [20], and a comprehensive study of zero first eigenvalue case on compact manifolds, possibly with boundary, with dimensions at least 3 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In this article, we apply a variation of the combination of monotone iteration scheme and local analysis to show the results of prescribed scalar and mean curvatures for the cases η1 > 0 and η1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We also develop a general monotone iteration scheme, which can handle nonlinear terms both in the PDE and on the boundary condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' this new monotone iteration scheme, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3, allows us to work on 2-dimensional case without using the calculus of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This systematic procedure is powerful, but unfortunately this direct method cannot be used to the classical manifold, the unit ball with spherical boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We will explain why this direct method does not work in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that Escobar [7] has proved a nontrivial Kazdan-Warner type obstruction of prescribed mean curvature functions for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This paper is organized as follows: In §2, we introduce the essential definitions and results that will be used throughout this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We assume the backgrounds of standard elliptic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We also introduced two versions of mono- tone iteration schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 is for the PDE (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 is more general, works for all second order semi-linear elliptic PDE with Robin boundary conditions, possibly with zeroth order nonlinear term on the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 works well for the PDE like (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In §3, we give results for prescribing scalar curvature function S and mean curvature function H within a conformal class [g] on ( ¯ M, g), n = dim ¯ M ⩾ 3, provided that η1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' When S < 0 everywhere and arbitrary H, the results are given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' When S < 0 somewhere and arbitrary H, the results are given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 with some restriction on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The monotone iteration scheme plays a central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In §4, we give results for prescribing scalar curvature function S and mean curvature function H for some metric conformally equivalent to g on ( ¯ M, g), n = dim ¯ M ⩾ 3, provided that η1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We conclude in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 that any S that is negative somewhere can be realized as a scalar curvature function of some metric conformally equivalent to g, with the mean curvature cH for small enough constant c > 0 and arbitrary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In §5, we discuss prescribing Gauss and geodesic curvature functions K, σ on compact Riemann surfaces with boundary for metrics conformally equivalent to the original metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We show in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 that any function K that is negative somewhere and satisfies some analytic condition can be realized as Gaussian curvature function for some metric conformally equivalent to g, the metric also has geodesic curvature cσ for some small enough constant c > 0 and arbitrary σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The result in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 says that when K < 0 everywhere on ¯ M, the metric can be chosen within a conformal class [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In §6, we give results for prescribing scalar function S and mean curvature function H for some metric conformally equivalent to g on ( ¯ M, g), n = dim ¯ M ⩾ 3, provided that η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We show that any function S that changes sign can be realized as a scalar curvature function some metric 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU conformally equivalent to g, with the mean curvature cH for small enough constant c > 0 and arbitrary H in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Obviously there is a trivial case S ≡ H ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In §7, we consider the prescribing scalar and mean curvature problem for η1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The results in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 and Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 are for the case S > 0 somewhere and arbitrary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We also explain why our method cannot work on closed Euclidean ball with some nontrivial mean curvature on the boundary Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The Preliminaries and The Monotone Iteration Scheme In this section, we first introduce the necessary definitions and essential results we need for the later sections, then introduce a general version of the monotone iteration scheme given in [18], other than many variations we have used in [16, 17, 19, 20, 21], with respect to the following Yamabe equation with Robin boundary condition − a∆gu + Rgu = Sup−1 in M, ∂u ∂ν + 2 p − 2hgu = 2 p − 2Hu p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (3) for given functions S, H ∈ C∞( ¯ M), and n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Lastly we introduce a W s,q-type regularity for elliptic PDE with Robin boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' First of all, we give definitions of Sobolev spaces, a local version and a global version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let Ω be a connected, bounded, open subset of Rn with smooth boundary ∂Ω equipped with some Riemannian metric g that can be extended smoothly to ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We call (Ω, g) a Riemannian domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Throughout this article, we denote the space of smooth functions with compact support by C∞ c , smooth functions by C∞, and continuous functions by C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let (Ω, g) be a Riemannian domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let (M, g) be a closed Riemannian n- manifold, and ( ¯ M, g) be a compact Riemannian n-manifold with non-empty smooth boundary, with volume density dVolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let u be a real valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ⟨v, w⟩g and |v|g = ⟨v, v⟩1/2 g denote the inner product and norm with respect to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (i) For 1 ⩽ p < ∞, we define the Lebesgue spaces on Ω and ¯ M to be Lp(Ω) is the completion of � u ∈ C∞ c (Ω) : ∥u∥p p := ˆ Ω |u|pdx < ∞ � , Lp(Ω, g) is the completion of � u ∈ C∞ c (Ω) : ∥u∥p p,g := ˆ Ω |u|p dVolg < ∞ � , Lp(M, g) is the completion of � u ∈ C∞(M) : ∥u∥p p,g := ˆ M |u|p dVolg < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (ii) For ∇u the Levi-Civita connection of g, and for u ∈ C∞(Ω) or u ∈ C∞( ¯ M), |∇ku|2 g := (∇α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ∇αku)(∇α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ∇αku).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (4) In particular, |∇0u|2 g = |u|2 and |∇1u|2 g = |∇u|2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 5 (iii) For s ∈ N, 1 ⩽ p < ∞, we define the (s, p)-type Sobolev spaces on Ω and ¯ M to be W s,p(Ω) = \uf8f1 \uf8f2 \uf8f3u ∈ Lp(Ω) : ∥u∥p W s,p(Ω) := ˆ Ω s � j=0 ��Dju ��p dx < ∞ \uf8fc \uf8fd \uf8fe , (5) W s,p(Ω, g) = \uf8f1 \uf8f2 \uf8f3u ∈ Lp(Ω, g) : ∥u∥p W s,p(Ω,g) = s � j=0 ˆ Ω ��∇ju ��p g dVolg < ∞ \uf8fc \uf8fd \uf8fe , W s,p(M, g) = \uf8f1 \uf8f2 \uf8f3u ∈ Lp(M, g) : ∥u∥p W s,p(M,g) = s � j=0 ˆ M ��∇ju ��p g dVolg < ∞ \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Here |Dju|p := � |α|=j|∂αu|p in the weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Similarly, W s,p 0 (Ω) is the completion of C∞ c (Ω) with respect to the W s,p-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In particular, Hs(Ω) := W s,2(Ω) and Hs(Ω, g) := W s,2(Ω, g), Hs(M, g) := W s,2(M, g) are the usual Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We similarly define Hs 0(Ω), Hs 0(Ω, g) and Hs 0(M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (iv) On closed manifolds (M, g), we say that a function u ∈ Hs(M, g) if u ∈ L2(M, g) , and for any coordinate chart U ⊂ M, any ψ ∈ C∞ c (U), the function ψu ∈ Hs(U, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We assume the background of the standard elliptic theory, including the solvability of standard linear elliptic PDEs, elliptic regularity of Hs-type, trace theorem, Sobolev embedding, Schauder estimates, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We introduce a W s,q-type elliptic regularity for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' [17, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2] Let ( ¯ M, g) be a compact manifold with smooth boundary ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ν be the unit outward normal vector along ∂M and q > n = dim ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let L : C∞( ¯ M) → C∞( ¯ M) be a uniform second order elliptic operator on M with smooth coefficients up to ∂M and can be extended to L : W 2,q(M, g) → Lq(M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let f ∈ Lq(M, g), ˜f ∈ W 1,q(M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let u ∈ H1(M, g) be a weak solution of the following boundary value problem Lu = f in M, Bu = ∂u ∂ν + c(x)u = ˜f on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (6) Here c ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume also that Ker(L) = {0} associated with the homogeneous Robin boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If, in addition, u ∈ Lq(M, g), then u ∈ W 2,q(M, g) with the following estimates ∥u∥W 2,q(M,g) ⩽ γ′ � ∥Lu∥Lq(M,g) + ∥Bu∥W 1,q(M,g) � (7) Here γ′ depends on L, q, c and the manifold ( ¯ M, g) and is independent of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We then introduce the first eigenvalue of conformal Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that a = 4(n−1) n−2 and p = 2n n−2, hence it only makes sense when n ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We denote η1 be the first eigenvalue of conformal Laplacian with its corresponding eigenfunction ϕ > 0 if and only if the following PDE holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' − a∆gϕ + Rgϕ = η1ϕ in M, ∂ϕ ∂ν + 2 p − 2hgϕ = 0 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (8) We now introduce a variation of the monotone iteration scheme we used in [16], [17] and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In particular, we do require hg = h > 0 to be some positive constant on ∂M, this can be done due to the proof of the Han-Li conjecture in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We will also use other versions of monotone iteration schemes introduced in eariler work [16, 17, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='4] Let ( ¯ M, g) be a compact manifold with smooth boundary ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ν be the unit outward normal vector along ∂M and q > dim ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S ∈ C∞( ¯ M) and H ∈ C∞( ¯ M) 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU be given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let the mean curvature hg = h > 0 be some positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In addition, we assume that sup ¯ M|H| is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Suppose that there exist u− ∈ C0( ¯ M) ∩ H1(M, g) and u+ ∈ W 2,q(M, g) ∩ C0( ¯ M), 0 ⩽ u− ⩽ u+, u− ̸≡ 0 on ¯ M, some constants θ1 ⩽ 0, θ2 ⩾ 0 such that −a∆gu− + Rgu− − Sup−1 − ⩽ 0 in M, ∂u− ∂ν + 2 p − 2hgu− ⩽ θ1u− ⩽ 2 p − 2Hu p 2 − on ∂M −a∆gu+ + Rgu+ − Sup−1 + ⩾ 0 in M, ∂u+ ∂ν + 2 p − 2hgu+ ⩾ θ2u+ ⩾ 2 p − 2Hu p 2 + on ∂M (9) holds weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In particular, θ1 can be zero if H ⩾ 0 on ∂M, and θ1 must be negative if H < 0 somewhere on ∂M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' similarly, θ2 can be zero if H ⩽ 0 on ∂M, and θ2 must be positive if H > 0 somewhere on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Then there exists a real, positive solution u ∈ C∞(M) ∩ C1,α( ¯ M) of □gu = −a∆gu + Rgu = Sup−1 in M, Bgu = ∂u ∂ν + 2 p − 2hgu = 2 p − 2Hu p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (10) The following two results are necessary, which shows the existence of the solution of some local Yamabe-type problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' When the manifold is not locally conformally flat, we need Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2] Let (Ω, g) be a Riemannian domain in Rn, n ⩾ 3, not locally conformally flat, with C∞ boundary, with Volg(Ω) and the Euclidean diameter of Ω sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let f ∈ Ω′ ⊃ Ω be a positive, smooth function in some open region Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In addition, we assume that the first eigenvalue of Laplace-Beltrami operator −∆g on Ω with Dirichlet condition satisfies λ1 → ∞ as Ω shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume Rg < 0 within the small enough closed domain ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Then the Dirichlet problem − a∆gu + Rgu = fup−1 in Ω, u ≡ 0 on ∂Ω (11) has a real, positive, smooth solution u ∈ C∞(Ω) ∩ H1 0(Ω, g) ∩ C0(¯Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The size of Ω is depending on the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' When the manifold is locally conformally flat, we give the local solution of (11) provided that Ω is not topologically trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' [19, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='5] Let (Ω, g) be a Riemannian domain in Rn, n ⩾ 3, with C∞ boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let the metric g be locally conformally flat on some open subset Ω′ ⊃ ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For any point ρ ∈ Ω and any positive constant ǫ, denote the region Ωǫ to be Ωǫ = {x ∈ Ω||x − ρ| > ǫ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume that Q ∈ C2(¯Ω), minx∈¯Ω Q(x) > 0 and ∇Q(ρ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Then there exists some ǫ0 such that for every ǫ ∈ (0, ǫ0) the Dirichlet problem − a∆gu + Rgu = Qup−1 in Ωǫ, u = 0 on ∂Ωǫ (12) has a real, positive, smooth solution u ∈ C∞(Ωǫ) ∩ H1 0(Ωǫ, g) ∩ C0( ¯Ωǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It is straightforward to see that under conformal change ˜g = φp−2g, we have ˜g = φp−2g ⇒ −a∆˜g + R˜g = φ− n+2 n−2 (−a∆g + Rg) φ ⇔ □˜g = φ1−p□gφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (13) We call (13) the conformal invariance of the conformal Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 and (13) that if the manifold ( ¯ M, g) is locally conformally flat in the interior, the equation (12) is equivalent to − a∆geu = Qup−1 in Ωǫ, u = 0 on ∂Ωǫ (14) which admits a positive solution u ∈ C∞(Ωǫ) ∩ H1 0(Ωǫ, g) ∩ C0( ¯Ωǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' As a prerequisite, we also need a result in terms of the perturbation of negative first eigenvalue of conformal Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact Riemannian manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let β > 0 be a small enough constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If η′ 1 < 0, then the quantity η′ 1,β = inf u̸=0 a ´ M|∇gu|2dVolg + ´ M Rgu2dVolg + 2a p−2 ´ ∂M(hg + β)u2dS ´ M u2dVolg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In particular, η′ 1,β satisfies − a∆gϕ + Rgϕ = η′ 1,βϕ in M, ∂ϕ ∂ν + 2 p − 2(hg + β)ϕ = 0 on ∂M (15) with some positive function ϕ ∈ C∞( ¯ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Since η′ 1 < 0, the normalized first eigenfunction ϕ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ´ M ϕ2 1dVolg = 1, satisfies η′ 1 = a ˆ M |∇gϕ1|2dVolg + ˆ M Rgϕ2 1dVolg + 2a p − 2 ˆ ∂M hgϕ2 1dS By characterization of η′ 1,β, we have η′ 1,β ⩽ a ˆ M |∇gϕ1|2dVolg + ˆ M Rgϕ2 1dVolg + 2a p − 2 ˆ ∂M (hg + β)ϕ2 1dS = η′ 1 + β ˆ ∂M ϕ2 1dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Since ϕ1 is fixed, it follows that η′ 1,β < 0 if β > 0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ When n = 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' M or ¯ M is a compact Riemann surface (possibly with boundary), all tools above are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We thus need a new version of the monotone iteration scheme for compact Riemann surfaces with non-empty smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We point out that the monotone iteration scheme below works for all compact manifolds with non-empty boundary, with dimensions at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim M ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let q > n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let F(·, ·), G(·, ·) : ¯ M × R → R be smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ν be the unit outward normal vector along ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let σ be some nonnegative, small enough constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If (i) there exists two functions u+ ∈ C∞( ¯ M) and u− ∈ C0( ¯ M) ∩ H1(M, g) such that −∆gu+ ⩾ F(·, u) in M, ∂u ∂ν + σu ⩾ G(·, u+) on ∂M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' −∆gu− ⩽ F(·, u) in M, ∂u ∂ν + σu ⩽ G(·, u−) on ∂M, (16) where the sub-solution may hold in the weak sense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' and (ii) in addition, sup ¯ M|G(·, u+)|, sup ¯ M|∇G(·, u+)| are small enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (iii) furthermore, u+ ⩾ u− pointwise on ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' then there exists a smooth function u ∈ C∞( ¯ M) with u− ⩽ u ⩽ u+ such that − ∆gu = F(·, u) in M, ∂u ∂ν + σu = G(·, u) on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (17) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 is essentially the same as the proof of [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='4], except some minor change, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' here we use general smooth functions F and G but not specific Yamabe equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We therefore will give a relatively concise proof for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ¯ M is compact, so extremal values of continuous functions u+, u− can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Choose positive constant A and nonnegative constant B such that A ⩾ −∂F ∂u (x, u(x)), ∀x ∈ ¯ M, u(x) ∈ [min ¯ M u−, max ¯ M u+];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' B ⩾ σ − ∂G ∂u (x, u(x)), ∀x ∈ ¯ M, u(x) ∈ [min ¯ M u−, max ¯ M u+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (18) Denote u0 = u+ ∈ C∞( ¯ M), and consider the iteration scheme −∆guk + Auk = Auk−1 + F(·, uk−1) in M, k ∈ N, ∂uk ∂ν + Buk = Buk−1 − σuk−1 + G(·, uk−1) on ∂M, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (19) Since A > 0, B ⩾ 0, the operator � −∆g + A, ∂ ∂ν + B � is invertible due to the standard argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Clearly when k = 1, the first iteration step in (19) gives a unique smooth solution u1 ∈ C∞( ¯ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The regularity argument is also standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We show that u− ⩽ u ⩽ u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For u− ⩽ u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' we have to use the sub-solution in the weak sense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' since u0 = u+ ⩾ u−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' we pair (19) for k = 1 with arbitrary non-negative function v ∈ C∞( ¯ M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' and subtract this with the sub-solution (adding Au− and Bu− on both sides of the PDE and boundary conditions respectively) in the weak sense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' we have ˆ M (A (u0 − u−) + F (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' u0) − F (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' u−)) vdVolg ⩽ ˆ M (−∆g (u1 − u−) + A (u1 − u−)) vdVolg ⩽ ˆ ∂M B (u1 − u−) vdS − ˆ ∂M (B (u0 − u−) − σ (u0 − u−) + G (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' u0) − G (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' u−)) vdS + ˆ M A (u1 − u−) vdVolg + ˆ M ∇g (u1 − u−) · ∇gvdVolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Taking v = w := max (u− − u1, 0), and applying the mean value theorem for F, G, due to the definitions of A, B in (18), we observe that ˆ M |∇gw|2 + ˆ ∂M Bw2 + ˆ M Aw2 ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows that w = 0, therefore u− ⩽ u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By a very similar argument in terms of the subtraction between (19) and the super-solution, we conclude that u+ ⩾ u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Inductively, we may assume the existence of the solutions u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' , uk with u− ⩽ uk ⩽ uk−1 ⩽ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ⩽ u1 ⩽ u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By the same argument in the first iteration step, we conclude the existence of uk+1 ∈ C∞( ¯ M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' in addition, uk+1 satisfies u− ⩽ uk+1 ⩽ uk ⩽ uk−1 ⩽ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ⩽ u1 ⩽ u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Therefore we show the existence of the sequence of solutions of (19) with the monotonicity u− ⩽ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ⩽ uk+1 ⩽ uk ⩽ uk−1 ⩽ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ⩽ u0, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (20) We now show the uniform boundedness of ∥uk∥C1,α( ¯ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Since q > n, showing the uniform boundedness of ∥uk∥C1,α( ¯ M) is equivalent to show the uniform boundedness of ∥uk∥W 2,q(M,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We have mentioned that the operator is invertible and thus the W s,q-type estimates (7) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We L and the boundary condition c to be the operators here with associated constant γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Mimicking the TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 9 boundedness proof in [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='4], we should require σ and sup ¯ M|G(·, u)|, and sup ¯ M|∇G(·, u)| to be small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Denote C = sup x∈ ¯ M,u(x)∈[min ¯ M u−,max ¯ M u+] |F(x, u(x))|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' D1 = sup x∈ ¯ M,u(x)∈[min ¯ M u−,max ¯ M u+] |G(x, u(x))| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' D2 = sup x∈ ¯ M,u(x)∈[min ¯ M u−,max ¯ M u+] |∇G(x, u(x))| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (21) We require that G(·, u+), D1, D2 satisfies ∥(B − σ) u+ + G (·, u+)∥W 1,q(M,g) ⩽ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (B − σ) · γ′ �� A max ¯ M (|u+|, |u−|) + C � Volg(M) 1 q + 1 � + D1 · Volg(M) 1 q + D2 · γ′ �� A max ¯ M (|u+|, |u−|) + C � Volg(M) 1 q + 1 � ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (22) By (7) and the first inequality in (22), we observe from the PDE (19) with k = 1 that ∥u1∥W 2,q(M,g) ⩽ γ′ � ∥Au+ + F(·, u+)∥Lq(M,g) + ∥(B − σ) u+ + G (·, u+)∥W 1,q(M,g) � ⩽ γ′ �� A max ¯ M |u+| + C � Volg(M) 1 q + 1 � ⩽ γ′ �� A max ¯ M (|u+|, |u−|) + C � Volg(M) 1 q + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Inductively, assume that ∥uk∥W 2,q(M,g) ⩽ γ′ �� A max ¯ M (|u+|, |u−|) + C � Volg(M) 1 q + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (23) To check ∥uk+1∥W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='q(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' we apply the W s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='q-type elliptic estimate with the solution of (19) again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ∥uk+1∥W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='q(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g) ⩽ γ′ � ∥Auk + F(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' uk)∥Lq(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g) + ∥(B − σ) uk + G (·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' uk)∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='q(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g) � ⩽ γ′ �� A max ¯ M (|u+|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' |u−|) + C � Volg(M) 1 q � + γ′ � (B − σ)∥uk∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='q(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g) + ∥G(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' uk)∥Lq(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g) + ∥∇G(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' uk)∥Lq(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g) � ⩽ γ′ �� A max ¯ M (|u+|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' |u−|) + C � Volg(M) 1 q � + � γ′�2 (B − σ) � A max ¯ M ((|u+|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' |u−|) + C) · Volg(M) 1 q + 1 � + γ′D1 · Volg(M) 1 q + � γ′�2 D2 � A max ¯ M ((|u+|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' |u−|) + C) · Volg(M) 1 q + 1 � ⩽ γ′ �� A max ¯ M (|u+|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' |u−|) + C � Volg(M) 1 q + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It turns that ∥uk∥W 2,q(M,g) is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The rest of the argument, in applying Arzela- Ascoli, the monotonicity of the sequence, and the elliptic regularity, is essentially the same as in [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We omit the details here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In conclusion, the sequence uk converges classically to a smooth function u which solves (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In addition, u− ⩽ u ⩽ u+ pointwise on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 is a special case of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 by taking F(·, u) = −Rgu+Sup−1 and G(·, u) = − 2 p−2hgu + 2 p−2Hu p 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Prescribed Scalar and Mean Curvature Functions under Pointwise Conformal Deformation When η1 < 0 Recall the Yamabe equation with Robin condition − a∆gu + Rgu = Sup−1 in M, ∂u ∂ν + 2 p − 2hgu = 2 p − 2Hu p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (24) In this section, we consider the existence of the solution of (24) for given functions S, H ∈ C∞( ¯ M), provided that η1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In particular, we will discuss the following cases: (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' S < 0 in M, and H ⩽ 0 everywhere on ∂M, H ̸≡ 0, with η1 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' S < 0 in M, and H > 0 somewhere on ∂M, with η1 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' S changes sign in M, and H is arbitrary on ∂M, with η1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that the Case (ii) above covers the possibilities when H > 0 everywhere on ∂M, or ´ ∂M HdS > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note also that the case S < 0 everywhere in M and H = 0 on ∂M has been discussed in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For Case (iii), obviously we have to impose some restrictions on S and H, as we shall see later;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' there is no free choice of S especially, due to Kazdan and Warner [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The first result concerns the Case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S1 < 0 be any smooth function on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let H1 ∈ C∞( ¯ M) such that H1 < 0 everywhere on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If η1 < 0, then there exists a small enough constant c > 0 such that (24) admits a positive solution u ∈ C∞( ¯ M) with S = S1 and H = cH1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S1 and h˜g = cH1 ���� ∂M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Due to the proof of Han-Li conjecture [17, Theorem], we may assume that hg = h > 0 and Rg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Since η1 < 0, it follows that η1,β < 0 with small enough positive constant β > 0, due to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Any constant multiple of ϕ solves (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Denote φ = δϕ, we choose the constant δ > 0 small enough so that η1,β inf ¯ M ϕ ⩾ δp−2 · inf ¯ M S1 · sup ¯ M ϕp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This can be done since both η1,β and S1 are negative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows that −a∆gφ + Rgφ = η1,βφ ⩽ S1φp−1 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Fix this δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We check the boundary condition −∂φ ∂ν + 2 p − 2hgφ = −β · 2 p − 2φ ⩽ 2 p − 2 · (cH1) φ p 2 for small enough positive constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Again it works since both −β and H1 are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We set u− := φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (25) The argument above shows that u− is a sub-solution of (24) with S = S1 and H = cH1 for small enough c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For super-solution, we set u+ := C ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (26) When C large enough, we have −a∆gu+ + Rgu+ = RgC ⩾ S1Cp−1 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 11 Since H1 < 0, it is straightforward to check that for any c > 0, we have −∂u+ ∂ν + 2 p − 2hgu+ ⩾ 0 > 2 p − 2 (cH1) u p 2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We can enlarge C so that C ⩾ sup ¯ M u−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Lastly we shrink c if necessary since we require the smallness of the sup-norm of the prescribing mean curvature function in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Since 0 < u− ⩽ u+ and both u+ and u− are smooth functions, we conclude by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 that (24) has a positive solution u ∈ C∞( ¯ M) with S = S1 and H = cH1 for small enough c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ We now consider the Case (ii) at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Actually the proof is very similar to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S2 < 0 be any smooth function on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let H2 ∈ C∞( ¯ M) such that H2 > 0 somewhere on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If η1 < 0, then there exists a small enough constant c > 0 such that (24) admits a positive solution u ∈ C∞( ¯ M) with S = S2 and H = cH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S2 and h˜g = cH2 ���� ∂M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The choice of the sub-solution is exactly the same as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' When we fix the sub- solution u−, we choose u+ = C ≫ 1 with C ⩾ u−, also large enough so that the same argument in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Fix this C from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The only difference is that since H2 > 0 somewhere, we may need to shrink c, if necessary, so that ∂C ∂ν + 2 p − 2hgC ⩾ 2 p − 2 · sup ∂M (cH2)C p 2 The rest of the argument is exactly the same as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The method of monotone iteration scheme has its limits, as we cannot obtain the prescribed mean curvature to be H, due to the technical issue, see [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We now discuss the Case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The following argument is inspired by Kazdan and Warner [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' When η1 < 0, Kazdan and Warner showed that the key is to get the super-solution of (24), if we are not using the variational method but instead the monotone iteration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Next result shows that a super-solution of (24) can be converted to another relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We point out that the following result is not specific for η1 < 0 case only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S, H ∈ C∞( ¯ M) be given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Then there exists some positive function u ∈ C∞( ¯ M) satisfying − a∆gu + Rgu ⩾ Sup−1 in M, ∂u ∂ν + 2 p − 2hgu ⩾ 2 p − 2Hu p 2 on ∂M (27) if and only if there exists some positive function w ∈ C∞( ¯ M) satisfying − a∆gw + (2 − p)Rgw + (p − 1)a p − 2 |∇gw|2 w ⩽ (2 − p)S in M, ∂w ∂ν − 2hgw ⩽ −2Hw 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (28) Moreover, the equality in (27) holds if and only if the equality in (28) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume that there is a positive function u ∈ C∞(M) that satisfies (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Define w = u2−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that 2 − p = − 4 n−2 < 0 since n ⩾ 3 by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We compute that ∇w = (2 − p)u1−p∇u ⇔ ∇u = up−1(2 − p)−1∇w, 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU and ∆gw = (2 − p)u1−p∆gu + (2 − p)(1 − p)u−p|∇gu|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By the inequality (27), we have a∆gw = (2 − p)u1−p (a∆gu) + a(2 − p)(1 − p)u−p|∇gu|2 ⩾ (p − 2)u1−p � −Rgu + Sup−1� + a(2 − p)(1 − p)(2 − p)−2u2p−2u−p|∇gv|2 = (p − 2)S + (2 − p)Rgu1−p + a(p − 1) p − 2 up−2|∇gv|2 = (p − 2)S + (2 − p)Rgw + a(p − 1) p − 2 |∇gw|2 w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Shifting (p − 2)S to the left side and a∆gw to the right side, we get the first part of the inequality (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For the boundary condition, recall that u = w 1 2−p and p = 2n n−2, it follows that ∂u ∂ν + 2 p − 2Hu p 2 ⩾ 2 p − 2hgu ⇔ 1 2 − pw 1 2−p −1 ∂w ∂ν + 2 p − 2hgw 1 2−p ⩾ 2 p − 2Hw p 2(2−p) ⇔ − n − 2 4 w− n 4 − 1 2 ∂w ∂ν + n − 2 2 hgw− n 4 + 1 2 ⩾ n − 2 2 Hw− n 4 ⇔∂w ∂ν − 2hgw ⩽ −2Hw 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Hence the second part of (28) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It is clear that the equality holds if an only if all inequalities above are equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If we assume (28) for some w, we just define u = w 1 2−p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The argument is very similar and we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ We now introduce the result of prescribing scalar and mean curvature functions for Case (iii), with a technical restriction very similar to the condition given by Kazdan and Warner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This technical condition, in principle, is to show the positivity of the function that satisfies (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Due to the Han-Li conjecture [17, Theorem], we may assume that the initial metric g has Rg = λ < 0 and hg = ζ > 0, since η1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Before we start with the special case, recall that if there exists a constant q > n, and some function u ∈ C∞( ¯ M) satisfies ∥u∥W 2,q(M,g) ⩽ γ′ � ∥F1∥Lq(M,g) + ∥F2∥W 1,q(M,g) � for some functions F1 ∈ Lq(M, g) and F2 ∈ W 1,q(M, g), the H¨older estimates implies that ∥u∥L∞( ¯ M) + ∥∇u∥L∞( ¯ M) ⩽ γ � ∥F1∥Lq(M,g) + ∥F2∥W 1,q(M,g) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (29) This inequality is due to the Sobolev embedding in [1, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume that η1 < 0, Rg = λ < 0 and hg = ζ > 0 for some constants λ and ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S3, H3 ∈ C∞( ¯ M) and q > n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let γ be the constant in the estimate (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Set D = (p−1)a p−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If there exists a function F ∈ C∞( ¯ M) and a positive constant A > 0, such that (2 − p)S3 ⩾ F on ∂M, ∥F − A∥Lq(M,g) ⩽ A 2γ (1 + (D + 1) (2 − p)λ), (30) then there exists a small enough constant c > 0 such that (24) admits a positive solution u ∈ C∞( ¯ M) with S = S3 and H = cH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S3 and h˜g = cH3 ���� ∂M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In this proof, we always denote Rg = λ and hg = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We construct the super-solution of (24) first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1, it is equivalent to show the existence of some positive function w ∈ C∞( ¯ M) such that (28) holds for S = S3 and H = cH3 for some constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Take δ := A (1 + (D + 1)(2 − p)λ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We also choose some negative constant δ′ = − δ 2γVolg(M) 1 q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By standard elliptic theory, there exists a unique solution w of the following PDE −a∆gw + (2 − p)λw = F − δ in M, ∂w ∂ν = δ′ on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The uniqueness comes from the fact that (2−p)λ > 0, which implies the invertibility of the operator � −a∆g + (2 − p)λ, ∂ ∂ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Clearly the constant (D + 1)δ solves the PDE −a∆g((D + 1)δ) + (2 − p)λ · ((D + 1)δ) = (2 − p)λ · ((D + 1)δ) in ∂M, ∂((D + 1)δ) ∂ν = 0 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Denote w0 := w − (D + 1)δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The function w0 satisfies −a∆gw0 + (2 − p)λw0 = F − δ − (D + 1)(2 − p)λδ = F − A in M, ∂w ∂ν = δ′ on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (31) The first line in (31) is due to the definition of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Since the differential operator with the boundary operator is invertible, we apply W s,q-type elliptic estimates (7) as well as the estimates of (29), ∥w0∥L∞( ¯ M) + ∥∇w0∥L∞( ¯ M) ⩽ γ � ∥F − A∥Lq(M,g) + ∥δ′∥W 1,q(M,g) � ⩽ γ � A 2γ (1 + (D + 1)(2 − p)λ) + |δ′| · Volg(M) 1 q � ⩽ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (32) The last inequality is due to the definitions of δ and δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By definition of w0, the inequality (32) implies ∥w − (D + 1)δ∥L∞( ¯ M) ⩽ δ, ∥∇w∥L∞(¯Ω) ⩽ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows that 0 < Dδ ⩽ w ⩽ (D + 2)w on ¯ M, sup ¯ M |∇w| ⩽ δ ⇒ (p − 1)a p − 2 |∇w|2 w ⩽ δ on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (33) With (36), (33), we have − a∆gw + (2 − p)λw + (p − 1)a p − 2 |∇w|2 w = F − δ + (p − 1)a p − 2 |∇w|2 w ⩽(2 − p)S3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' ∂w ∂ν − 2hgw = δ′ − 2hgw ⩽ −2 · (cH3) w 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The last inequality holds for small enough constant c > 0, regardless of the sign of H3 since δ′ − 2hgw < 0 by set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By (33) again, we conclude that w > 0 on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1, the positive, smooth function u = w 1 2−p 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU is a super-solution of (24) with S = S3 and H = cH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that u is still a super-solution if we make c smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For sub-solution, we apply the perturbed eigenvalue problem in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' There exists a small enough constant β > 0 such that − a∆gϕ + λϕ = η1,βϕ in M, ∂ϕ ∂ν + 2 p − 2 (ζ + β) ϕ = 0 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (34) Any scaling of ϕ solves (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Set the positive constant ξ ≪ 1 such that φ := ξϕ ⩽ u on ¯ M for the fixed super-solution u defined just above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We shrink ξ further, if necessary, such that η1,β (ξϕ) ⩽ S3 (ξϕ)p−1 in M, − 2 p − 2 · β (ξϕ) ⩽ 2 p − 2 · (cH3) · (ξϕ) p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (35) Note that the boundary condition holds for every c, as long as we take ξ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We point out that the choice of the constant c depends on the construction of the super-solution as well as the technical condition of the monotone iteration scheme, which only depends on the super-solution but not the sub-solution, see Equation (19) in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Thus we can choose c first, then determine ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows that φ is a sub-solution of (24) with S = S3 and H = cH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Furthermore, 0 < φ ⩽ u on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2, we conclude that there exists a positive function u ∈ C∞( ¯ M) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ The general case when η1 < 0 is a straightforward consequence of the result above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S4, H4 ∈ C∞( ¯ M) and q > n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let γ be the constant in the estimate (29) and λ be some negative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Set D = (p−1)a p−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume that η1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If there exists a function F ∈ C∞( ¯ M) and a positive constant A > 0, such that (2 − p)S4 ⩾ F on ∂M, ∥F − A∥Lq(M,g) ⩽ A 2γ (1 + (D + 1) (2 − p)λ), (36) then there exists a small enough constant c > 0 such that (24) admits a positive solution u ∈ C∞( ¯ M) with S = S4 and H = cH4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S4 and h˜g = cH4 ���� ∂M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By the result of the Han-Li conjecture [17, Theorem], there exists a conformal metric g1 = vp−2g such that Rg1 = λ and hg1 = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We then apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 for the metric g1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' there exists ˜g = up−2g1 with R˜g = S4 and h˜g = cH4 with small enough c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The conformal change ˜g = (uv)p−2 g is the desired metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Prescribed Scalar and Mean Curvature Functions for Conformal Equivalent Metrics When η1 < 0 Inspired by the “Trichotomy Theorem” on closed manifolds, we would like to discuss the pre- scribing scalar and mean curvature problem on ( ¯ M, g), n = dim ¯ M ⩾ 3, but not restricted in a conformal class [g] only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Instead, we are interested in the conformally equivalent metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 15 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, we say that a metric ˜g is conformally equivalent to the metric g if there exists a positive, smooth function u ∈ C∞( ¯ M) and a diffeomorphism φ : ¯ M → ¯ M such that φ∗˜g = up−2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Within in a conformal class, the prescribing scalar and mean curvature problem for given func- tions S, H ∈ C∞( ¯ M) is reduced to the PDE (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For conformlaly equivalent metrics, the prescribing scalar and mean curvature problem is reduced to the existence of a positive, smooth solution of the following PDE − a∆gu + Rgu = (S ◦ φ) up−1 in M, ∂u ∂ν + 2 p − 2hgu = 2 p − 2 · (H ◦ φ) · u p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (37) Our next result extends the result of prescribing scalar curvature problem on closed manifolds with dimensions at least 3 [10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3] to compact manifolds with non-empty smooth boundaries, provided that the first eigenvalue η1 of the conformal Laplacian with Robin boundary condition is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The method is essentially due to Kazdan and Warner [8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S5 be any smooth function on ¯ M that is negative somewhere in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let H5 ∈ C∞( ¯ M) and q > n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If η1 < 0, then there exists a small enough constant c > 0 and a diffeomorphism φ : ¯ M → ¯ M such that (37) admits a positive solution u ∈ C∞( ¯ M) with S = S5 and H = cH5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, there exists a conformally equivalent metric ˜g = � φ−1�∗ � up−2g � such that R˜g = S5 and h˜g = cH5 ���� ∂M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By Han-Li conjecture [17, Theorem], we may assume that Rg = λ < 0 and hg = ζ > 0 for some constants λ, ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Fix some constant q > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Due to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3, it suffices to find a diffeomorphism φ : ¯ M → ¯ M, a smooth function F ∈ C∞( ¯ M), a positive constant A > 0 and a small enough positive constant c such that (2 − p)S5 ◦ φ ⩾ F on ¯ M, ∥F − A∥Lq(M,g) ⩽ A 2γ (1 + (D + 1)(2 − p)λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (38) in addition, sup ¯ M c (H ◦ φ) is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Here γ is the constant in the estimate (29), the constant D is defined to be D = (p − 1)a p − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We determine φ, F and A first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If S5 < 0 everywhere on ¯ M, we just choose φ to be the identity map and set F = A = (2 − p) max ¯ M S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It is straightforward to check that (38) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If S5 ⩾ 0 somewhere and changes sign, we choose A first to be any positive constant such that 0 < A < (2 − p) min ¯ M S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (39) Just note that (2 − p) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We pick interior open submanifolds U, V ⊂ M such that V ⊂ ¯V ⊂ U ⊂ M ⊂ ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In particular, we require that Volg(U − V ) ⩽ \uf8eb \uf8ed A 2γ (1 + (D + 1)(2 − p)λ) · � (2 − p)∥S3∥L∞( ¯ M) − A � \uf8f6 \uf8f8 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (40) 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU We select the diffeomorphism φ such that (2 − p)S3 ◦ φ > A in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (41) We then take the function F to be F = A in V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (2 − p) max ¯ M S3 ◦ φ ⩽ F ⩽ A in U − V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' F = (2 − p) max ¯ M S3 ◦ φ in ¯ M − U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (42) Clearly F ⩽ (2 − p)S3 ◦ φ on ¯ M by (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The function F only differs with A in U − V , by (40), it is immediate to check that the second inequality in (38) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Lastly we choose c so that the condition in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 holds for the function S3 ◦ φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' c sup ¯ M|H5| is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The same c applies for the smallness of c sup ¯ M|H5 ◦ φ| since the diffeomorphism does not change the extremal values of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Therefore the function S3 ◦ φ and cH5 ◦ φ can be realized as prescribed scalar and mean curvature functions, respectively, for some metric φ∗˜g = up−2g where u is positive and smooth on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, S5 and cH5 can be realized as prescribed scalar and mean curvature functions, respectively, for some metric ˜g = � φ−1�∗ up−2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 indicates that on ( ¯ M, g) with n = dim ¯ M ⩾ 3, any function that is negative somewhere can be realized as a scalar curvature function of some metric g, meanwhile the mean curvature function of g can be some small enough scaling of any smooth function, provided that the manifold admits a metric with negative first eigenvalue of the conformal Laplacian, or equivalently, negative Yamabe invariant [6, §1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Prescribed Gauss and Geodesic Curvature Functions When χ( ¯ M) < 0 In this section, we discuss the prescribing Gauss and geodesic curvatures problem within a conformal class [g] of compact manifolds ( ¯ M, g) with non-empty smooth boundary ∂M, provided that χ( ¯ M) < 0 and n = dim ¯ M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This is a 2-dimensional analogy of prescribing scalar and mean curvatures problem with η1 < 0, provided that the dimension is at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let K, σ ∈ C∞( ¯ M) be given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This type of Kazdan-Warner problem is reduced to the existence of a smooth solution u of the following PDE − a∆gu + Kg = Ke2u in M, ∂u ∂ν + σg = σeu on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (43) Here Kg and σg are Gaussian and geodesic curvatures of g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The solvability of this PDE implies that the metric ˜g = e2ug has Gauss curvature K˜g = K and geodesic curvature σ˜g = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We mainly discuss to cases: (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' K ⩽ 0 everywhere in ¯ M, and arbitrary σ, with χ( ¯ M) < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' K > 0 somewhere in ¯ M and changes sign, σ is an arbitrary function, with χ( ¯ M) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We would like to apply the monotone iteration scheme to solve (43), it is equivalent to construct the sub- and super-solutions of (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The key is to construct the super-solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' As in §3, we convert the super-solution of (43) into another inequality involving derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let K, σ ∈ C∞( ¯ M) be given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Then there exists some function u ∈ C∞( ¯ M) satisfying − ∆gu + Kg ⩾ Ke2u in M, ∂u ∂ν + σg ⩾ σeu on ∂M (44) TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 17 if and only if there exists some positive function w ∈ C∞( ¯ M) satisfying − ∆gw − 2wKg + |∇gw|2 w ⩽ −2K in M, ∂w ∂ν − 2wσg ⩽ −2σw 1 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (45) Moreover, the equality in (44) holds if and only if the equality in (45) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' and the inequality in (44) is in the reverse direction if and only if the inequality in (45) is in the reverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume (44) for some function u first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Define w := e−2u We observe that ∇gw = −2e−2u∇gu ⇒ ∇gu = −1 2e2u∇gw, ∆gw = −2e−2u∆gu + 4e−2u|∇gu|2 = −2e−2u∆gu + e2u|∇gw|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Thus we have −∆gw = 2e−2u∆gu − e2u|∇gw|2 ⩽ 2e−2u � Kg − Ke2u� − |∇gw|2 w = 2wKg − 2K − |∇gw|2 w ⇒ − ∆gw − 2wKg + |∇gw|2 w ⩽ −2K in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For the boundary condition, we have ∂w ∂ν = ∂e−2u ∂ν = −2e−2u ∂u ∂ν ⩽ −2e−2u (−σg + σeu) = 2wσg − 2σw 1 2 ⇒∂w ∂ν − 2wσg ⩽ −2σw 1 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Therefore (45) holds for w = e−2u > 0 on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It is clear that equality holds when all inequalities above are equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It is also straightforward to see that the inequalities are in the reverse directions if and only if the inequalities are in the reverse directions in each step above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For the opposite direction, we assume (45) holds for some positive, smooth function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Define u = −1 2 log w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We can show that u satisfies (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The argument is quite similar to above and we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ Due to the uniformization theorem, we may assume Kg = −1 and σg = 0 in (43) from now on, as our model case up to some pointwise conformal change, provided that χ( ¯ M) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In 2-dimensional case, we also have the W s,q-type estimates from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We choose q = 3, the estimate in (7) plus the Sobolev embedding into H¨older space, the inequality in (29) becomes ∥u∥L∞( ¯ M) + ∥∇u∥L∞( ¯ M) ⩽ γ � ∥F1∥L3(M,g) + ∥F2∥W 1,3(M,g) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (46) Here F1, F2 and u comes from the PDE (6) with the operators L = −∆g + 2 and B = ∂ ∂ν , so is the constant γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Our main result of this section is the following, which covers both Case (i) and Case (ii) at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact Riemann surface with non-empty smooth boundary ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let K1, σ1 ∈ C∞( ¯ M) be given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let γ be the constant in the estimate (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume that χ( ¯ M) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If there exists a function F ∈ C∞( ¯ M) and a positive constant A > 0, such that − 2K1 ⩾ F on ∂M, ∥F − A∥L3(M,g) ⩽ A 6γ , (47) then there exists a small enough constant c > 0 such that (43) admits a positive solution u ∈ C∞( ¯ M) with K = K1 and σ = cσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, there exists a Yamabe metric ˜g = e2ug such that K˜g = K1 and σ˜g = cσ1 ���� ∂M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The proof is essentially the same as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By Lemma 43, the construction of the super-solution is equivalent to the construction of a function w that satisfies (45) for K1, σ1 and some small enough positive constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We set δ = A 3 , δ′ = − δ 2γVolg(M) 1 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (48) There is a unique solution for the PDE −∆gw + 2w = F − δ in M, ∂w ∂ν = δ′ on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Define w0 = w − 2δ, it follows that w0 satisfies the PDE − ∆gw0 + 2w0 = F − 3δ = F − A in M, ∂w0 ∂ν = δ′ on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (49) Apply the estimate (46) for w0 in (49), it follows that ∥w0∥L∞( ¯ M) + ∥∇w0∥L∞( ¯ M) ⩽ γ � ∥F − A∥L3(M,g) + ∥δ′∥W 1,3(M,g) � ⩽ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows from the definition of w0 that 0 < δ ⩽ w ⩽ 3δ on ¯ M, ∥∇w∥L∞( ¯ M) ⩽ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Therefore we conclude that −∆gw + 2w + |∇w|2 w = F − δ + |∇w|2 w ⩽ F ⩽ −2K1 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In addition, we take c small enough so that ∂w ∂ν = δ′ ⩽ −2cσ1w 1 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This can be done since δ′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows that the function u+ := −1 2 log w is a super-solution of (43) with K = K1 and σ = cσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Clearly u+ ∈ C∞( ¯ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We construct a sub-solution now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Consider the PDE −∆gu0 = 1 2 in M, ∂u0 ∂ν = C on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' By standard elliptic PDE theory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='7, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 4], the above PDE is solvable by some smooth function u0 ∈ C∞( ¯ M) if − ´ M 1 2dVolg = ´ ∂M CdSg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We choose the constant C < 0 so that the compatibility condition just mentioned holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Clearly u− := u0 + C1 TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 19 solves the PDE above also for any constant C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We just choose C1 to be very negative such that u− ⩽ u+ on ∂M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In addition, −∆gu− − 1 = −1 2 ⩽ K1e2u− = K1e2u0 · e2C1 in M, ∂u− ∂ν = C ⩽ cσ1eu− = cσ1eu0 · eC1 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' These can be done since the constants on the left sides of the inequalities are both negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that both the super-solution and sub-solution holds for smaller constant c by adjusting the constant C1 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that when F(·, u) = K1e2u + 1, G(·, u) = cσ1eu, the condition (22) is independent of the sub-solution u− as we can see the very similar case in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 for the Yamabe equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Thus we take c small enough so that the hypotheses in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows that there exists some smooth function u that solves (43) with K = K1 and σ = cσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ We can partially answer the two cases we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For Case (ii), not every function that changes sign can be a prescribed scalar curvature function unless it is not too positive too often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We show that every function that is negative everywhere can be realized as a scalar curvature function, meanwhile, a small enough scaling of any function can be realized as prescribed mean curvature function, under pointwise conformal deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This is Case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact Riemann surface with non-empty smooth boundary ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let K2, σ2 ∈ C∞( ¯ M) be given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume that K2 < 0 everywhere on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If χ( ¯ M) < 0, then there exists a small enough constant c, a smooth function u ∈ C∞( ¯ M) such that u solves (43) with K = K2 and σ = cσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It is equivalent to say that the metric ˜g = e2ug has Gauss curvature K˜g = K2 and geodesic curvature σ˜g = cσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We show that the condition (47) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Since K2 < 0 everywhere, we just choose F = A = −2 max ¯ M K2 ⇒ −2K2 ⩾ F, ∥F − A∥L3(M,g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We just need to choose a small enough c such that the hypotheses in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ For Case (ii), we can get a more comprehensive answer by considering the class of conformally equivalent metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Analogous to §4, we are looking for a metric ˜g = � φ−1�∗ e2ug with some diffeomorphism φ : ¯ M → ¯ M and smooth function u ∈ C∞( ¯ M) such that the scalar and mean curvatures of ˜g are given functions K, σ ∈ C∞( ¯ M), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This problem is reduced to the PDE − ∆gu + Kg = (K ◦ φ) e2u in M, ∂u ∂ν + σgu = (σ ◦ φ) eu on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (50) Similar to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 for dimensions at least 3, we introduce the following result for compact Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact Riemann surface with non-empty smooth boundary ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let σ3 ∈ C∞( ¯ M) be any function and K3 ∈ C∞( ¯ M) be a function that is negative somewhere in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If χ( ¯ M) < 0, then there exists a small enough constant c, a smooth function u ∈ C∞( ¯ M) and a diffeomorphism φ : ¯ M → ¯ M such that u solves (50) with K = K3 and σ = cσ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It is equivalent to say that the metric ˜g = � φ−1�∗ e2ug has Gauss curvature K˜g = K3 and geodesic curvature σ˜g = cσ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The proof is essentially the same as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We determine φ, F, A first so that (47) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' then determine the constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We may assume that K3 is negative somewhere but not everywhere since otherwise it is reduced to the result of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We choose A first to be any positive constant such that 0 < A < −2 min ¯ M K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (51) We pick interior open submanifolds U, V ⊂ M such that V ⊂ ¯V ⊂ U ⊂ M ⊂ ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In particular, we require that Volg(U − V ) ⩽ \uf8eb \uf8ed A 6γ · � 2∥K3∥L∞( ¯ M) − A � \uf8f6 \uf8f8 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (52) We select the diffeomorphism φ such that − 2K3 ◦ φ > A in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (53) We then take the function F to be F = A in V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' − 2 max ¯ M K3 ◦ φ ⩽ F ⩽ A in U − V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' F = −2 max ¯ M K3 ◦ φ in ¯ M − U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (54) Clearly F ⩽ −2K3 ◦ φ on ¯ M by (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The function F only differs with A in U − V , by (52), it is immediate to check that the second inequality in (47) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Lastly we choose c so that the condition in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3 holds for the function K3 ◦ φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' c sup ¯ M|σ3| is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The same c applies for the smallness of c sup ¯ M|σ3 ◦ φ| since the diffeomorphism does not change the extremal values of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Therefore the function K3 ◦ φ and cσ3 ◦ φ can be realized as prescribed scalar and mean curvature functions, respectively, for some metric φ∗˜g = up−2g where u is positive and smooth on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, K3 and cσ3 can be realized as prescribed scalar and mean curvature functions, respectively, for some metric ˜g = � φ−1�∗ up−2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The result of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2, combining Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 indicate that on ( ¯ M, g) with n = dim ¯ M ⩾ 2, any function that is negative somewhere can be realized as a scalar/Gauss curvature function of some metric g, meanwhile the mean/geodesic curvature function of g can be some small enough scaling of any smooth function, provided that the manifold admits a metric with negative first eigenvalue of the conformal Laplacian, or negative Euler characteristics, respectively, depending on the dimension of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' This improve the result mentioned in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Prescribed Scalar and Mean Curvature Functions for Conformally Equivalent Metrics When η1 = 0 In this section, we discuss the prescribing scalar and mean curvatures problem for metrics con- formally equivalent to the metric g on compact manifolds ( ¯ M, g) with non-empty smooth boundary ∂M, provided that η1 = 0 and n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We gave a comprehensive study for manifolds with dimensions at least 3 in [18] for pointwise conformal change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Here we consider whether there exists some smooth function u ∈ C∞( ¯ M) and some diffeomorphism φ : ¯ M → ¯ M such that the metric ˜g = � φ−1�∗ up−2g has scalar curvature S and mean curvature H for some given functions TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 21 S, H ∈ C∞( ¯ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Since the model case for zero first eigenvalue case is Rg = hg = 0, the problem above is reduced to the existence of the solution of the following PDE − a∆gu = (S ◦ φ) · up−1 in M, ∂u ∂ν = 2 p − 2 · (H ◦ φ) · u p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (55) Recall the result of prescribing scalar and mean curvature problems for conformal metrics on ( ¯ M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='4] Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S, H ∈ C∞( ¯ M) be given nonzero functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume that η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If the function S satisfies S changes sign and ˆ M SdVolg < 0, then there exists a pointwise conformal metric ˜g ∈ [g] that has scalar curvature R˜g = S and h˜g = cH for some small enough positive constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The conformally equivalent case follows from the result of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1, we show it below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that the case S = H = 0 is the trivial case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S6, H6 ∈ C∞( ¯ M) be given nonzero functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Assume that η1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If the function S satisfies S6 changes sign, then there exists a diffeomorphism φ : ¯ M → ¯ M and a small enough constant c > 0 such that (55) has a smooth solution u ∈ C∞( ¯ M) for φ, S = S6 and H = cH6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It is equivalent to say that the conformally equivalent metric ˜g = � φ−1�∗ up−2g has scalar curvature R˜g = S6 and mean curvature h˜g = cH6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Due to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1, it suffices to show that there exist a diffeomorphism φ : ¯ M → ¯ M such that ˆ M (S6 ◦ φ) dVolg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Due to the same reason in [9, 8], it is straightforward that such a diffeomorphism does exist since S6 changes sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The smallness of c is then determined by S6 ◦ φ, sup ¯ M|H6| as well as the choice of sub- and super-solutions in the proofs of [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that any diffeomorphism φ will not change the supremum of |H6| on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The result of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 indicates that on ( ¯ M, g) with n = dim ¯ M ⩾ 3, any function that changes sign or identically zero can be realized as a scalar curvature function of some metric g, meanwhile the mean curvature function of g can be some small enough scaling of any smooth function or zero function, respectively, provided that the manifold admits a metric with zero first eigenvalue of the conformal Laplacian, or equivalently, zero Yamabe invariant [6, §1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Prescribed Scalar and Mean Curvature Functions When η1 > 0 In this section, we seek for a positive, smooth solution of the following PDE − a∆gu + Rgu = Sup−1 in M, ∂u ∂ν + 2 p − 2hgu = 2 p − 2Hu p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (56) on compact manifolds ( ¯ M, g) with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3, for given functions S, H ∈ C∞( ¯ M), provided that η1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' As we have shown in [16], [17] and [19], we need to use local analysis, gluing a super-solution, and then apply monotone iteration scheme here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' According to the “Trichotomy Theorem” in [20], we expect few restrictions on prescribed scalar and mean curvature functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We will discuss the following case: 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' XU (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' S > 0 somewhere in M, and H > 0 somewhere on ∂M, with η1 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' S > 0 somewhere in M, and H ⩽ 0 everywhere on ∂M but H ̸≡ 0, with η1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Note that we have discussed the case S > 0 somewhere and H ≡ 0 in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Currently we do not see how to apply our method to the case mentioned in [7], − ∆eu = 0 in Bn, ∂u ∂ν + 2 p − 2hgu = 2 p − 2Hu p 2 on ∂Bn, u > 0 (57) for some given function H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Escobar showed that there is an obstruction for the choice of H ˆ ∂Bn X · ∇gHdS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Here X is some conformal Killing field on ∂Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' With standard Euclidean metric in Bn and the induced metric on ∂Bn, the first eigenvalue of conformal Laplacian with Robin condition is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' However, since the right side is zero, we are not able to get a nontrivial local solution of the Dirichlet problem −∆eu = 0 in Ω, u = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Therefore we may need some alternative method to resolve this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' However, we can get some interesting results provided that S ̸≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' According to the detailed analysis in [19, §5], we know that there will be obstructions for the choices of prescribed scalar curvature functions on Sn/Γ for some Kleinian group Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The map Sn → Sn/Γ must be a covering map since otherwise Sn/Γ cannnot be a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows that Sn/Γ has empty boundary, which follows that there will be no obstruction for the choice of prescribed scalar curvature functions on ( ¯ M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The first result concerns the Case (i) above: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S7 > 0 somewhere be any smooth function on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let H7 ∈ C∞( ¯ M) such that H7 > 0 somewhere on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If η1 > 0, then there exists a small enough constant c > 0 such that (56) admits a positive solution u ∈ C∞( ¯ M) with S = S7 and H = cH7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S7 and h˜g = cH7 ���� ∂M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Without loss of generality, we may assume that Sg > 0 and hg = h > 0 with positive constant h, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='3, we fix some β < 0 small enough so that η1,β > 0 and satisfies − a∆gϕ + Rgϕ = η1,βϕ in M, ∂ϕ ∂ν + 2 p − 2hgϕ = 0 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (58) Here ϕ > 0 on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Any scaling of ϕ solves (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Denote φ = δϕ for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We choose δ > 0 small enough so that η1,β inf ¯ M ϕ ⩾ δp−2 sup ¯ M S7 sup ¯ M ϕp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows that −a∆gφ + Rgφ ⩾ S7φp−1 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Fix this δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We then choose c > 0 small enough so that βφ ⩾ (cH7) φ p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' It follows that ∂φ ∂ν + 2 p − 2hgφ ⩾ 2 p − 2 (cH7) φ p 2 on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' (59) Note that (59) still holds for any smaller c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' For the sub-solution, we apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 or Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2, depending on the vanishing of the Weyl tensor in the interior M, to construct local TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES 23 solution u0 of the Yamabe equation with Dirichlet boundary condition on some domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 in [19], we can construct a local super-solution f of the Yamabe equation in Ω such that f = φ near ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' We then define u− = � u0 in Ω 0 in M\\Ω u+ := � f in Ω φ in M\\Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Since u− ≡ 0 on ∂M, it follows from the same argument in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 in [19] that u− is a sub- solution of the (56) with S = S7 and H = cH7 for any constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' According to the construction in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 of [19], we conclude that 0 ⩽ u− ⩽ u+, u− ̸≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' In addition, u− ∈ H1(M, g) ∩ C0( ¯ M), and u+ ∈ C∞( ¯ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' According to (59), we have seen that u+ is a super-solution of the (56) with S = S7 and H = cH7 for small enough c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Shrinking c, if necessary, so that the hypotheses of smallness of c sup ¯ M|H7| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' A direct application of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 indicates the existence of a positive solution u ∈ C∞( ¯ M) with S = S7 and H = cH7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ The proof of the Case (ii) is very similar as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let ( ¯ M, g) be a compact manifold with non-empty smooth boundary ∂M, n = dim ¯ M ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let S8 > 0 somewhere be any smooth function on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Let H8 ∈ C∞( ¯ M) such that H8 ⩽ 0 everywhere on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' If η1 > 0, then there exists a small enough constant c > 0 such that (56) admits a positive solution u ∈ C∞( ¯ M) with S = S8 and H = cH8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S8 and h˜g = cH8 ���� ∂M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Everything is exactly the same as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1, except at (59), there is no restriction for the choice of the constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' However, c should be small enough so that the hypotheses in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' The result of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='1 and Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content='2 indicate that on ( ¯ M, g) with n = dim ¯ M ⩾ 3, any function that is positive somewhere can be realized as a scalar curvature function of some metric g, meanwhile the mean curvature function of g can be some small enough scaling of any smooth function, provided that the manifold admits a metric with positive first eigenvalue of the conformal Laplacian, or equivalently, positive Yamabe invariant [6, §1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' References [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Aubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Nonlinear Analysis on Manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Monge-Amp´ere Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfFvoN/content/2301.01014v1.pdf'} +page_content=' Grundlehren der mathematischen Wis- senschaften.' metadata={'source': 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sha256:1084dc181450401a60e63f63d0862cee85a54b7973e67f4477505d38c3be184e +size 39772 diff --git a/3tE4T4oBgHgl3EQf0g0b/content/tmp_files/2301.05282v1.pdf.txt b/3tE4T4oBgHgl3EQf0g0b/content/tmp_files/2301.05282v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..510274f4788a49faa839de298e7674f0869e5861 --- /dev/null +++ b/3tE4T4oBgHgl3EQf0g0b/content/tmp_files/2301.05282v1.pdf.txt @@ -0,0 +1,1481 @@ +1 + +Non-centrosymmetric Sr2IrO4 obtained under High Pressure + +Haozhe Wang1‡, Madalynn Marshall2‡, Zhen Wang3, Kemp W. Plumb4, Martha Greenblatt2, +Yimei Zhu3, David Walker5, Weiwei Xie1* + + +1. Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA +2. Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey +08854, USA +3. Condensed Matter Physics and Materials Science Department, Brookhaven National +Laboratory, Upton, New York 11973, USA +4. Department of Physics, Brown University, Providence, Rhode Island 02912, USA +5. Lamont Doherty Earth Observatory, Columbia University, Palisades, New York 10964, USA + + +‡ H.W. and M.M. contributed equally. * Email: xieweiwe@msu.edu + + + +Abstract + +Sr2IrO4 with strong spin-orbit coupling (SOC) and Hubbard repulsion (U) hosts Mott +insulating states. The similar crystal structure, magnetic and electronic properties, particularly the +d-wave gap observed in Sr2IrO4 enhanced the analogies to cuprate high-Tc superconductor, +La2CuO4. The incomplete analogy was due to the lack of broken inversion symmetry phases +observed in Sr2IrO4. Here, under high pressure and high temperature conditions, we report a non- +centrosymmetric Sr2IrO4. The crystal structure and its noncentrosymmetric character were +determined by single crystal X-ray diffraction and high-resolution scanning transmission electron +microscopy (HR-STEM). The magnetic characterization confirms the Ir4+ with S = 1/2 at low +temperature in Sr2IrO4 with magnetic ordering occurred at around 86 K, where a larger moment is +observed than the ambient pressure Sr2IrO4. Moreover, the resistivity measurement shows three- +dimensional Mott variable-range hopping existed in the system. This non-centrosymmetric Sr2IrO4 +phase appears to be a unique material to offer further understanding of high-Tc superconductivity. + + + +2 + +Introduction + +Iridates with strong spin-orbit coupling effects can generate exotic quantum phenomena, +such as quantum spin liquid phases, Kitaev magnetism, and possible superconductivity.1-4 +Different from most 3d transition metal oxides in which the spin and orbit can be distinct in the +energy scale, the spin and orbit interact heavily in 5d transition metal oxides. Among the Mott +insulating 5d transition metal oxides, Sr2IrO45-9 has attracted significant of attention due to its +similarity to cuprate high-temperature superconductor, La2CuO410-12. As a single-layer +Ruddlesden–Popper compound, Sr2IrO4 crystallizes in a tetragonal lattice with an inversion center +(I41/acd, #142) at ambient pressure. Sr2IrO4 contains stacked IrO2 square lattices where the unit +cell is doubled compared to the CuO2 square lattices in high-Tc cuprates as a result of a staggered +rotation of IrO6 octahedron. Although superconductivity is not yet confirmed, many phenomena +characteristic of the superconducting cuprates have been observed in electron-and hole-doped +iridates including pseudogaps, Fermi arcs, and d-wave gaps .13-15 The Ir-d5 electrons in regular +IrO6 octahedron occupy the t2g orbitals, which can be approximated as two fully filled spin-orbital +coupled Jeff = 3/2 bands and one half-filled Jeff = 1/2 band. The Jeff band is split into an upper and +lower Hubbard band by on-site Coulomb interaction. According to a previous study, as Sr2IrO4 is +cooled below its Néel temperature (TN, ~230 K), the spin-orbit coupled Jeff = 1/2 moments order +into a basal plane commensurate Néel state. Octahedral rotations in Sr2IrO4 allow for non-zero +Dzyaloshinskii-Moriya (DM) interactions that results in a canting of the ordered moments away +from the crystallographic axis and a weak ferromagnetic moment per layer.16 Such a magnetic +transition maintains the inversion symmetry but lowers the rotational symmetry of the system from +C4 to C2. However, no additional symmetry breaking has been observed by neutron or X-ray +diffraction, which makes the comparison of the iridate to cuprate phenomenology incomplete. To +date, multiple methods have been used to tune the Mott insulating states in Sr2IrO4, for example, +isovalent Rh doping on the Ir site.5,17-23 After partially substituting Ir with Rh, an insulator-to-metal +transition can be detected. However, high pressure was also used to tune the electronic states up +to 55 GPa without observing any metallic state in Sr2IrO4.24,25 + +In this report, we applied the high-pressure (6 GPa) high-temperature (1400 °C) method +for synthesizing Sr2IrO4. Under such extreme conditions, the obtained Sr2IrO4 remains in a +tetragonal structure but without an inversion center. The space group was determined by single- + +3 + +crystal X-ray diffraction (SC-XRD) as I4mm (#107). Unlike the ambient pressure phase, the high- +pressure phase consists of the single layered IrO2 square lattice, just like CuO2 square in cuprate. +Magnetic susceptibility measurement on high pressure Sr2IrO4 indicate a magnetic ordering +temperature of approximately 86 K, which is dramatically lower than ambient pressure Sr2IrO4. +Interestingly, the resistivity data shows three-dimensional Mott variable-range hopping of charge +carriers between states localized by disorder with negligible long-range Coulomb interactions. +Discovering the non-centrosymmetric phase in Sr2IrO4 may accelerate the realization of +superconductivity and unravel the puzzle in cuprate high-Tc superconductors. + + + + +4 + +Experimental Section + +High-Pressure Synthesis. The ambient pressure Sr2IrO4 phase was prepared accordingly by +thoroughly mixing and pelletizing the materials SrCO3 and IrO2 and subsequently heating them to +900 °C then regrinding and reannealing at 1000 °C and subsequently reannealing at 1100 °C.26 +The ambient pressure Sr2IrO4 was pressurized to 6 GPa in 24 hours. After that, the sample was +heated up to 1400 °C and stayed at 1400 °C for 4 hours. Another sample was heated to 1400 °C +and stayed up to 28 hours to explore the optimal condition. The sample was cooled down to room +temperature before depressurizing to the ambient pressure. The high-pressure synthesis was +performed by statically compressing the sample using the Walker type multi-anvil press27 where +the original Sr2IrO4 was placed in a Pt capsule inside an Al2O3 crucible that was inserted into a +Cermacast 646 octahedra pressure medium lined on the inside with a LaCrO3 heater. + +Phase Analysis and Chemical Composition Determinations. The phase identity and purity were +examined using a Bruker D2 Phaser powder X-ray diffractometer with Cu K������������ radiation (������������ = +1.5406 Å). Room temperature measurements were performed with a step size of 0.004° at a scan +speed of 0.55°/min over a Bragg angle (2������������) range of 5–90°. FullProf Suite software28,29 was +utilized to analyze the phase information and lattice parameters from a Rietveld refinement. + +Structure Determination. The room temperature and low temperature (100 K) crystal structure +was determined using a Bruker D8 Quest Eco single crystal X-ray diffractometer, equipped with +Mo radiation (������������������������������������ = 0.71073 Å) with an ������������ of 2.0° per scan and an exposure time of 10 s per frame. +A SHELXTL package with the direct methods and full-matrix least-squares on the F2 model was +used to determine the crystal structure of Sr2IrO4.30,31 To confirm the crystal structure, high- +resolution scanning transmission electron microscopy (HR-STEM) images were collected and +electron diffraction was conducted using a 200 kV JEOL ARM electron microscope equipped with +double aberration correctors. Samples for TEM analysis were crushed in an agate mortar and +deposited directly onto a holey carbon copper grid. + +Physical Properties Measurement. Temperature and field-dependent magnetization, resistivity, +and heat capacity measurements were performed with a Quantum Design physical property +measurement system (PPMS) under a temperature range of 1.85–300 K and applied fields up to 9 + +5 + +T. Electrical resistivity measurements were accomplished with a four-probe method using +platinum wires on a pelletized sample of Sr2IrO4. The polycrystalline Sr2IrO4 was pressed up to 6 +GPa and heated at a lower temperature (100 °C) to eliminate the contribution of grain boundary +effect but also keep the phase stable. + + +6 + +Results and Discussions + +Exploring New Phase. The new Sr2IrO4 phase (I4mm, #107) was formed at 6 GPa from the +starting material, ambient pressure Sr2IrO4 (I41/acd, #142). The synthesis temperatures were set +up at 1200 °C and 1400 °C. The high pressure Sr2IrO4 phase was only produced at 1400 °C. To +increase the yield and grow larger crystals, the longer heating duration of 28 hours was tested. +However, the secondary tetragonal phase Sr3Ir2O7 simultaneously forms once the heating duration +was increased. As a result, only 4 hours heating process can produce the specimen consisting +mostly of pure phase. The resulting Le Bail fitting of the PXRD patterns for the high-pressure +phase Sr2IrO4 is shown in Fig. 1. An overlay of the PXRD patterns in Fig. S1 demonstrates the +formation of the secondary Sr3Ir2O7 phase. The pure phase synthesized at 1400 °C for 4 hours was +used for the physical property measurements below. + + +Fig. 1 Powder X-ray diffraction pattern of the high-pressure Sr2IrO4 phase. The experimental +data (red dots) was modeled with a Rietveld refinement (black line). The blue line indicates the +corresponding residual pattern (difference between observed and calculated patterns) along with +Bragg peak positions for Sr2IrO4 (green) and Al2O3 (purple) represented by the vertical tick marks. + + + + +Calc +Diff +Obs +Intensity (a.u.) +Sr2lrO4 +Al203 +10 +30 +50 +70 +90 +Sr,IrO. +20 (degree) +14mm (#107)7 + +Crystal Structure and Phase Determination. After 4 hours of treatment at 6 GPa and 1400 °C, +single crystals of Sr2IrO4 were formed, subsequently selected, and measured at both 300 K and 100 +K using the single crystal X-ray diffractometer. High-pressure Sr2IrO4 crystallizes with good +agreement into the tetragonal space group I4mm, as indicated by the single crystal X-ray diffraction +(SCXRD) refinement information listed in Table S1. Similar to ambient pressure Sr2IrO4, the high +pressure Sr2IrO4 phase contains the layers of IrO6 octahedra with intercalated Sr atoms. The +differences between these two are half-c lattice, the disappearance of the inversion center because +of the nonsymmetric distortion of IrO6 octahedra, and the disappearance of IrO6 octahedral +rotations in the ab-plane in high-pressure Sr2IrO4 compared to the ambient pressure phase. Shown +in Fig. 2 are crystal structures and IrO6 octahedra stacking view of ambient pressure Sr2IrO4 +(I41/acd), high-pressure Sr2IrO4 (I4mm), and previously reported La2CuO4 (I4/mmm), with Ir-O +atomic distance in the IrO6 octahedra highlighted. Atomic site vacancies and site disorder were +considered and refined to reveal the O3 atomic site is slightly displaced from the closer ideal 4b +site (1/2, 0, z) to the 8d site (x, 0, z) having a statistical occupancy of 0.5. The disordered model +yielded a more reasonable refinement with an R factor of 4.35 and goodness of fit (GOF) of 1.177 +while having only one O3 atomic site resulted in an R factor of 4.62 and GOF of 1.305. As such +an angle ������������ can be determined from (1/2 ± ������������, 0, z) with respect to an IrO6 octahedra where the O3 +atoms occupy the 4b site. This structural disorder has been thoroughly discussed for the ambient +pressure Sr2IrO4 structure.32 Additionally, the high-pressure Sr2IrO4 phase possesses a +nonsymmetric IrO6 octahedra elongation along the c axis, ranging in Ir-O atomic distance from +1.94(6)–2.27(6) Å, as indicated in Fig. 2b, which is in fact the cause of noncentrosymmetric +structural character. This behavior is kind of similar to the prominent feature of ambient pressure +Sr2IrO4 that has been speculated to originate from a Jahn Teller distortion.33-35 Previous studies +under high-pressure have revealed an increase in the IrO6 octahedra elongation with +pressurization.36,37 Compared to ambient pressure Sr2IrO4, one Ir-O along the c -axis is +significantly elongated, with the other almost remains the same, i.e., one oxygen atom is driven +away from the Ir atom, and thus the repulsion between Ir and the oxygen ligand is reduced. This +will lower the energy of orbitals that contains z contribution and split eg and t2g orbitals, making +the crystal field split of Ir d orbitals even more complicated. Together with spin-orbit coupling, +this may further remove orbital degeneracies. Moreover, as pressure applied for Sr2IrO4, the Ir-O- + +8 + +Ir angle was pushed close to 180°, which is the angle in Cu-O-Cu in La2CuO4. The structural +disorder was further confirmed at 100 K and the SCXRD refinement details can be found in SI. + +Fig. 2 Crystal structure illustration. Crystal structures, octahedra stacking view along a axis, +and along c axis of (a) ambient pressure Sr2IrO4, (b) as-synthesized high pressure Sr2IrO4, and (c) +previously reported La2CuO4, with Ir(Cu)O6 octahedra and Ir(Cu)-O atomic distances presented. +Green, blue, dark green, dark blue, and red atoms represent Sr, Ir, La, Cu, and O atoms, +respectively. Single-layer square net is also highlighted. + + + +(a) +(b) +1.98(1) A +2.27(6) A +1.95(1) A +1.93(1) A +1.94(1) A +1.94(6) A +2.11(1) A +Sr,IrO4 +Sr,lrO4 +La,CuO +14,/acd (#142) +14mm (#107) +14/mmm(#139)9 + +Transmission Electron Microscopy. The non-centrosymmetric space group and loss of IrO6 +octahedral rotation, as well as the oxygen distortion and defects in Sr2IrO4, can at first, be +surprisingly interesting, thus high-pressure Sr2IrO4 was investigated by transmission electron +microscopy (TEM) to characterize its crystallographic nature. The High-angle annular dark-field +scanning transmission electron microscopy (HAADF-STEM) image was obtained along the a axis +shown in Fig. 3a. The TEM diffraction patterns projected down the crystalline [100] axis (Fig. 3b) +allowed for the determination of the orientation of the images through the d002 spacing. The c-axis +parameter is ~12.8 Å, agreeing with the single crystal XRD results. The electron diffraction and +imaging study confirmed the high quality of the nanoscale ordering in the specimen. However, the +fractional spots 1/2 (110)/(1-10) were observed by TEM electron diffraction in Fig. 3d. As is +known that IrO6 tilt/rotation along the c-axis would not introduce these fractional spots. Such +fractional reflection spots are related to the ordering of oxygen vacancy, which is consistent with +single crystal X-ray diffraction results in Fig. 3e. + + +Fig. 3 Transmission electron microscopy study of high pressure Sr2IrO4 phase. (a) HAADF- +STEM image taken along a axis from a large area showing the high quality of the crystal Sr2IrO4. +(b) The zoom-in HAADF image shows the projected structure in the [100] direction, with a crystal +model superimposed, where Sr (green), Ir (blue), and O (orange). (c) The diffraction pattern took +along the [100] direction which is consistent with the simulated pattern (Fig. 3e) based on the +crystal model determined by single crystal X-ray diffraction. (d) SAED pattern along the [001] +direction showing fractional spots of 1/2 (110)/(1-10). (e) Simulated diffraction pattern and (f) +projected crystal structure along the [001] direction based on the crystal structure determined by + +220 +020 +X +200 +000 +003 +220 +020 +20 +[100] +110 +200 +10 +[001]10 + +SCXRD. The fractional spots observed in TEM were marked in red. The single crystal structure +of Sr2IrO4 with oxygen distortion was confirmed by both single crystal X-ray diffraction and TEM. + + +Weak Ferromagnetic Ordering. To study the magnetic properties of the high pressure Sr2IrO4 +phase, the temperature-dependent susceptibility was measured under field cooled warming (FCW) +and field cooled cooling (FCC) mode at 0.1 T shown in Fig. 4a. No significant differences between +FCW and FCC were observed. At about 150 K, the susceptibility goes below 0, indicating a +diamagnetic contribution in the system, which suggests the possible breakdown of Curie-Weiss +behavior at high temperatures in the system. The data between 80–140 K was modeled with the +modified Curie-Weiss law (Eqn. 1), shown in Fig. 4b and Fig. S2b, +������������ = ������������0 + +������������ +������������ − ������������cw +(1) +where ������������������������������������ is the paramagnetic Curie temperature, ������������0 is the temperature independent susceptibility +and ������������ is the Curie constant. From the fitting, the Curie temperature, ������������������������������������, of 86(7) K was found to +be comparable to the magnetic ordering temperature ������������������������ ~84 K, as determined from the minimum +in the temperature derivative of ������������ (See Fig. S2a for details). The magnetic ordering temperature, +consequently, decreases when compared to ambient pressure Sr2IrO4, which has a ������������������������ ~240 K.39,40 +On the other hand, it can be assumed that the Tc significantly decreases as the angle of Ir-O-Ir is +more close to 180 °, which is the one observed in Cu-O-Cu in high Tc superconductor La2CuO4. +The fitting also gave a negative ������������0 of -2.9(9)×10-3 emu mol-1 Oe-1, which provided a potential +opportunity to extrapolate our Curie-Weiss fit to higher temperature. Finally, up to 160 K was +included (Fig. S2c) and the fit yielded the effective moment ������������eff = 1.2(2) µB/Ir, which is more +agreeable with the Hund’s-rule value of 1.73 µB/Ir for S = 1/2 than the reported ������������eff = 0.33 µB/Ir +for ambient pressure Sr2IrO4. + +Furthermore, the magnetization of high pressure Sr2IrO4 was measured as shown in Fig. +4c up to 9 T at different temperatures. It appears to saturate at ~3 T at which the magnetic saturation +moment (������������������������������������������������) was determined to be ~0.046 µB/Ir. This value is significantly lower than the +theoretical value of 1/3 µB f.u−1, however, similar to the previously reported moment for the +ambient pressure Sr2IrO4 phase, which originates from spin canted antiferromagnetic (AFM) +order.39 This could also explain why the weak ferromagnetic behavior observed in the temperature + +11 + +dependence of magnetic susceptibility gives such a low value of moment. However, unlike the +ambient pressure Sr2IrO4 phase, the magnetization reaches a maximum at around 3 T at which +point the magnetization decreases. It turned out that diamagnetic transition was observed under +higher fields at the respective temperatures (e.g., see the 50 K and 100 K data). At 300 K, a +complete diamagnetic behavior was shown, consistent with ������������ < 0 shown in Fig. 4c. Subtracting +this by linearly fitting data from 7–9 T, the ������������������������������������������������ was modified to be 0.067 µB/Ir at 2 K and 0.014 +µB/Ir at 100 K, as presented in Fig. 4d and 4e. Magnetic hysteresis was observed in the system +under 2 K from -0.6 T to 0.6 T, presented in Fig. S3, which could be interpreted as small canting +of the moments existed in the system. + +12 + + +Fig. 4 Magnetization in the dependence of temperature and field. (a) Temperature dependence +of magnetic susceptibility ������������ at 1000 Oe under FCW and FCC mode ranging from 2–300 K. No +significant difference was observed. (b) The modified inverse magnetic susceptibility data (FCW, +80–140 K, blue hollow circle) fitted with the modified Curie-Weiss model (orange line). (c) Field +dependence of magnetization up to 9 T at different temperatures. (d) Derivation of ������������sat at 2 K by +linearly fitting the magnetization data from 7–9 T. (e) Derivation of ������������sat at 100 K. + +No Magnetically Induced Anomalies Observed in Specific Heat Measurement. To confirm the +magnetic transition, the specific heat over the temperature range of 2–200 K was measured under +0 T with a polycrystalline pelletized sample of Sr2IrO4, as presented in Fig. 5a. Measurements + +(a) +(b) +1e-2 +1e3 +2 +FCW +Cw fit +Oe) +FCC +0 +FCW +mol +0 +8 +oo +0 +0 +0 +1 +090 +0 +4 +8 +08 +X +8 +0 +0 +0 +100 +200 +300 +80 +130 +180 +230 +280 +T (K) +T (K) +(c) +(d) +(e) +1e-2 +8 +0 +10 +20 K +(μB per Ir ion) +4 +0 +.4 +M +50 K +-8 +100 K +0 2K +100K +300 K +"2 K" +"100 K" +-9 +-6-3 +3 +6 +9 +0 +36 +9 +0 +36 +9 +μoH (T) +μoH (T) +μoH (T)13 + +under applied fields of 0.05 T and 1 T in Fig. S3 were additionally tested to conclude no significant +deviation from the 0 T specific heat. No ������������ shape anomalies were observed at the whole temperature +regime studied, which may result from higher temperature regions being heavily dominated by the +phonon contribution. The specific heat data were fitted by the Debye model (Eqn. 2), and Einstein +model (Eqn. 3), shown in Fig. S4a and b. The Debye and Einstein temperatures could then be +determined as 417(2) K and 306(2) K, respectively. However, neither of these two described the +experimental data well. +������������D = 9������������������������ � ������������ +������������D +� +3 +� +������������4������������������������ +(������������������������ − 1)2 ������������������������ +������������D ������������ +⁄ +0 +(2) +where ������������ is the number of atoms per formula unit, ������������ is the gas constant, and ������������������������ is the Debye +temperature. +������������E = 3������������������������ �������������E +������������ � +2 +������������ +������������E +������������ ������������� +������������E +������������ − 1� +−2 +(3) +where ������������ is the number of atoms per formula unit, ������������ is the gas constant, and ������������������������ is the Einstein +temperature. + +The specific heat data was further fitted with two Debye model (Eqn. 4) and weighted +Debye model (Eqn. 5), with and without the electronic contribution included, shown in Fig. 5a +and Fig. S4c, d, and e. The data was found to be described well with two Debye model (Fig. 5a), +and the Debye temperatures, ������������������������1 of 235(1) K, ������������������������2 of 708(5) K was obtained. At low temperatures, +the first Debye mode has a larger contribution to the specific heat. Within the temperature regime +studied, the expected Dulong-Petit value of 3������������������������ is not recovered, and this can be explained by the +high value of ������������������������2, which means that the specific heat will plateau at ������������ ≫ ������������������������2. The fitting also yields +������������������������1 of 3.20(3) and ������������������������2 of 4.51(2). The sum of these two seems a little larger than the expected +value of 7 for Sr2IrO4, which may be attributed to the impurity of Srn+1IrnO3n+1, lack of electron +contribution, or overestimation of photon contribution in the model. Once the electron contribution +term was included, ������������������������1 was slightly shifted to 238(2) K and the sum of ������������������������1 and ������������������������2 went down to +7.52(11). +������������ = 9������������D1������������ � ������������ +������������D1 +� +3 +� +������������4������������������������ +(������������������������ − 1)2 ������������������������ +������������D1 ������������ +⁄ +0 ++ 9������������D2������������ � ������������ +������������D2 +� +3 +� +������������4������������������������ +(������������������������ − 1)2 ������������������������ +������������D2 ������������ +⁄ +0 +(+������������������������) +(4) + +14 + +where ������������������������1 and ������������������������2 are Debye temperatures, ������������������������1 and ������������������������2 are the oscillator strengths, and ������������������������ is the +electron contribution. +������������ = 9������������D������������ � ������������ +������������D +� +3 +� +������������4������������������������ +(������������������������ − 1)2 ������������������������ +������������D ������������ +⁄ +0 ++ 3������������E������������ �������������E +������������ � +2 +������������ +������������E +������������ ������������� +������������E +������������ − 1� +−2 +(+������������������������) +(5) +where ������������������������ and ������������������������ are the Debye and Einstein temperatures, ������������������������ and ������������������������ are the oscillator strengths. + +It should be noted that the magnetic contribution cannot be quantitatively extracted from +the specific heat data as the phonon contribution cannot be distinguished from the magnetic +contribution due to the lack of a nonmagnetic analog. + +At a low-temperature regime, of 2–20 K, the specific heat was measured, as shown in Fig. +S5. The data ranging from 2–3.2 K was fitted with Eqn. 6, shown in Fig. 5b. +������������p +������������ = ������������ + ������������������������2 +(6) +From this fitting, a ������������ and ������������ value of 0.0153(2) J mol-1 K-2 and 7.1(2) × 10-4 J mol-1 K-3 +corresponding to the electronic and phonon contributions to the specific heat, respectively, could +be obtained. The ������������ value recovered the Debye temperature (Eqn. 7) to be 268(2) K, which is much +closer to ������������������������1 rather than ������������������������2. It falls out of the temperature interval, 300–350 K, where iridates +most commonly exhibit Debye temperatures.41 +������������D = �12������������4 +5������������ ������������������������� +1 +3 +(7) + + +15 + + +Fig. 5 Specific heat data fitting of high pressure Sr2IrO4. (a) Temperature dependence of +specific heat over temperature (������������p ������������ +⁄ ) for high-pressure Sr2IrO4 fitted by two Debye model in +orange. Green and red dotted lines refer to the 1st and 2nd Debye model. (b) ������������p ������������ +⁄ vs ������������2 between +2–3.2 K fitted with Eqn. 6 (orange dotted line). + +Mott Variable-range Hopping (VRH). It is critical to investigate the electrical conductivity in +the high pressure Sr2IrO4 phase to compare to the Mott insulator ambient pressure Sr2IrO4. +Temperature-dependent resistivity measurements were performed from 2–300 K with an applied +field up to 9 T on a pelletized polycrystalline sample of the high pressure Sr2IrO4 phase, shown in +Fig. 6a. No significant field dependence was observed, which indicates the insignificance of +magnetoresistance for the high pressure Sr2IrO4 phase. This may be not unexpected considering +the small saturation moment under fields (see the discussion above). At room temperature and 0 +T, the resistivity is relatively low, only around 4 Ω cm. However, the resistivity is increases by 6 +orders of magnitude upon cooling, indicating the semiconducting character of the high-pressure +Sr2IrO4 phase. + +To further analyze its behavior, we first tried to model the temperature dependence of ������������ with the +Arrhenius law (Eqn. 8), +������������ = ������������0������������������������������������ ������������������������ +⁄ +(8) + +(a) +(b) +1e-1 +1e-2 + two Debye +OT +... y+βT2 +OOT +8 +Cp/T ( mol-1 K-2) +8 +2.4 +6 +4 +2.0 +6 +2 +Debyel +Debye2 +0 +0 +50 +100 +150 +200 +4 +8 +12 +16 +T (K) +T2 (K2)16 + +where ������������0 is the residual resistivity, ������������������������ is the activation energy, and ������������ is the Boltzmann constant. +However, ������������ could not be fitted well to a ������������������������, shown in Fig. S7a, i.e., the Arrhenius law is not well +obeyed. Then its temperature dependence was fitted by law in the form (Eqn. 9) with ������������ of 1/2 and +1/4, +������������ = ������������0������������(������������0 ������������ +⁄ )������������ +(9) +where ������������0 is the residual resistivity, and ������������0 is the characteristic temperature. The fitting results were +presented in Fig. 6b, and Fig. S8, with parameters summarized in Table S4. The value ������������ of 1/4 is +favored over 1/2. While both of them indicate three-dimensional Mott variable-range hopping of +charge carriers between localized states, the weaker temperature dependence with ������������ of 1/4 implies +negligible long-range Coulomb interactions between localized electrons in the temperature regime +studied. This behavior is also reported in the ambient pressure Sr2IrO4.42 To explore the harboring +quantum states in the high-pressure Sr2IrO4 phase, further examination of its transport properties +is warranted. + + + +Fig. 6 Details of field and temperature dependent resistivity. (a) Temperature dependence of +resistivity data for high-pressure phase Sr2IrO4 under fields up to 9 T. No significant derivation +was observed. (b) The resistivity ������������ (blue hollow circle) ranging from 80–300 K was fitted by Eqn. +9 with ������������ of 1/4 (orange line). A linear relationship was obtained. + + + + +(a) +(b) +107 +OT +fit +1 T +o data +3 T +8 +105 +5 T +In(p/(2 cm)) +p (Q cm) +7 T +9T +103 +4 +101 +0 +0 +100 +200 +300 +0.2 +0.3 +0.4 +0.5 +T (K) +T-1/4 (K-1/4)17 + +Conclusion + +In summary, we reported the non-centrosymmetric Sr2IrO4 phase obtained under high +pressure and high temperature conditions. The ferromagnetic ordering temperature decreases +significantly to ������������c ~86 K from ~240 K in the ambient pressure Sr2IrO4, while there may be a +possible breakdown of the Curie-Weiss law under higher temperatures. Diamagnetism was +observed under room temperature and higher fields. No anomalies indicating magnetic ordering +were observed in the specific heat measurements, where a greater photon contribution was +obtained from the low-temperature regime. Temperature-dependent resistivity revealed three- +dimensional Mott variable-range hopping of charge carriers between states localized by disorder +with negligible long-range Coulomb repulsions. Further transport measurements, together with +first-principal calculation, are expected to explore the electronic properties of the high-pressure +Sr2IrO4 phase. Such a system may offer a promising platform to unravel the mystery of high-Tc +superconductivity in cuprates. + + + +Acknowledgments + +The work at Rutgers was supported by U.S. DOE-BES under Contract DE-SC0022156. +The electron microscopy work at BNL was supported by U.S. DOE-BES, Materials Sciences and +Engineering Division under Contract No. DESC0012704. + +Supporting Information + +Single crystal X-ray diffraction data at room temperature and 100 K; Anisotropic +displacement parameters; Atomic coordinates and equivalent isotropic displacement parameters; +PXRD overlay of Sr2IrO4; Magnetic susceptibility and Curie-Weiss fitting; Magnetic hysteresis; +Field dependence of specific heat; Specific heat data fitted by Debye and Einstein model; Low +temperature specific heat data (2–20 K); Temperature dependence of resistivity; Resistivity data +fitted by Eqn. 9 with ������������ of 1/2 and 1/4; Summary of fitting parameters for resistivity data. + + + + +18 + +References +1. 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First-order structural transition +and pressure-induced lattice/phonon anomalies in Sr2IrO4. Phys. Rev. B 2018, 98, 094101. +38. Longo, J. M.; Raccah, P. M. The structure of La2CuO4 and LaSrVO4. J. Solid State Chem. +1973, 6, 526-531. +39. Ye, F.; Chi, S.; Chakoumakos, B. C.; Fernandez-Baca, J. A.; Qi, T.; Cao, G. Magnetic and +crystal structures of Sr2IrO4: A neutron diffraction study. Phys. Rev. B 2013, 87, 140406. +40. Kini, N. S.; Strydom, A. M.; Jeevan, H. S.; Geibel, C.; Ramakrishnan, S. Transport and thermal +properties of weakly ferromagnetic Sr2IrO4. J. Phys.: Condens. Matter 2006, 18, 8205-8216. +41. Pallecchi, I.; Buscaglia, M. T.; Buscaglia, V.; Gilioli, E.; Lamura, G.; Telesio, F.; Cimberle, +M. R.; Marré, D. Thermoelectric behavior of Ruddlesden–Popper series iridates. J. Phys.: Condens. +Matter 2016, 28, 065601. +42. Cao, G.; Bolivar, J.; McCall, S.; Crow, J. E.; Guertin, R. P. Weak ferromagnetism, metal-to- +nonmetal transition, and negative differential resistivity in single-crystal Sr2IrO4. Phys. Rev. B +1998, 57, R11039-R11042. + + + + +21 + + Non-centrosymmetric Sr2IrO4 obtained under high pressure + + +Haozhe Wang1‡, Madalynn Marshall2‡, Zhen Wang3, Kemp W. Plumb4, Martha Greenblatt2, +Yimei Zhu3, David Walker5, Weiwei Xie1* + +1. Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA +2. Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey +08854, USA +3. Condensed Matter Physics and Materials Science Department, Brookhaven National +Laboratory, Upton, New York 11973, USA +4. Department of Physics, Brown University, Providence, Rhode Island 02912, USA +5. Lamont Doherty Earth Observatory, Columbia University, Palisades, New York 10964, USA + + +‡ H.W. and M.M. contributed equally. * Email: xieweiwe@msu.edu + + + +Supporting Information + +Table S1 Single crystal X-ray diffraction data at room temperature and 100 K .......................... S2 +Table S2 Anisotropic displacement parameters ........................................................................... S3 +Table S3 Atomic coordinates and equivalent isotropic displacement parameters ....................... S4 +Figure S1 PXRD overlay of Sr2IrO4 ............................................................................................ S5 +Figure S2 Magnetic susceptibility and Curie-Weiss fitting ......................................................... S6 +Figure S3 Magnetic hysteresis ..................................................................................................... S7 +Figure S4 Field dependence of specific heat ............................................................................... S8 +Figure S5 Specific heat data fitted by Debye and Einstein model ............................................... S9 +Figure S6 Low temperature specific heat data (2–20 K) ........................................................... S10 +Figure S7 Temperature dependence of resistivity ...................................................................... S11 +Figure S8 Resistivity data fitted by Equation 9 with ������������ of 1/2 and 1/4 ..................................... S12 +Table S4 Summary of fitting parameters for resistivity data ..................................................... S13 + + +22 + +Table S1 Single crystal X-ray diffraction data at room temperature and 100 K. +Temperature +Room Temperature +100 K +Refined formula +Sr2IrO4 +Sr2IrO4 +FW (g/mol) +431.44 +431.44 +Space group +I4mm +I4mm +a (Å) +3.8860(5) +3.8777(5) +c (Å) +12.826(2) +12.825(2) +V (Å3) +193.69(6) +192.85(6) +Extinction Coefficient +N/A +N/A +������������ range (°) +3.177–33.030 +3.177–33.075 +# of reflections; Rint +1088; 0.0627 +1286; 0.0591 +# of independent reflections +267 +264 +# of parameters +23 +23 +R1; ωR2 (������������ > ������������������������(������������)) +0.0409; 0.0651 +0.0312; 0.0443 +Goodness of fit (GOF) +1.177 +1.125 +Diffraction peak and hole (e-/ Å3) +3.658, -3.492 +2.359, -1.96 + + + + +23 + +Table S2 Anisotropic displacement parameters for Sr2IrO4 at room temperature and 100 K. +Sr2IrO4 at Room Temperature +Atom +U11 +U22 +U33 +U23 +U13 +U12 +Ir1 +-0.0018(6) +-0.0018(6) +-0.0021(6) +0 +0 +0 +Sr1 +0.026(7) +0.026(7) +0.007(7) +0 +0 +0 +Sr2 +-0.001(4) +-0.001(4) +0.005(6) +0 +0 +0 +O1 +0.03(2) +0.03(2) +-0.02(2) +0 +0 +0 +O2 +-0.006(10) +-0.006(10) +-0.023(18) +0 +0 +0 +O3 +0.04(3) +0.003(11) +-0.01(3) +0 +0.02(3) +0 + +Sr2IrO4 at 100 K +Atom +U11 +U22 +U33 +U23 +U13 +U12 +Ir1 +-0.0004(3) +-0.0004(3) +0.0012(7) +0 +0 +0 +Sr1 +0.005(8) +0.005(8) +0.005(4) +0 +0 +0 +Sr2 +0.002(8) +0.002(8) +0.000(4) +0 +0 +0 +O1 +0.005(8) +0.005(8) +-0.033(15) +0 +0 +0 +O2 +0.012(10) +0.012(10) +-0.033(14) +0 +0 +0 +O3 +0.009(10) +0.005(7) +0.006(8) +0 +0.01(3) +0 + + + + +24 + +Table S3 Atomic coordinates and equivalent isotropic displacement parameters for Sr2IrO4 at room +temperature and 100 K. (Ueq is defined as one-third of the trace of the orthogonalized Uij tensor (Å2)). +Sr2IrO4 at Room Temperature +Atom +Wyck. +x +y +z +Occ. +Ueq +Ir1 +2a +0 +0 +0.1513(13) +1 +-0.0019(4) +Sr2 +2a +0 +0 +0.5044(4) +1 +0.020(4) +Sr1 +2a +0 +0 +0.79985(2) +1 +0.001(3) +O1 +2a +0 +0 +0.328(4) +1 +0.013(18) +O2 +2a +0 +0 +0.000(4) +1 +-0.011(7) +O3 +8d +0.419(9) +0 +0.661(7) +0.5 +0.010(15) + +Sr2IrO4 at 100 K +Atom +Wyck. +x +y +z +Occ. +Ueq +Ir1 +2a +0 +0 +0.1489(7) +1 +0.0001(3) +Sr2 +2a +0 +0 +0.5019(4) +1 +0.005(5) +Sr1 +2a +0 +0 +0.7978(2) +1 +0.002(5) +O1 +2a +0 +0 +0.321(3) +1 +-0.008(6) +O2 +2a +0 +0 +0.000(3) +1 +-0.003(8) +O3 +8d +0.412(4) +0 +0.649(6) +0.5 +0.007(4) + + + + +25 + +Figure S1 Powder X-ray diffraction pattern overlay. The experimental data of high pressure Sr2IrO4 +phase synthesized at 1400 °C for ~4 hrs (black line) and ~28 hrs (red line) were presented. Bragg peak +positions are indicated as Sr2IrO4 and Sr3Ir2O7 with green and purple vertical tick marks, respectively. + + + + + + +TT-1412 ~4 hrs +GG-1418 -28 hrs +Intensity (a.u.) +Sr2lrO4 +Sr3lr2Q7 +10 +30 +50 +70 +90 +2e (degree)26 + +Figure S2 Magnetic susceptibility and Curie-Weiss fitting. (a) Temperature derivative of magnetic +susceptibility ������������ at 1000 Oe under FCW mode. The minimum at around 84 K was highlighted by red circle +and an arrow. (b) The inverse magnetic susceptibility data (FCW, 80–140 K, blue hollow circle) fitted with +the modified Curie-Weiss model (orange line). (c) The Curie-Weiss fit was further extrapolated to 160 K. + + + + + + +(a) +(b) +1e-3 +1e3 +K-1) +Cw fit +1.2 - +FCW +-1 +00 +0.6 - +8 +-2 +1/x +000 +0.0 - +000 +50 +60 +70 +80 ° +90 +100 +80 +100 +120 +140 +T (K) +T (K) +(c) +1e2 +mol Oe) +4 +CW fit +FCW +N +0% +00 +0 +80 +100 +120 +140 +160 +T (K)27 + +Figure S3 Magnetic hysteresis. Magnetic hysteresis observed in the high pressure Sr2IrO4 phase at 2 K +ranging from -0.6 T to 0.6 T. + + + + + + +1e-2 +0 2K +1 +M (μB per Ir ion) +-1 +-0.6 +-0.3 +0.0 +0.3 +0.6 +μoH (T)28 + +Figure S4 Field dependence of specific heat. Temperature dependence of specific heat data over +temperature (������������p/������������) for high pressure Sr2IrO4 phase, under 0 T (blue), 0.05 T (orange), and 1 T (green). No +significant differences were observed. No ������������ shape anomalies emerged in the whole temperature regime +studied under either case. + + + + + + +le-1 +8 +6 +4 +O T +2 +0.05 T +0 +1 T +0 +0 +50 +100 +150 +200 +T (K)29 + +Figure S5 Specific heat data fitted by Debye and Einstein model. Temperature dependence of +specific heat data over temperature under 0 T (������������p/������������, blue hollow circle) for high pressure Sr2IrO4 phase, +fitted by (a) Debye model, (b) Einstein model, (c) two Debye model with the electronic contribution +included, and weighted Debye model (d) without and (e) with the electronic contribution included. + + + + + +(a) +(b) +le-1 +le-1 +8 +K-2) +6 +6 +4 +4 +Debye +2 +Einstein +0 +10 +10 +0 +0 +0 +50 +100 +150 +200 +0 +50 +100 +150 +200 +T (K) +T (K) +(c) +le-1 +(d) +le-1 +two Debye + yT +OOT +weightedDebye +10 +VT +8 +8 - +Cp/T (I mol-1 K-2) +K-2) +6 +6 +4 +4 +2 +2 +Debyel +Debye +Debye2 +Einstein +0 +0 +0 +50 +100 +150 +200 +0 +50 +100 +150 +200 +T (K) +T (K) +(e) +le-1 +weighted Debye + yT o O T +Cp/T (I mol-1 K-2) +8 +6 +Debye +2 +Einstein +yT +0 +0 +50 +100 +150 +200 +T (K)30 + +Figure S6 Low temperature specific heat data (2–20 K). Specific heat data over temperature (������������p/������������) +plotted versus ������������2 under low temperature regime, 2–20 K, providing the possibility to derivate the +Sommerfeld parameter, ������������. + + + + + + +le-1 +1oo +2 +0 +0 +0 +0 +1 : +0 +0 +0 +888 +0 +. +0 +1 +2 +3 +4 +T2 (K2) +1e231 + +Figure S7 Temperature dependence of resistivity. Temperature dependence of the resistivity data ������������ +plotted as ln ������������ versus (a) ������������−1, (b) ������������−1/2, and (c) ������������−1/4 under 0 T. + + + + + + +(a) +(b) +16 +16 +OT +OT +8 +12 +12 +In(p/(α2 cm)) +In(p/(Ω2 cm)) +8 +8 - +4 +4 - +0 +0.0 +0.1 +0.2 +0.0 +0.2 +0.4 +T-1 (K-1) +T-1/2 (K-1/2) +(c) +16 +O T +8 +12 +In(p/(2 cm)) +8 +4 - +0 +0.2 +0.4 +0.6 +T-1/4 (K-1/4)32 + +Figure S8 Resistivity data fitted by Equation 9 with ������������ of 1/2 and 1/4. (a) The resistivity data ������������ +(blue hollow circle) ranging from 110–300 K fitted by Equation 9 with ������������ of 1/2 (orange line). (b) The +resistivity data ������������ in the low temperature regime ranging from 8–20 K fitted by Equation 9 with ������������ of 1/2. +(c) The resistivity data ������������ in the low temperature regime ranging from 10–20 K fitted by Equation 9 with ������������ +of 1/4. Fitting parameters were summarized in Table S4. The value ������������ of 1/4 is favored over 1/2. + + + + + + +(a) +(b) +16 +fit +fit +00 +data +data +8 - +In(p/(α2 cm)) +In(p/(Ω cm) +12 +4 +8 +0 +4 +0.1 +0.2 +0.1 +0.2 +0.3 +0.4 +T-1/2 (K-1/2) +T-1/2 (K-1/2) +(c) +16 +fit +8 +data +12 +In(p/(2 cm)) +8 - +4 - +0.3 +0.5 +0.7 +T-1/4 (K-1/4)33 + + +Table S4. Summary of fitting parameters for resistivity data. Summary of fitting parameters for +the resistivity data ������������ by Equation 9. R2 is the coefficient of determination. + +������������ = 1 2 +⁄ +Temperature Range / K +������������0 / (Ω cm) +������������0 / K +R2 +110–300 +3.01(4) × 10-5 +6682 +0.9996 +8–20 +79.6(37) +583 +0.9996 + +������������ = 1 4 +⁄ +Temperature Range / K +������������0 / (Ω cm) +������������0 / K +R2 +80–300 +7.82(12) × 10-5 +3.83 × 106 +0.9998 +10–20 +-2.23(4) +4.10 × 105 +0.9999 + + + + + + diff --git a/3tE4T4oBgHgl3EQf0g0b/content/tmp_files/load_file.txt b/3tE4T4oBgHgl3EQf0g0b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..91f955500f175410e10ce211a1ee63f1cf7ac4c2 --- /dev/null +++ b/3tE4T4oBgHgl3EQf0g0b/content/tmp_files/load_file.txt @@ -0,0 +1,1761 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf,len=1760 +page_content='1 Non-centrosymmetric Sr2IrO4 obtained under High Pressure Haozhe Wang1‡, Madalynn Marshall2‡, Zhen Wang3, Kemp W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Plumb4, Martha Greenblatt2, Yimei Zhu3, David Walker5, Weiwei Xie1* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey 08854, USA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Department of Physics, Brown University, Providence, Rhode Island 02912, USA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Lamont Doherty Earth Observatory, Columbia University, Palisades, New York 10964, USA ‡ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' * Email: xieweiwe@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='edu Abstract Sr2IrO4 with strong spin-orbit coupling (SOC) and Hubbard repulsion (U) hosts Mott insulating states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The similar crystal structure, magnetic and electronic properties, particularly the d-wave gap observed in Sr2IrO4 enhanced the analogies to cuprate high-Tc superconductor, La2CuO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The incomplete analogy was due to the lack of broken inversion symmetry phases observed in Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Here, under high pressure and high temperature conditions, we report a non- centrosymmetric Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The crystal structure and its noncentrosymmetric character were determined by single crystal X-ray diffraction and high-resolution scanning transmission electron microscopy (HR-STEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The magnetic characterization confirms the Ir4+ with S = 1/2 at low temperature in Sr2IrO4 with magnetic ordering occurred at around 86 K, where a larger moment is observed than the ambient pressure Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Moreover, the resistivity measurement shows three- dimensional Mott variable-range hopping existed in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' This non-centrosymmetric Sr2IrO4 phase appears to be a unique material to offer further understanding of high-Tc superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 2 Introduction Iridates with strong spin-orbit coupling effects can generate exotic quantum phenomena, such as quantum spin liquid phases, Kitaev magnetism, and possible superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='1-4 Different from most 3d transition metal oxides in which the spin and orbit can be distinct in the energy scale, the spin and orbit interact heavily in 5d transition metal oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Among the Mott insulating 5d transition metal oxides, Sr2IrO45-9 has attracted significant of attention due to its similarity to cuprate high-temperature superconductor, La2CuO410-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' As a single-layer Ruddlesden–Popper compound, Sr2IrO4 crystallizes in a tetragonal lattice with an inversion center (I41/acd, #142) at ambient pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Sr2IrO4 contains stacked IrO2 square lattices where the unit cell is doubled compared to the CuO2 square lattices in high-Tc cuprates as a result of a staggered rotation of IrO6 octahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Although superconductivity is not yet confirmed, many phenomena characteristic of the superconducting cuprates have been observed in electron-and hole-doped iridates including pseudogaps, Fermi arcs, and d-wave gaps .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='13-15 The Ir-d5 electrons in regular IrO6 octahedron occupy the t2g orbitals, which can be approximated as two fully filled spin-orbital coupled Jeff = 3/2 bands and one half-filled Jeff = 1/2 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The Jeff band is split into an upper and lower Hubbard band by on-site Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' According to a previous study, as Sr2IrO4 is cooled below its Néel temperature (TN, ~230 K), the spin-orbit coupled Jeff = 1/2 moments order into a basal plane commensurate Néel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Octahedral rotations in Sr2IrO4 allow for non-zero Dzyaloshinskii-Moriya (DM) interactions that results in a canting of the ordered moments away from the crystallographic axis and a weak ferromagnetic moment per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='16 Such a magnetic transition maintains the inversion symmetry but lowers the rotational symmetry of the system from C4 to C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' However, no additional symmetry breaking has been observed by neutron or X-ray diffraction, which makes the comparison of the iridate to cuprate phenomenology incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' To date, multiple methods have been used to tune the Mott insulating states in Sr2IrO4, for example, isovalent Rh doping on the Ir site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='5,17-23 After partially substituting Ir with Rh, an insulator-to-metal transition can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' However, high pressure was also used to tune the electronic states up to 55 GPa without observing any metallic state in Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='24,25 In this report, we applied the high-pressure (6 GPa) high-temperature (1400 °C) method for synthesizing Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Under such extreme conditions, the obtained Sr2IrO4 remains in a tetragonal structure but without an inversion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The space group was determined by single- 3 crystal X-ray diffraction (SC-XRD) as I4mm (#107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Unlike the ambient pressure phase, the high- pressure phase consists of the single layered IrO2 square lattice, just like CuO2 square in cuprate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Magnetic susceptibility measurement on high pressure Sr2IrO4 indicate a magnetic ordering temperature of approximately 86 K, which is dramatically lower than ambient pressure Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Interestingly, the resistivity data shows three-dimensional Mott variable-range hopping of charge carriers between states localized by disorder with negligible long-range Coulomb interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Discovering the non-centrosymmetric phase in Sr2IrO4 may accelerate the realization of superconductivity and unravel the puzzle in cuprate high-Tc superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 4 Experimental Section High-Pressure Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The ambient pressure Sr2IrO4 phase was prepared accordingly by thoroughly mixing and pelletizing the materials SrCO3 and IrO2 and subsequently heating them to 900 °C then regrinding and reannealing at 1000 °C and subsequently reannealing at 1100 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='26 The ambient pressure Sr2IrO4 was pressurized to 6 GPa in 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' After that, the sample was heated up to 1400 °C and stayed at 1400 °C for 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Another sample was heated to 1400 °C and stayed up to 28 hours to explore the optimal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The sample was cooled down to room temperature before depressurizing to the ambient pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The high-pressure synthesis was performed by statically compressing the sample using the Walker type multi-anvil press27 where the original Sr2IrO4 was placed in a Pt capsule inside an Al2O3 crucible that was inserted into a Cermacast 646 octahedra pressure medium lined on the inside with a LaCrO3 heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Phase Analysis and Chemical Composition Determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The phase identity and purity were examined using a Bruker D2 Phaser powder X-ray diffractometer with Cu K������������ radiation (������������ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='5406 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Room temperature measurements were performed with a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='004° at a scan speed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='55°/min over a Bragg angle (2������������) range of 5–90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' FullProf Suite software28,29 was utilized to analyze the phase information and lattice parameters from a Rietveld refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Structure Determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The room temperature and low temperature (100 K) crystal structure was determined using a Bruker D8 Quest Eco single crystal X-ray diffractometer, equipped with Mo radiation (������������������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='71073 Å) with an ������������ of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0° per scan and an exposure time of 10 s per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' A SHELXTL package with the direct methods and full-matrix least-squares on the F2 model was used to determine the crystal structure of Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='30,31 To confirm the crystal structure, high- resolution scanning transmission electron microscopy (HR-STEM) images were collected and electron diffraction was conducted using a 200 kV JEOL ARM electron microscope equipped with double aberration correctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Samples for TEM analysis were crushed in an agate mortar and deposited directly onto a holey carbon copper grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Physical Properties Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Temperature and field-dependent magnetization, resistivity, and heat capacity measurements were performed with a Quantum Design physical property measurement system (PPMS) under a temperature range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='85–300 K and applied fields up to 9 5 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Electrical resistivity measurements were accomplished with a four-probe method using platinum wires on a pelletized sample of Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The polycrystalline Sr2IrO4 was pressed up to 6 GPa and heated at a lower temperature (100 °C) to eliminate the contribution of grain boundary effect but also keep the phase stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 6 Results and Discussions Exploring New Phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The new Sr2IrO4 phase (I4mm, #107) was formed at 6 GPa from the starting material, ambient pressure Sr2IrO4 (I41/acd, #142).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The synthesis temperatures were set up at 1200 °C and 1400 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The high pressure Sr2IrO4 phase was only produced at 1400 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' To increase the yield and grow larger crystals, the longer heating duration of 28 hours was tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' However, the secondary tetragonal phase Sr3Ir2O7 simultaneously forms once the heating duration was increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' As a result, only 4 hours heating process can produce the specimen consisting mostly of pure phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The resulting Le Bail fitting of the PXRD patterns for the high-pressure phase Sr2IrO4 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' An overlay of the PXRD patterns in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S1 demonstrates the formation of the secondary Sr3Ir2O7 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The pure phase synthesized at 1400 °C for 4 hours was used for the physical property measurements below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 1 Powder X-ray diffraction pattern of the high-pressure Sr2IrO4 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The experimental data (red dots) was modeled with a Rietveld refinement (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The blue line indicates the corresponding residual pattern (difference between observed and calculated patterns) along with Bragg peak positions for Sr2IrO4 (green) and Al2O3 (purple) represented by the vertical tick marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Calc Diff Obs Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=') Sr2lrO4 Al203 10 30 50 70 90 Sr,IrO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 20 (degree) 14mm (#107)7 Crystal Structure and Phase Determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' After 4 hours of treatment at 6 GPa and 1400 °C, single crystals of Sr2IrO4 were formed, subsequently selected, and measured at both 300 K and 100 K using the single crystal X-ray diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' High-pressure Sr2IrO4 crystallizes with good agreement into the tetragonal space group I4mm, as indicated by the single crystal X-ray diffraction (SCXRD) refinement information listed in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Similar to ambient pressure Sr2IrO4, the high pressure Sr2IrO4 phase contains the layers of IrO6 octahedra with intercalated Sr atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The differences between these two are half-c lattice, the disappearance of the inversion center because of the nonsymmetric distortion of IrO6 octahedra, and the disappearance of IrO6 octahedral rotations in the ab-plane in high-pressure Sr2IrO4 compared to the ambient pressure phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 2 are crystal structures and IrO6 octahedra stacking view of ambient pressure Sr2IrO4 (I41/acd), high-pressure Sr2IrO4 (I4mm), and previously reported La2CuO4 (I4/mmm), with Ir-O atomic distance in the IrO6 octahedra highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Atomic site vacancies and site disorder were considered and refined to reveal the O3 atomic site is slightly displaced from the closer ideal 4b site (1/2, 0, z) to the 8d site (x, 0, z) having a statistical occupancy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The disordered model yielded a more reasonable refinement with an R factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='35 and goodness of fit (GOF) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='177 while having only one O3 atomic site resulted in an R factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='62 and GOF of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' As such an angle ������������ can be determined from (1/2 ± ������������, 0, z) with respect to an IrO6 octahedra where the O3 atoms occupy the 4b site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' This structural disorder has been thoroughly discussed for the ambient pressure Sr2IrO4 structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='32 Additionally, the high-pressure Sr2IrO4 phase possesses a nonsymmetric IrO6 octahedra elongation along the c axis, ranging in Ir-O atomic distance from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='94(6)–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='27(6) Å, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 2b, which is in fact the cause of noncentrosymmetric structural character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' This behavior is kind of similar to the prominent feature of ambient pressure Sr2IrO4 that has been speculated to originate from a Jahn Teller distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='33-35 Previous studies under high-pressure have revealed an increase in the IrO6 octahedra elongation with pressurization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='36,37 Compared to ambient pressure Sr2IrO4, one Ir-O along the c -axis is significantly elongated, with the other almost remains the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=', one oxygen atom is driven away from the Ir atom, and thus the repulsion between Ir and the oxygen ligand is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' This will lower the energy of orbitals that contains z contribution and split eg and t2g orbitals, making the crystal field split of Ir d orbitals even more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Together with spin-orbit coupling, this may further remove orbital degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Moreover, as pressure applied for Sr2IrO4, the Ir-O- 8 Ir angle was pushed close to 180°, which is the angle in Cu-O-Cu in La2CuO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The structural disorder was further confirmed at 100 K and the SCXRD refinement details can be found in SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 2 Crystal structure illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Crystal structures, octahedra stacking view along a axis, and along c axis of (a) ambient pressure Sr2IrO4, (b) as-synthesized high pressure Sr2IrO4, and (c) previously reported La2CuO4, with Ir(Cu)O6 octahedra and Ir(Cu)-O atomic distances presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Green, blue, dark green, dark blue, and red atoms represent Sr, Ir, La, Cu, and O atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Single-layer square net is also highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='98(1) A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='27(6) A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='95(1) A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='93(1) A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='94(1) A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='94(6) A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='11(1) A Sr,IrO4 Sr,lrO4 La,CuO 14,/acd (#142) 14mm (#107) 14/mmm(#139)9 Transmission Electron Microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The non-centrosymmetric space group and loss of IrO6 octahedral rotation, as well as the oxygen distortion and defects in Sr2IrO4, can at first, be surprisingly interesting, thus high-pressure Sr2IrO4 was investigated by transmission electron microscopy (TEM) to characterize its crystallographic nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image was obtained along the a axis shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The TEM diffraction patterns projected down the crystalline [100] axis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 3b) allowed for the determination of the orientation of the images through the d002 spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The c-axis parameter is ~12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='8 Å, agreeing with the single crystal XRD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The electron diffraction and imaging study confirmed the high quality of the nanoscale ordering in the specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' However, the fractional spots 1/2 (110)/(1-10) were observed by TEM electron diffraction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' As is known that IrO6 tilt/rotation along the c-axis would not introduce these fractional spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Such fractional reflection spots are related to the ordering of oxygen vacancy, which is consistent with single crystal X-ray diffraction results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 3 Transmission electron microscopy study of high pressure Sr2IrO4 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) HAADF- STEM image taken along a axis from a large area showing the high quality of the crystal Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (b) The zoom-in HAADF image shows the projected structure in the [100] direction, with a crystal model superimposed, where Sr (green), Ir (blue), and O (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (c) The diffraction pattern took along the [100] direction which is consistent with the simulated pattern (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 3e) based on the crystal model determined by single crystal X-ray diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (d) SAED pattern along the [001] direction showing fractional spots of 1/2 (110)/(1-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (e) Simulated diffraction pattern and (f) projected crystal structure along the [001] direction based on the crystal structure determined by 220 020 X 200 000 003 220 020 20 [100] 110 200 10 [001]10 SCXRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The fractional spots observed in TEM were marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The single crystal structure of Sr2IrO4 with oxygen distortion was confirmed by both single crystal X-ray diffraction and TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Weak Ferromagnetic Ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' To study the magnetic properties of the high pressure Sr2IrO4 phase, the temperature-dependent susceptibility was measured under field cooled warming (FCW) and field cooled cooling (FCC) mode at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='1 T shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' No significant differences between FCW and FCC were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' At about 150 K, the susceptibility goes below 0, indicating a diamagnetic contribution in the system, which suggests the possible breakdown of Curie-Weiss behavior at high temperatures in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The data between 80–140 K was modeled with the modified Curie-Weiss law (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 1), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 4b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S2b, ������������ = ������������0 + ������������ ������������ − ������������cw (1) where ������������������������������������ is the paramagnetic Curie temperature, ������������0 is the temperature independent susceptibility and ������������ is the Curie constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' From the fitting, the Curie temperature, ������������������������������������, of 86(7) K was found to be comparable to the magnetic ordering temperature ������������������������ ~84 K, as determined from the minimum in the temperature derivative of ������������ (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S2a for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The magnetic ordering temperature, consequently, decreases when compared to ambient pressure Sr2IrO4, which has a ������������������������ ~240 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='39,40 On the other hand, it can be assumed that the Tc significantly decreases as the angle of Ir-O-Ir is more close to 180 °, which is the one observed in Cu-O-Cu in high Tc superconductor La2CuO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The fitting also gave a negative ������������0 of -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='9(9)×10-3 emu mol-1 Oe-1, which provided a potential opportunity to extrapolate our Curie-Weiss fit to higher temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Finally, up to 160 K was included (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S2c) and the fit yielded the effective moment ������������eff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2(2) µB/Ir, which is more agreeable with the Hund’s-rule value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='73 µB/Ir for S = 1/2 than the reported ������������eff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='33 µB/Ir for ambient pressure Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Furthermore, the magnetization of high pressure Sr2IrO4 was measured as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 4c up to 9 T at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' It appears to saturate at ~3 T at which the magnetic saturation moment (������������������������������������������������) was determined to be ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='046 µB/Ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' This value is significantly lower than the theoretical value of 1/3 µB f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='u−1, however, similar to the previously reported moment for the ambient pressure Sr2IrO4 phase, which originates from spin canted antiferromagnetic (AFM) order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='39 This could also explain why the weak ferromagnetic behavior observed in the temperature 11 dependence of magnetic susceptibility gives such a low value of moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' However, unlike the ambient pressure Sr2IrO4 phase, the magnetization reaches a maximum at around 3 T at which point the magnetization decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' It turned out that diamagnetic transition was observed under higher fields at the respective temperatures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=', see the 50 K and 100 K data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' At 300 K, a complete diamagnetic behavior was shown, consistent with ������������ < 0 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Subtracting this by linearly fitting data from 7–9 T, the ������������������������������������������������ was modified to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='067 µB/Ir at 2 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='014 µB/Ir at 100 K, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 4d and 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Magnetic hysteresis was observed in the system under 2 K from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='6 T to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='6 T, presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S3, which could be interpreted as small canting of the moments existed in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 4 Magnetization in the dependence of temperature and field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) Temperature dependence of magnetic susceptibility ������������ at 1000 Oe under FCW and FCC mode ranging from 2–300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' No significant difference was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (b) The modified inverse magnetic susceptibility data (FCW, 80–140 K, blue hollow circle) fitted with the modified Curie-Weiss model (orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (c) Field dependence of magnetization up to 9 T at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (d) Derivation of ������������sat at 2 K by linearly fitting the magnetization data from 7–9 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (e) Derivation of ������������sat at 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' No Magnetically Induced Anomalies Observed in Specific Heat Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' To confirm the magnetic transition, the specific heat over the temperature range of 2–200 K was measured under 0 T with a polycrystalline pelletized sample of Sr2IrO4, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Measurements (a) (b) 1e 2 1e3 2 FCW Cw fit Oe) FCC 0 FCW mol 0 8 oo 0 0 0 1 090 0 4 8 08 X 8 0 0 0 100 200 300 80 130 180 230 280 T (K) T (K) (c) (d) (e) 1e 2 8 0 10 20 K (μB per Ir ion) 4 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='4 M 50 K 8 100 K 0 2K 100K 300 K "2 K" "100 K" 9 6 3 3 6 9 0 36 9 0 36 9 μoH (T) μoH (T) μoH (T)13 under applied fields of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='05 T and 1 T in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S3 were additionally tested to conclude no significant deviation from the 0 T specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' No ������������ shape anomalies were observed at the whole temperature regime studied, which may result from higher temperature regions being heavily dominated by the phonon contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The specific heat data were fitted by the Debye model (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 2), and Einstein model (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 3), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S4a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The Debye and Einstein temperatures could then be determined as 417(2) K and 306(2) K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' However, neither of these two described the experimental data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' ������������D = 9������������������������ � ������������ ������������D � 3 � ������������4������������������������ (������������������������ − 1)2 ������������������������ ������������D ������������ ⁄ 0 (2) where ������������ is the number of atoms per formula unit, ������������ is the gas constant, and ������������������������ is the Debye temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' ������������E = 3������������������������ �������������E ������������ � 2 ������������ ������������E ������������ ������������� ������������E ������������ − 1� −2 (3) where ������������ is the number of atoms per formula unit, ������������ is the gas constant, and ������������������������ is the Einstein temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The specific heat data was further fitted with two Debye model (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 4) and weighted Debye model (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 5), with and without the electronic contribution included, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 5a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S4c, d, and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The data was found to be described well with two Debye model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 5a), and the Debye temperatures, ������������������������1 of 235(1) K, ������������������������2 of 708(5) K was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' At low temperatures, the first Debye mode has a larger contribution to the specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Within the temperature regime studied, the expected Dulong-Petit value of 3������������������������ is not recovered, and this can be explained by the high value of ������������������������2, which means that the specific heat will plateau at ������������ ≫ ������������������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The fitting also yields ������������������������1 of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='20(3) and ������������������������2 of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='51(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The sum of these two seems a little larger than the expected value of 7 for Sr2IrO4, which may be attributed to the impurity of Srn+1IrnO3n+1, lack of electron contribution, or overestimation of photon contribution in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Once the electron contribution term was included, ������������������������1 was slightly shifted to 238(2) K and the sum of ������������������������1 and ������������������������2 went down to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='52(11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='������������ = 9������������D1������������ � ������������ ������������D1 � 3 � ������������4������������������������ (������������������������ − 1)2 ������������������������ ������������D1 ������������ ⁄ 0 + 9������������D2������������ � ������������ ������������D2 � 3 � ������������4������������������������ (������������������������ − 1)2 ������������������������ ������������D2 ������������ ⁄ 0 (+������������������������) (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='where ������������������������1 and ������������������������2 are Debye temperatures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' ������������������������1 and ������������������������2 are the oscillator strengths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' and ������������������������ is the electron contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' ������������ = 9������������D������������ � ������������ ������������D � 3 � ������������4������������������������ (������������������������ − 1)2 ������������������������ ������������D ������������ ⁄ 0 + 3������������E������������ �������������E ������������ � 2 ������������ ������������E ������������ ������������� ������������E ������������ − 1� −2 (+������������������������) (5) where ������������������������ and ������������������������ are the Debye and Einstein temperatures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' ������������������������ and ������������������������ are the oscillator strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' It should be noted that the magnetic contribution cannot be quantitatively extracted from the specific heat data as the phonon contribution cannot be distinguished from the magnetic contribution due to the lack of a nonmagnetic analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' At a low-temperature regime, of 2–20 K, the specific heat was measured, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The data ranging from 2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2 K was fitted with Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 6, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' ������������p ������������ = ������������ + ������������������������2 (6) From this fitting, a ������������ and ������������ value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0153(2) J mol-1 K-2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='1(2) × 10-4 J mol-1 K-3 corresponding to the electronic and phonon contributions to the specific heat, respectively, could be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The ������������ value recovered the Debye temperature (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 7) to be 268(2) K, which is much closer to ������������������������1 rather than ������������������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' It falls out of the temperature interval, 300–350 K, where iridates most commonly exhibit Debye temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='41 ������������D = �12������������4 5������������ ������������������������� 1 3 (7) 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 5 Specific heat data fitting of high pressure Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) Temperature dependence of specific heat over temperature (������������p ������������ ⁄ ) for high-pressure Sr2IrO4 fitted by two Debye model in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Green and red dotted lines refer to the 1st and 2nd Debye model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (b) ������������p ������������ ⁄ vs ������������2 between 2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2 K fitted with Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 6 (orange dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Mott Variable-range Hopping (VRH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' It is critical to investigate the electrical conductivity in the high pressure Sr2IrO4 phase to compare to the Mott insulator ambient pressure Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Temperature-dependent resistivity measurements were performed from 2–300 K with an applied field up to 9 T on a pelletized polycrystalline sample of the high pressure Sr2IrO4 phase, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' No significant field dependence was observed, which indicates the insignificance of magnetoresistance for the high pressure Sr2IrO4 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' This may be not unexpected considering the small saturation moment under fields (see the discussion above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' At room temperature and 0 T, the resistivity is relatively low, only around 4 Ω cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' However, the resistivity is increases by 6 orders of magnitude upon cooling, indicating the semiconducting character of the high-pressure Sr2IrO4 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' To further analyze its behavior, we first tried to model the temperature dependence of ������������ with the Arrhenius law (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 8), ������������ = ������������0������������������������������������ ������������������������ ⁄ (8) (a) (b) 1e 1 1e 2 two Debye OT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' y+βT2 OOT 8 Cp/T ( mol 1 K 2) 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='4 6 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0 6 2 Debyel Debye2 0 0 50 100 150 200 4 8 12 16 T (K) T2 (K2)16 where ������������0 is the residual resistivity, ������������������������ is the activation energy, and ������������ is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' However, ������������ could not be fitted well to a ������������������������, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S7a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=', the Arrhenius law is not well obeyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Then its temperature dependence was fitted by law in the form (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 9) with ������������ of 1/2 and 1/4, ������������ = ������������0������������(������������0 ������������ ⁄ )������������ (9) where ������������0 is the residual resistivity, and ������������0 is the characteristic temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The fitting results were presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 6b, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S8, with parameters summarized in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The value ������������ of 1/4 is favored over 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' While both of them indicate three-dimensional Mott variable-range hopping of charge carriers between localized states, the weaker temperature dependence with ������������ of 1/4 implies negligible long-range Coulomb interactions between localized electrons in the temperature regime studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' This behavior is also reported in the ambient pressure Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='42 To explore the harboring quantum states in the high-pressure Sr2IrO4 phase, further examination of its transport properties is warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 6 Details of field and temperature dependent resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) Temperature dependence of resistivity data for high-pressure phase Sr2IrO4 under fields up to 9 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' No significant derivation was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (b) The resistivity ������������ (blue hollow circle) ranging from 80–300 K was fitted by Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 9 with ������������ of 1/4 (orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' A linear relationship was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) (b) 107 OT fit 1 T o data 3 T 8 105 5 T In(p/(2 cm)) p (Q cm) 7 T 9T 103 4 101 0 0 100 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='5 T (K) T 1/4 (K 1/4)17 Conclusion In summary, we reported the non-centrosymmetric Sr2IrO4 phase obtained under high pressure and high temperature conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The ferromagnetic ordering temperature decreases significantly to ������������c ~86 K from ~240 K in the ambient pressure Sr2IrO4, while there may be a possible breakdown of the Curie-Weiss law under higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Diamagnetism was observed under room temperature and higher fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' No anomalies indicating magnetic ordering were observed in the specific heat measurements, where a greater photon contribution was obtained from the low-temperature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Temperature-dependent resistivity revealed three- dimensional Mott variable-range hopping of charge carriers between states localized by disorder with negligible long-range Coulomb repulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Further transport measurements, together with first-principal calculation, are expected to explore the electronic properties of the high-pressure Sr2IrO4 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Such a system may offer a promising platform to unravel the mystery of high-Tc superconductivity in cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Acknowledgments The work at Rutgers was supported by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' DOE-BES under Contract DE-SC0022156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The electron microscopy work at BNL was supported by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' DOE-BES, Materials Sciences and Engineering Division under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' DESC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Supporting Information Single crystal X-ray diffraction data at room temperature and 100 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Anisotropic displacement parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Atomic coordinates and equivalent isotropic displacement parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' PXRD overlay of Sr2IrO4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Magnetic susceptibility and Curie-Weiss fitting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Magnetic hysteresis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Field dependence of specific heat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Specific heat data fitted by Debye and Einstein model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Low temperature specific heat data (2–20 K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Temperature dependence of resistivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Resistivity data fitted by Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 9 with ������������ of 1/2 and 1/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Summary of fitting parameters for resistivity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 18 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Takagi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Takayama, T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Grüninger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Fingerprints of Kitaev physics in the magnetic excitations of honeycomb iridates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 2020, 2, 043094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Zhou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Possible superconductivity in Sr2IrO4 probed by quasiparticle interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 2015, 5, 9251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Mitchell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Sr2IrO4: Gateway to cuprate superconductivity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' APL Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 2015, 3, 062404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Reber, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Parham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Plumb, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Qi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Cao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Hsieh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Evidence of an odd-parity hidden order in a spin–orbit coupled correlated iridate.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Cao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Structure symmetry determination and magnetic evolution in Sr2Ir1-xRhxO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' B 2015, 92, 201112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 22.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Jin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Cao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' van Veenendaal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Pressure Tuning of the 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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Matter 2006, 18, 8205-8216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Pallecchi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Buscaglia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Buscaglia, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Gilioli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Lamura, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Telesio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Cimberle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Marré, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Thermoelectric behavior of Ruddlesden–Popper series iridates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Matter 2016, 28, 065601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Cao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Bolivar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' McCall, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Crow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Guertin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Weak ferromagnetism, metal-to- nonmetal transition, and negative differential resistivity in single-crystal Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' B 1998, 57, R11039-R11042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 21 Non-centrosymmetric Sr2IrO4 obtained under high pressure Haozhe Wang1‡, Madalynn Marshall2‡, Zhen Wang3, Kemp W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Plumb4, Martha Greenblatt2, Yimei Zhu3, David Walker5, Weiwei Xie1* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey 08854, USA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Department of Physics, Brown University, Providence, Rhode Island 02912, USA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Lamont Doherty Earth Observatory, Columbia University, Palisades, New York 10964, USA ‡ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' * Email: xieweiwe@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='edu Supporting Information Table S1 Single crystal X-ray diffraction data at room temperature and 100 K .' metadata={'source': 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S2 Table S2 Anisotropic displacement parameters .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S3 Table S3 Atomic coordinates and equivalent isotropic displacement parameters .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S4 Figure S1 PXRD overlay of Sr2IrO4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': 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S5 Figure S2 Magnetic susceptibility and Curie-Weiss fitting .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S7 Figure S4 Field dependence of specific heat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S8 Figure S5 Specific heat data fitted by Debye and Einstein model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S9 Figure S6 Low temperature specific heat data (2–20 K) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S10 Figure S7 Temperature dependence of resistivity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='. S11 Figure S8 Resistivity data fitted by Equation 9 with ������������ of 1/2 and 1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S12 Table S4 Summary of fitting parameters for resistivity data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' S13 22 Table S1 Single crystal X-ray diffraction data at room temperature and 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Temperature Room Temperature 100 K Refined formula Sr2IrO4 Sr2IrO4 FW (g/mol) 431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='44 431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='44 Space group I4mm I4mm a (Å) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='8860(5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='8777(5) c (Å) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='826(2) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='825(2) V (Å3) 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='69(6) 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='85(6) Extinction Coefficient N/A N/A ������������ range (°) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='177–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='030 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='177–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='075 # of reflections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Rint 1088;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0627 1286;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0591 # of independent reflections 267 264 # of parameters 23 23 R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' ωR2 (������������ > ������������������������(������������)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0409;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0312;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0443 Goodness of fit (GOF) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='177 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='125 Diffraction peak and hole (e-/ Å3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='658, -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='492 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='359, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='96 23 Table S2 Anisotropic displacement parameters for Sr2IrO4 at room temperature and 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Sr2IrO4 at Room Temperature Atom U11 U22 U33 U23 U13 U12 Ir1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0018(6) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0018(6) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0021(6) 0 0 0 Sr1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='026(7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='026(7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='007(7) 0 0 0 Sr2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='001(4) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='001(4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='005(6) 0 0 0 O1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='03(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='03(2) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='02(2) 0 0 0 O2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='006(10) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='006(10) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='023(18) 0 0 0 O3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='04(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='003(11) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='01(3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='02(3) 0 Sr2IrO4 at 100 K Atom U11 U22 U33 U23 U13 U12 Ir1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0004(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0004(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0012(7) 0 0 0 Sr1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='005(8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='005(8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='005(4) 0 0 0 Sr2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='002(8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='002(8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='000(4) 0 0 0 O1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='005(8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='005(8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='033(15) 0 0 0 O2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='012(10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='012(10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='033(14) 0 0 0 O3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='009(10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='005(7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='006(8) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='01(3) 0 24 Table S3 Atomic coordinates and equivalent isotropic displacement parameters for Sr2IrO4 at room temperature and 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (Ueq is defined as one-third of the trace of the orthogonalized Uij tensor (Å2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Sr2IrO4 at Room Temperature Atom Wyck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' x y z Occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Ueq Ir1 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='1513(13) 1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0019(4) Sr2 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='5044(4) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='020(4) Sr1 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='79985(2) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='001(3) O1 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='328(4) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='013(18) O2 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='000(4) 1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='011(7) O3 8d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='419(9) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='661(7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='010(15) Sr2IrO4 at 100 K Atom Wyck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' x y z Occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Ueq Ir1 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='1489(7) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0001(3) Sr2 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='5019(4) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='005(5) Sr1 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='7978(2) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='002(5) O1 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='321(3) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='008(6) O2 2a 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='000(3) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='003(8) O3 8d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='412(4) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='649(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='007(4) 25 Figure S1 Powder X-ray diffraction pattern overlay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The experimental data of high pressure Sr2IrO4 phase synthesized at 1400 °C for ~4 hrs (black line) and ~28 hrs (red line) were presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Bragg peak positions are indicated as Sr2IrO4 and Sr3Ir2O7 with green and purple vertical tick marks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' TT 1412 ~4 hrs GG 1418 28 hrs Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=') Sr2lrO4 Sr3lr2Q7 10 30 50 70 90 2e (degree)26 Figure S2 Magnetic susceptibility and Curie-Weiss fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) Temperature derivative of magnetic susceptibility ������������ at 1000 Oe under FCW mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The minimum at around 84 K was highlighted by red circle and an arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (b) The inverse magnetic susceptibility data (FCW, 80–140 K, blue hollow circle) fitted with the modified Curie-Weiss model (orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (c) The Curie-Weiss fit was further extrapolated to 160 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) (b) 1e 3 1e3 K 1) Cw fit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2 FCW 1 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='6 8 2 1/x 000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0 000 50 60 70 80 ° 90 100 80 100 120 140 T (K) T (K) (c) 1e2 mol Oe) 4 CW fit FCW N 0% 00 0 80 100 120 140 160 T (K)27 Figure S3 Magnetic hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Magnetic hysteresis observed in the high pressure Sr2IrO4 phase at 2 K ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='6 T to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 1e-2 0 2K 1 M (μB per Ir ion) -1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='6 μoH (T)28 Figure S4 Field dependence of specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Temperature dependence of specific heat data over temperature (������������p/������������) for high pressure Sr2IrO4 phase, under 0 T (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='05 T (orange), and 1 T (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' No significant differences were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' No ������������ shape anomalies emerged in the whole temperature regime studied under either case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' le 1 8 6 4 O T 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='05 T 0 1 T 0 0 50 100 150 200 T (K)29 Figure S5 Specific heat data fitted by Debye and Einstein model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Temperature dependence of specific heat data over temperature under 0 T (������������p/������������, blue hollow circle) for high pressure Sr2IrO4 phase, fitted by (a) Debye model, (b) Einstein model, (c) two Debye model with the electronic contribution included, and weighted Debye model (d) without and (e) with the electronic contribution included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) (b) le-1 le-1 8 K-2) 6 6 4 4 Debye 2 Einstein 0 10 10 0 0 0 50 100 150 200 0 50 100 150 200 T (K) T (K) (c) le-1 (d) le-1 two Debye + yT OOT weightedDebye 10 VT 8 8 - Cp/T (I mol-1 K-2) K-2) 6 6 4 4 2 2 Debyel Debye Debye2 Einstein 0 0 0 50 100 150 200 0 50 100 150 200 T (K) T (K) (e) le-1 weighted Debye + yT o O T Cp/T (I mol-1 K-2) 8 6 Debye 2 Einstein yT 0 0 50 100 150 200 T (K)30 Figure S6 Low temperature specific heat data (2–20 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Specific heat data over temperature (������������p/������������) plotted versus ������������2 under low temperature regime, 2–20 K, providing the possibility to derivate the Sommerfeld parameter, ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' le 1 1oo 2 0 0 0 0 1 : 0 0 0 888 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' 0 1 2 3 4 T2 (K2) 1e231 Figure S7 Temperature dependence of resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Temperature dependence of the resistivity data ������������ plotted as ln ������������ versus (a) ������������−1, (b) ������������−1/2, and (c) ������������−1/4 under 0 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) (b) 16 16 OT OT 8 12 12 In(p/(α2 cm)) In(p/(Ω2 cm)) 8 8 4 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='4 T 1 (K 1) T 1/2 (K 1/2) (c) 16 O T 8 12 In(p/(2 cm)) 8 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='6 T 1/4 (K 1/4)32 Figure S8 Resistivity data fitted by Equation 9 with ������������ of 1/2 and 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) The resistivity data ������������ (blue hollow circle) ranging from 110–300 K fitted by Equation 9 with ������������ of 1/2 (orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (b) The resistivity data ������������ in the low temperature regime ranging from 8–20 K fitted by Equation 9 with ������������ of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (c) The resistivity data ������������ in the low temperature regime ranging from 10–20 K fitted by Equation 9 with ������������ of 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Fitting parameters were summarized in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' The value ������������ of 1/4 is favored over 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' (a) (b) 16 fit fit 00 data data 8 In(p/(α2 cm)) In(p/(Ω cm) 12 4 8 0 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='4 T 1/2 (K 1/2) T 1/2 (K 1/2) (c) 16 fit 8 data 12 In(p/(2 cm)) 8 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='7 T 1/4 (K 1/4)33 Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Summary of fitting parameters for resistivity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' Summary of fitting parameters for the resistivity data ������������ by Equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' R2 is the coefficient of determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content=' ������������ = 1 2 ⁄ Temperature Range / K ������������0 / (Ω cm) ������������0 / K R2 110–300 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='01(4) × 10 5 6682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='9996 8–20 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='6(37) 583 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='9996 ������������ = 1 4 ⁄ Temperature Range / K ������������0 / (Ω cm) ������������0 / K R2 80–300 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='82(12) × 10 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='83 × 106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='9998 10–20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='23(4) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='10 × 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} +page_content='9999' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf'} diff --git a/49E2T4oBgHgl3EQfkAeM/content/tmp_files/2301.03974v1.pdf.txt b/49E2T4oBgHgl3EQfkAeM/content/tmp_files/2301.03974v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fadc96b214bd0b5a7cca8df73d60270f83a21b1e --- /dev/null +++ b/49E2T4oBgHgl3EQfkAeM/content/tmp_files/2301.03974v1.pdf.txt @@ -0,0 +1,979 @@ +Hydrogen storage in Li functionalized [2,2,2]paracyclophane +at cryogenic to room temperatures: A computational quest +Rakesh K. Sahoo, Sridhar Sahu + +Computational Materials Research Lab, Department of Physics, Indian Institute of Technology +(Indian School of Mines) Dhanbad, India +Abstract +In this work, we have studied the hydrogen adsorption-desorption properties, and storage capacities of Li +functionalized [2,2,2]paracyclophane (PCP222) using dispersion-corrected density functional theory and +molecular dynamic simulation. The Li atom was found bonded strongly with the benzene ring of PCP222 +via Dewar interaction. Subsequently, the calculation of the diffusion energy barrier revealed a significantly +high energy barrier of 1.38 eV, preventing the Li clustering on PCP222. The host material, PCP222-3Li +adsorbed up to 15H2 molecules via charge polarization mechanism with an average adsorption energy of +0.145 eV/5H2, suggesting physisorption type of adsorption. The PCP222 functionalized with three Li atom +showed maximum hydrogen uptake capacity up to 8.32 wt% which was fairly above the US-DOE criterion. +The practical H2storage estimation revealed that the PCP222-3Li desorbed 100% of adsorbed H2 molecules +at the temperature range of 260 K-300 K and pressure range of 1-10 bar. The maximum H2 desorption +temperature estimated by the Vant-Hoff relation was found to be 219 K and 266 K at 1 bar and 5 bar, +respectively. The ADMP molecular dynamics simulations assured the reversibility of adsorbed H2 and the +structural integrity of the host material at sufficiently above the desorption temperature (300K and 500K). +Therefore, the Li-functionalized PCP222 can be considered as a thermodynamically viable and potentially +reversible H2 storage material below room temperature. +Keywords: Hydrogen storage, DFT, Van’t-Hoff equation, ADMP, [2,2,2]paracyclophane, +PCP222, ESP +1 Introduction +The excessive consumption of traditional fossil fuels has not only led to the depletion of the energy +supplies but also has emerged as the prime cause of environmental pollution. The global +consumption of petroleum and other traditional fossil fuel is anticipated to expand up to 56% by +the year 2040 and the crude oil supply is expected to endure until 2060 if the current demand trend +continues[1]. Thus, it is essential to develop alternative energy sources that are free from the +drawbacks of traditional fossil fuels. To meet the world’s energy demand and reduce the pollution +caused by fossil fuels, hydrogen has been considered as a plausible alternative due to its natural +abundance, environmental friendliness, and regenerative properties. One of the distinctive quality + +of hydrogen is that it produces a large amount of energy per unit mass (120 MJ/kg) without +releasing any pollutant by-products [2, 3]. Despite these benefits, however, the use of hydrogen in +practice is limited due to the obstacle of finding the most appropriate and affordable way to store +and deliver hydrogen under normal environmental conditions. As per the criteria proposed by the +United State department of energy (DOE-US) an effective hydrogen storage material should have +a minimum storage capacity of up to 5.5 wt% by the year 2025 at moderate thermodynamics[5, 6]. +In addition, as reported by many authors, the adsorption energy of hydrogen molecules of an +effective storage materials should be in the range of 0.1 eV/H2 to 0.6 eV/H2[4]. +Though, numerous varieties of materials such as; metal hydrides [7, 8], graphene [9, 10], +metal alloys [11, 12], metal-organic frameworks (MOF) [13, 14], covalent-organic frameworks +(COF) [15] and carbon nanostructures [16, 17] etc have been investigated both theoretically and +experimentally as potential hydrogen storage materials, but there are many drawbacks and +unsolved issues to handle. The metal hydrides and complex hydrides store hydrogen via +chemisorption process which is highly irreversible and prevents easy desorption of hydrogen [18]. +For example, Al(BH4)3, which yields hydrogen uptake capacity of up to 16.9 wt%, has high +desorption temperature (about 1000 K) that makes the material non-effective practical reversible +hydrogen storage applications[19]. Under ambient conditions, Mg-metal hydrides have a storage +capacity up to 7.6 wt%; however, it can only be used for 2-3 cycles[20]. Tavhare et al. studied the +hetero atom substituted Ti-benzene and reported an H2 uptake capacity up to 5.85 wt%, but at +relatively high desorption temperature (1193 K)[21]. Furthermore, MOF and COF applications are +constrained in the practical H2 storage field due to the difficulties of their heavy structure and +challenging step-wise production [22]. +The efficient use of carbonaceous materials as hydrogen storage media was initially +reported by Dillon et al. [23]. Carbonaceous materials are appropriate for H2 storage due to their +unique qualities such as, large surface area, high porosity, better stabilities, and low densities. +However, the early findings have shown that these pure materials are weakly interact with the +hydrogen molecules (with BE ~4-5 kJ/mol), thus impractical for realistic hydrogen storage at +ambient environment [24, 25]. Meanwhile, carbon-based pure substrates are excellent materials +for hydrogen storage at cryogenic temperatures. For instance, pure single wall carbon nanotube + +(SWCNT) can store hydrogen molecules up to 8.25 wt%, with a substantially lower desorption +temperature of 80 K [26]. +It has been reported that the H2 interaction strength and the desorption temperature can be +tuned by integrating pure carbon substrates with alkali metal (AM)(Li, Na, and K), alkali earth +metals (Be, Mg, Ca), and transition metals (TM)(Sc, Ti, V, Y.)[27, 28, 29]. Numerous theoretical +investigations showed that integrating AMs and TMs with the carbon/borane substrates can bind +H2 molecules via charge polarization and the Kubas mechanism [30, 31]. The metallic atom +decorated fullerenes were first explored to investigate the impact of metal integration on pure +carbon substrates. According to studies by Sun et al., Li decorated fullerene could show a storage +capacity of 9 wt%; however, the hydrogen adsorption energy was estimated to be 0.075 eV/H2, +which is much lower than the DOE criterion [32]. The Li and Na-loaded C60 revealed H2 uptake +capacities of 4.5 wt% and 4 wt%, respectively, that were significantly below the target of DoE [33]. +Experimental studies of transition metals like V and Pd decorated CNT reveal 0.66 wt% and 0.69 +wt% of hydrogen capacity respectively, while pure CNT has 0.53 wt% of storage capacity [34]. +Sahoo et al. reported storage of H2 on Li and Sc doped C8N8 cage via Niu-Rao-Jena and Kubas +interaction and estimated a desorption temperature of 286 K and 456 K, respectively [35]. The Li +and Na decorated on C24 fullerene could adsorb H2 molecules, with average hydrogen binding +energies of 0.198 eV/H2 and 0.164 eV/H2 and led to storage capacity up to 12.7 wt% and 10 wt %, +respectively [36]. Recently, we have investigated the H2 storage on alkali metal decorated +C20 fullerene and found the molecular hydrogen are physisorbed on host material via charge +polarization mechanism with desorption temperature of 182 -191 K [37]. Each Li and Na atom on +C20 could uptake up to 5H2 molecules with a total gravimetric storage capacity of 13.08 wt % and +10.82 wt%, respectively, and the H2 binding energies found in the range of 0.12 eV—0.13 eV/H2. +Other carbonaceous materials such as functionalized organometallic compounds, +macrocyclic compounds have also been reported recently as potential candidates for hydrogen +storage. For example, Mahamiya et al. revealed the H2 storage capacities of 11. 9 wt % in K and +Ca decorated biphenylene with an average adsorption energy of 0.24-0.33 eV [38]. Y atom doped +zeolite shows high capacity adsorption of H2 with binding energy 0.35 eV/H2 and the desorption +energy of 437K for fuel cells[39]. Lithium-doped Calixarenes show an excellent hydrogen storage +behaviour but at very low up to 100 K [40]. Calix[4]arene functionalized with Li metal reveals 10 + +wt% storage capacity via Kubas—Niu—Rao—Jena interaction, and all most all H2 desorbed at a +temperature of 273 K [41]. +Macrocyclic compounds such as, paracyclophane (PCP), a subgroup derivative of +cyclophanes, contains aromatic benzene rings, and their nomenclature is established on the arene +substitution pattern. For a [n,n]paracyclophane, the number of -CH2- moiety connecting the +successive benzene rings is indicated by the number in the square bracket [42]. Due to the +existence of aromatic benzene rings in the geometry, PCPs are easy to synthesize experimentally +and can be functionalized with metal atoms, making them a viable choice for hydrogen storage +candidates. A report on Li and Sc functionalized [4,4]paracyclophane revealed the hydrogen +uptake capacity up to 11.8 wt% and 13.7 wt% with an average adsorption energy of 0.08 eV/H2 and +0.3 eV/H2 respectively [43]. Sahoo et al. recently studied the H2 storage capacity of +[1,1]paracyclophane functionalized with Sc and Y metals and found an H2 gravimetric storage +capacity of 8.22 wt% and 6.33 wt%, respectively, with an average adsorption energy 0.36 +eV/H2[44]. They reported the H2 desorption temperature of 439 K and 412 K for Sc and Y doped +PCP11, respectively, at 1 atm. The hydrogen molecules are physisorbed on Li, and Sc decorated +paracyclophane via Kubas-Niu-Jena interaction and show a storage capacity of 10.3 wt%, as +reported by Sathe et al. [45]. Many more alkali metal-doped macrocyclic compounds have also +been investigated for hydrogen storage candidates and found the storage capacity above the DOE +target; however very few reported the practical H2 capacity at various thermodynamic +conditions[46, 47]. Though few of PCP-based hydrogen storage systems are available in literature, +the [2,2,2]paracyclophane (PCP222) which is experimentally synthesized by Tabushi et al.[48] is +yet to be explored as hydrogen storage material. Because Li, the lightest alkali metal atom and can +hold H2 molecules via charge polarization mechanism, it can serve as better sorption center on +PCP. +Therefore, in the current work, we intend to investigate the hydrogen storage properties +and potential of Li functionalized [2,2,2]paracyclophane (PCP222). We chose the PCP222 for +hydrogen storage because it is already experimentally synthesized and can be decorated with metal +atoms to form a hydrogen storage media with a high hydrogen uptake capacity. The Li atoms are +functionalized as sorption centers; this is because the light-weight metal doping method is an +effective way to increase the capacity of H2 storage. Li being the lightest alkali metal atom, + +received a lot of attention to for hydrogen sorption application. Though there are few reports +available based on hydrogen adsorption mechanism on metal doped macrocyclic organic +molecules and other Li decorated nanostructures, our work is the first to reveal the efficiency of +Li functionalized PCP222 using the atomistic MD simulation, practical storage capacity and +diffusion energy barrier estimation +2 Theory and Computation +The theoretical computations are carried out on [2.2.2] paracyclophane (PCP222) and their +hydrogenated derivatives within the framework of density functional theory (DFT)[49]. The +modern range separated hybrid functional wB97Xd is used, and molecular orbitals (MO) are +defined as linear combination of atom centered basis functions, with all atoms using the valence +diffuse and polarization function 6-311+G(d,p) basis sets. The wB97Xd, a long range separated +form of Becke’s 97 functional, also adds Grimme’s D2 dispersion correction[50, 51]. It is worth +mentioning that the wB97Xd is a reliable approach to investigate the non-covalent interaction of +metal doped organic molecules and their thermochemistry. The harmonic frequencies of all the +studied structures are calculated to confirm that they are truly in the ground state on the potential +surface. +Some of the crucial quantitative metrics, including, binding energy of metal atom on host, +average H2 adsorption energy and successive H2 desorption energy must be determined in order to +analyze the mechanism of hydrogen storage. +The binding strength of Li atom on the PCP222 is calculated by the following expression[44]; +𝐸𝑏 = +1 +𝑚 [𝐸𝑃𝐶𝑃222 + 𝑚𝐸𝐿𝑖 − 𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖] + + (1) +Where 𝐸𝑃𝐶𝑃222, 𝐸𝐿𝑖, and 𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖 are symbolize for the total energy of PCP222, energy of +single isolated Li atom and energy of Li-decorated PCP222 respectively. m denotes for the number +of Li atoms used to functionalized the PCP222. +The average adsorption energy of H2 molecules with Li functionalized PCP222 is estimated +as [52]; +𝐸𝑎𝑑𝑠 = +1 +𝑛 [𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖 + 𝑛𝐸𝐻2 − 𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖+𝑛𝐻2] + (2) + +Where, EH2, and EPCP222+mLi+nH2 represents the energy of isolated single H2 molecule and +hydrogen adsorbed PCP222+mLi, respectively. n is the number of H2 molecules adsorbed in each +Li functionalized PCP222. +The successive desorption energy of adsorbed H2 molecules is estimated using following +equation[52]. +𝐸𝑑𝑒𝑠 = +1 +𝑛 [𝐸𝐻2 + 𝐸𝐻𝑜𝑠𝑡+(𝑛−1)𝐻2 − 𝐸𝐻𝑜𝑠𝑡+𝑛𝐻2] + + (3) +where 𝐸𝐻𝑜𝑠𝑡+(𝑛−1)𝐻2is the energy of previous H2 molecules adsorbed 𝐸𝐻𝑜𝑠𝑡+𝑛𝐻2. +The energy gap between the highest occupied molecular orbital (HOMO) and the lowest +unoccupied molecular orbital (LUMO) is calculated to ensure the kinetic stability of the Li +functionalized PCP222 and their hydrogen derivatives. The Hirshfeld charges and electrostatic +potential map (ESP) was used to study electronic charge transfer and interaction mechanism. +Further, to understand the metal and hydrogen interaction we have performed the partial density +of states (PDOS), and topological using the Bader’s quantum theory of atoms in molecules +(QTAIM). To investigate the structural integrity of the host material and H2 reversibility of the +system, atomistic molecular dynamic simulations were carried out using the expanded lagrangian +approach, atom-centered density matrix propagation (ADMP). +To determine the H2 adsorption capacity, gravimetric density (wt%) of hydrogen can be calculated +using the following expression [53]: +𝐻2(𝑤𝑡%) = +𝑀𝐻2 +𝑀𝐻2+𝑀𝐻𝑜𝑠𝑡 × 100 + + + + (4) +Here MH2 represent the mass of the total number of H2molecules adsorbed and MHost represent the +mass of Li functionalized PCP222. +3 Results and Discussion +3.1 Structural properties of PCP222 +Figure 1 depicts the ground state geometrical structure of PCP222. The PCP222 comprises three +benzene rings, that are linked via two CH2 moiety as bridge between the adjacent rings. The lengths +of the nearest CH2-CH2, and the CH2 across the benzene rings are observed to be 1.5 and 5.84 Å , + +respectively, that agrees with the empirically reported value by Cohen-Addad et al [54]. To +confirm the aromaticity of the relaxed PCP222, we calculated the Nucleus Independent Chemical +Shift (NICS) from center to to 3 Å above the benzene ring by increment of 1 Å. The NICS(1) is +found to have negative maximum (-10.1 ppm), demonstrating the aromatic character of +PCP222[55, 56]. This suggest that the cyclic rings of PCP222 are -electron rich and most +probably can bind the metal atom above (outside of PC222) the benzene rings. The Li atom then +functionalized above the benzene rings and on every possible site of PCP222 and allowed to relax +as discussed below. + +Figure 1: (a) Optimized structure of PCP222 with adsorption site marked with red-colored +text, (b) Li functionalized PCP222. +3.2 Functionalization of Li atom on PCP222 +To explore the hydrogen adsorption capacity in Li-functionalized PCP222, we must first carefully +examine the suitable adsorption site for Li atoms on the PCP222. In order to do this, we +investigated several PCP222 adsorption site, including the C-C bridge of benzene ring (B1), +CH2 moiety and benzene bridge (B2), CH2 - CH2 bridge (B3), and above the center of benzene +(Rc). All the possible Li adsorption sites of PCP222 are depicted in Figure 1(a). A single Li atom +is placed nearly 2 Å above the several probable adsorption sites of PCP222 and the structure is +allowed to get optimized. It is observed that functionalization of Li atom over B1 and B2 sites, it +migrate towards the Rc site following the optimization. On optimization of Li atom over B3 site, +the it moves away from the PCP222 and does not bind to the surface. We found that the Li atom +is stable on Rc site with binding energy of 0.32 eV that is 0.1 eV higher than that of Li on PCP44, +reported by Sathe et al. [43]. The Li atom is supposed to be functionalized on PCP222 via Dewar +mechanism, in which is due to the electronic charge transfer between the p-complex and s- orbitals + +5.84 +5.858 +B1 +1.541 +1.543 +(a) +(b)of Li atom [43, 45]. After functionalization of Li, the estimated Hirshfeld charge on benzene ring +of PCP222 is increased to -0.08 e.u from -0.03 e.u (in bare PCP222). These charges are transferred +from the metal atom, with the Hirshfeld charges on Li atom being +0.35 e.u after functionalization, +which make the Li atom ionic. The ionic Li atom is exposed to the guest H2 molecules and can +bind them via charge polarization mechanism as proposed by Niu et al. [57]. No significant change +in geometrical bond distances is observed after the functionalization of Li. The thermal stability +of the structures (host) is discussed in the molecular dynamic simulations section (section 3.5). All +the hydrogen adsorption/desorption simulations are performed by functionalizing the Li atom +above the center of benzene ring of PCP222. +3.2.1 Diffusion energy barrier calculation + +Figure 2: Diffusion energy barrier plot between energy difference and diffusion coordinates of +Li atom on PCP222 + +The clustering of metal atoms on the substrate can reduce the hydrogen uptake capacity of the +system as reported earlier [17]. The barrier of metal atoms diffusion energy ultimately decides +whether or not the clustering will occur. With a small rise in temperature, if the Li atom migrated +from its adsorption location, the possibility of metal-metal clustering would increase. Since, the +binding energy of Li atom on the PCP222 is less than the cohesive energy of the isolated Li atom +(1.63eV), we calculate if there is an energy barrier for diffusion of Li atom on PCP222 that can +avoid the possibility of metal clustering. To calculate the energy barrier, we shift the Li atom over +its adsorption site (on the benzene ring) by a small distance along the path shown in the +Figure 2 and carried out the single point energy calculation. Then we exhibit the energy difference +between initial and current step energy with the diffusion coordinate as illustrated in Figure 2. The + +1.4- +AE=1.38eV +1.2- +1.0- +0.8- +ev +0.6- +4 +1-3) +0.4 +0.2 - +0.0 - +1 +- +0 +1 +2 +3 +4 +5 +Diffusion coordinatesfigure shows presence of an energy barrier of 1.38 eV, that is sufficient to stop the Li atom from +diffusing across the PCP222 and thus prevent the metal clustering. Therefore, our calculated +energy barrier for diffusion of Li atom is high enough to prevent metal clustering over the studided +PCP222 compound. +3.3 Interaction of H2 with PCP222-Li +3.3.1 Adsorption Energy + +Table 1: Average bond distance between carbon bridge (C-C), center of PCP222 benzene ring (Rc) +and Lithium atom (Rc-Li), Lithium and hydrogen molecules (Li-H2), and hydrogen Hydrogen +(H-H) in Å. Average adsorption energy and successive desorption energy of PCP222-Li- +nH2 (n=1-5) + +Name of complex +Bridge C-C Rc-Li +Li-H +H-H +Eads (eV) Edes (eV) +PCP222-Li +1.542 +1.735 +- +- +- +- +PCP222-Li-1H2 +1.542 +1.745 +2.124 +0.753 +0.171 +0.171 +PCP222-Li-2H2 +1.541 +1.742 +2.083 +0.757 +0.159 +0.147 +PCP222-Li-3H2 +1.541 +1.767 +2.159 +0.753 +0.148 +0.127 +PCP222-Li-4H2 +1.541 +1.811 +2.243 +0.752 +0.134 +0.089 +PCP222-Li-5H2 +1.541 +1.813 +2.478 +0.751 +0.113 +0.030 +To explore the storage capacity and characteristics of Li functionalized PCP222, we introduced +the H2 molecules in a sequential manner to PCP222-Li. Firstly we introduced a single H2 molecule +at around 2Å above the Li atom on PCP222 and allowed the structure to get relaxed. It is observed +that, the H2 molecule is adsorbed at a distance of 2.124 Å from the Li atom with an adsorption +energy of 0.171 eV and the H-H bond length elongated by 0.01 Å. Sathe et al. studied the hydrogen +storage capacity of Li functionalized PCP11 (PCP22) and reported the adsorption energy of first +H2 molecule ~0.13 eV (0.11 eV) [46, 45]. Our calculated adsorption energy is slightly higher, +which is important in alkali metal doped H2 storage material and leads to the increase in the +desorption temperature. Further, we optimized the structures by adding H2 molecules sequentially +onto the PCP222-Li. On addition of second H2 molecule to the system, the average H2 adsorption +energy calculated to be 0.159 eV/H2. In this way, adsorption of 3rd, 4th and 5th H2 molecules to + +PCP222-Li, the average H2 adsorption energy reduces to 0.148, 0.134 and 0.113 +eV/H2respectively. When of more than five H2 molecules are added to the system, they fly away +from the Li atom and adsorption energy fall below 0.1 eV. We observed that the average adsorption +energy decreases with increase in number of H2 molecules in the system which is due the steric +hindrance between the adsorbed H2 crowed around the sorption centers and the increase in Li- +H2 distances (Table 1). The estimated data of adsorption energy and geometrical parameters of all +the bare hydrogenated systems and presented in Table 1. + +Figure 3: Optimized geometry of hydrogenated Li functionalized PCP222, (a) PCP222-Li- +1H2, (b) PCP222-Li-2H2, (c) PCP222-Li-3H2, (d) PCP222-Li-4H2, (e) PCP222-Li-5H2. +3.3.2 Electrostatics potential and Hirshfeld charges +To get a qualitative picture of electronic charge distribution over the surface of Li functionalized +PCP222 and their hydrogen adsorbed systems during the hydrogen adsorbed, we generate and +plotted the electrostatic potential map (ESP map) on the total electron density as depicted in +Figure 4. The electronic charge distribution is used to identify the active adsorption site, where the +hydrogen molecules can be introduced. The red and blue regions in the ESP plot reflects the +aggregation and reduction of electronic charge density respectively. The variation in the charge +density is plotted with the sequence of color code as red (highest electron density)> orange > +yellow > green > blue (lowest electron density). The ESP map of PCP222-Li shows that the Li +atom has the deficiency of electronic charges as marked by the dark blue region over the Li atom, +this indicate that the Li atom is somewhat ionic and is prone to bind the guest H2 molecules. When +the first H2 molecule added to the Li atom, the colour of the region over the Li changes from dark + +(a) +b +C +(d) +eblue to light blue, demonstrating the charge transfer from C atom of PCP222 and adsorbed H2 to +the Li atom. Further sequential adsorption of H2 molecules to PCP222-Li changes the colour of Li +region from blue to light blue indicating additional charge transfer. The blue region over Li almost +disappears on the adsorption of 5th H2 molecules suggesting the saturation of hydrogen uptake and +more guest H2 are unlikely to be adsorbed. The exact charge transfer is determined by calculating +the hirshfeld charges as discussed below. + + +Figure 4: Electrostatics potential map of (a) PCP222-Li, (b) PCP222-Li-1H2, (c) PCP222-Li- +2H2, (d) PCP222-Li-3H2, (e) PCP222-Li-4H2, (f) PCP222-Li-5H2. +We have performed the hirshfeld charge analysis to quantify the charge transfer distributions on +the Li functionalized PCP222 and their H2 adsorbed systems. The computed average Hirshfeld +charges on C atoms of benzene ring (Li functionalized site), Li atom, and adsorbed H2 molecules +with the number of hydrogen molecules is depicted in Figure 5. The average charges on C atom +of benzene ring is noted to be -0.031 e which raises to -0.084 e with the functionalization of Li +atom. The charge on Li atom of PCP222-Li is noted to be +0.354 e, which illustrate the transfer of +charges from benzene ring to Li atom making the sorption center (Li) ionic and more suitable for +H2 adsorption. These results agree well with the aforesaid ESP analysis. On adsorption of the first +H2 molecule to PCP222-Li, the charge on C atom is reduced by 2.38% and at the same time the +charge on Li atom is increased by 16.7 %. Further addition of hydrogen molecules follows the +trend of decrease in charge on benzene ring and increase in charge on Li atom (Figure 5). These +observations suggest that, the ionic Li atom polarize the guest H2 molecules and the H2 molecules +are adsorbed to the sorption center via a charge polarization mechanism due to induced dipole +developed in H2 as suggested by the Neu-Rao-Jena [30]. It is noted that the electronic charge on + +- 4.000 e-2 ++ 4.000 e-2 +(a) +(b) +(c) +(d) +(e) +(f) +Sideview +Top viewLi atom is raised by 41.36 % after the adsorption of the 5th H2 molecule. The adsorbed +H2 molecules are found to have an average charge of 0.027e to 0.013 e. + +Figure 5: Hirshfeld charges before and after hydrogen adsorption on PCP222-Li +3.3.3 Bader’s topological analysis and PDOS +The nature of interaction between the Li functionalized PCP222 and the adsorbed hydrogen +molecules is analyzed using the topological Bader’s quantum theory of atoms in molecules +(QTAIM). The parameters of electron density distribution at the bond critical point (BCP), +including the electron density (BCP), total electron energy density (ℋBCP), and Laplacian (2BCP), +are computed and given in Table S1 (in Supporting Information). The electron density (r) on C-C, +and C-Li, of hydrogenated PCP222-Li estimated to be almost equal to that of bare host material, +suggesting the post-adsorption chemical stability of the material. Additionally, the +average BCP values on H-H in PCP222-Li-5H2 is 0.258 a.u which is same as that on isolated bare +H2 molecules (-0.263). This reveal that the adsorbed hydrogens are in molecular form during the +adsorption. According to Kumar et al., the positive value of 2BCP indicated an electron density +depletion in the region of bonding and implied a close-shell kind of interaction. We noticed there +is no BCP between the Li and H atoms which implies no chemical bond between the Li atom and +the adsorbed H2 molecules and the interaction is purely closed-shell type resulting from the charge +polarization as proposed by the Neu-Rao-Jena. +Figure 6 illustrate the density of state plot of Li and adsorbed H atoms of the hydrogenated +PCP222-Li including the first and last (5th) H2 molecules adsorbed on the system. When one +hydrogen molecule is bound to the sorption center (Li), the s-orbital of the H2 molecule appears +below the Fermi level (E = 0) and stays unaffected as in the case of bare H2in Figure S2. This + +0.6 +Ring CbeforeLi decoration +0.5 +Ring C after Li decoration +-Liatom +0.4- +-Hatom +Hirshfeld Charges (eu) +0.3 +0.2 +0.1 - +0.0 +0.1 +-0.2 +-0.3 +0.4 +- +* +* +0 +1 +2 +3 +4 +5 +Number of H, molecules, nsignifies that there is no hybridization between the Li and adsorbed H2. This implies that the +adsorption of H2 molecule is owing to the induced dipole produced by charge polarization in H2. +With the adsorption of 5H2 molecules on PCP222-Li, the orbital of H atom splits into multiple +peaks ranging from -16 eV to -4 eV. This implies that the adsorption weakens as the quantity of +H2 molecules increases in the host. + +Figure 6: Partial density of state on Li and H atoms of PCP222-Li-1H2 and PCP222-Li-5H2 +3.4 Thermodynamics and storage capacity +3.4.1 Storage Capacity + +Figure 7: Optimized geometry of (a) PCP222-3Li, (b) PCP222-3Li-3H2, (c) PCP222-3Li-6H2, +(d) PCP222-3Li-9H2, (e) PCP222-3Li-12H2, (f) PCP222-3Li-15H2. + + +3.0 +Li +PCP222-Li-1H2 +2.5 +H +2.0 +1.5 +1.0 +HOMO +LUMO +-7.97eV +0.51eV +W +0.5 +0.0 +-18 +16 +14 +-12 +-10 +-8 +-6 +-4 +. +-2 +0 +2 +4 +Energy (ev) +(a) +4.0 +Li +PCP222-Li-5H, +3.5 +H +3.0 +2.5 +PDOS +2.0 +1.5 +HOMO +LUMO +1.0 +-7.91eV +0.45eV +0.5 +0.0 +-18 +16 +14 +12 +10 +-8 +-6 +-4 +T +-2 +0 +2 +4 +Energy (ev) +(b)+3H2 +(a) +(b) +15H2 ++3H2 ++ 3H,To investigate the optimum hydrogen storage capacity of the studied system, we functionalized +the maximum possible number of Li atoms over each benzene ring of PCP222. The geometrical +structure of three Li functionalized PCP222 ( PCP222-3Li) is shown in Figure 7 Further, we +introduced H2 molecules to each Li atom of PCP222-3Li sequentially as discussed in previous +section (3.3.1). The computed average hydrogen adsorption energy and the geometrical parameters +of all the hydrogenated systems are provided in the Table S2 (in Supporting Information). It is +noticed that, the adsorption process of hydrogen molecules on PCP22-3Li is found similar to that +of on PCP222-Li. On saturation of H2 adsorption on PCP222-3Li, we found each Li atom can +adsorb a maximum of 5H2 molecules resulting in total gravimetric density of 8.32 wt%. The +estimated value of hydrogen storage capacity is fairly above the requirement of US-DOE for +effective hydrogen storage systems. Our results can be compared with earlier reported +H2 gravimetric density on metal decorated carbon-based materials for hydrogen storage, for +example, Li-decorated C41 allotrope (7.12 wt%) [58], Li doped MOF impregnated with Li-coated +fullerenes[59], Li-doped B4C3 monolayer (6.22 wt%) [4]. +To develop a realistically usable hydrogen storage system, a significant quantity of hydrogen +molecules must be adsorbed by the host material under achievable storage conditions. Further the +adsorbed hydrogen molecules must also be efficiently desorbed at suitable temperature (T) and +pressure (P). Thus, we estimated the quantity of adsorbed hydrogen that could be used at a +accessible range of temperature (T) and pressure (P). To calculate the number of H2 molecules +remain adsorbed on PCP222-3Li (Occupation number) at different T and P, we calculated the +empirical value of hydrogen gas chemical potential (µ). Then the occupation number (N) is +estimated by the following expression and plotted with various T and P in Figure 8(b)[60]. +𝑁 = +∑ +𝑛𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇] +𝑁𝑚𝑎𝑥 +𝑛=0 +∑ +𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇] +𝑛𝑚𝑎𝑥 +𝑛=0 + + + + +(5) +Here Nmax is the maximum number of H2 molecules adsorbed at each Li atom on +PCP222, n and gn represents the number of H2 molecules adsorbed and configurational +degeneracy for a n respectively. kB is the Boltzmann constant and -Eads indicates the average +adsorption energy of H2 molecules to PCP222-3Li. m is the empirical value of chemical potential +of hydrogen gas at specific T and P, and is obtained by using the following expression [61]. +𝜇 = 𝐻0(𝑇) − 𝐻0(0) − 𝑇𝑆0(𝑇) + 𝐾𝐵𝑇 ln ( +𝑃 +𝑃0) + + (6) + +Here H0(T), S0(T) are the enthalpy and entropy of H2 at pressure P0 (1 bar). +We can see in Figure 8(b) that the PCP222-3Li can adsorbed H2 molecules giving rise to +maximum hydrogen uptake capacity of ~8.32 wt% up to the temperature of 80 K and pressure of +30-60 bar. When the temperature rises beyond 80 K, the H2 molecules begin to desorb from the +PCP222-3Li and the gravimetric density closes to ~5.5 wt% (target of US-DOE by 2025) when +the temperature reaches 180 K under the pressure of 30-60 bar. Further rise in temperature, the +storage capacity of the PCP222-3Li fall below 4 wt% at 220 K and 40-bar. At a temperature range +of 260 K-300 K and pressure range of 1-10 bar, the studied system shows a 100% desorption of +hydrogen. Thus, we can propose the Li functionalized PCP222 as a low-temperature-adsorption +and room-temperature-desorption hydrogen storage material. Under the room temperature (300 +K), the studied system shows up to 8.32 wt % of usable hydrogen storage capacity with 100% +reversibility. Thus, we believe that, our studied material Li functionalized PCP222 can be used as +an efficient hydrogen storage material satisfying the criteria of US-DOE. + +Figure 8: Plot of Van’t-Hoff desorption temperature for Li functionalized PCP222 at different +temperature and pressure. +3.4.2 Desorption temperature +For a reversible hydrogen storage media, it is crucial to estimate the desorption temperature of +hydrogen molecules. We have estimated the desorption temperature (T D) of H2 for the Li +functionalized PCP222 using the Van’t Hoff equation [17]. +𝑇𝐷 = ( +𝐸𝑎𝑑𝑠 +𝐾𝐵 ) ( +∆𝑆 +𝑅 − ln 𝑝) +−1 + + + + + +(7) + +280 +81 +7,488 +6.656 +7- +260- +5.824 +6- +4.992 +Desorptiontemperature +240 +219K +5- +4.160 +220 +G,wt% +2.496 +200 +3 +1.664 +182K +2- +180 +0.000 +1 +160 +145K +averageT, +140- +一minT, +1.5 +2.0 +2.5 +3.0 +1.0 +3.5 +4.5 +5.0 +Pressure(atm) +300 +(a) +(b)Where, Eads represents the computed hydrogen adsorption energy, KB, and R denotes for the +Boltzmann constant and R the gas constant respectively. P represent s the equilibrium pressure +(we take a range of 1 to 5 atm with an increment of 0.5 atm) and △S is the entropy change of +hydrogen from its gaseous state to liquid state [62]. Using the highest and lowest adsorption energy +of system (with minimum and maximum H2 gravimetric density, respectively), the maximum and +minimum desorption temperatures (TDmax∕TDmin) are determined. While, TDmin denotes the +minimum temperature necessary to initiate the desorption of H2 molecules, the TDmax is the +temperature required for complete desorption process. The estimated desorption temperatures +along with the equilibrium pressure is depicted in Figure 8(a). The minimum and maximum +temperatures for H2 desorption are determined to be 145 K and 219 K, respectively, at 1 atm +pressure. The estimated average TD of Li functionalized PCP222 is 182 K at 1 atm. This result +reveals that, the system can adsorb its full capacity H2 at cryogenic temperature and desorb all the +H2 molecules bellow room temperature at 1 atm pressure. However, the desorption temperature +can be increases by increase in the equilibrium pressure as presented in Figure 8(a) and as +discussed above. +3.5 Molecular dynamics simulations + +Figure 9: (a) Potential energy trajectories of hydrogenated PCP222-3Li and (b) Time +evolution trajectory of average bond length between the Li atom and C atoms of PCP222 at +300K and 500K, + +968.88 +968.90 +300K +(Hartree +500K +968.92 +968.94 +energy +968.96 +968.98 +-969.00 +Potential +-969.02 +969.04 +-969.06 +-969.08 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +Time (fs) +(a) +2.6 +C-Lidistance@300K +Average C-Li distance (A) +2.5 +C-Lidistance@50oK +2.4 +2.3 +2.2 +2.1 +2.0 +1.9 +1.8 +1.7 +1.6 +300K,1000fs +500K,1000fs +1.5 +0 +100 +200 +300 +400 +500 +600 +700 +800 +006 +1000 +Time (fs) +(b) +To validate the reversibility of hydrogen molecules on PCP222-3Li estimated by the DFT +computation, we have carried out molecular dynamics (MD) simulations using the atomistic +density matrix propagation (ADMP). ADMP is an extended Lagrangian approach to MD, that uses +the gaussian basis function and propagates the density matrix. The ADMP-MD simulations is +performed on system with highest storage capacity (PCP222-3Li-15H2), at two different +temperatures of 300K and 500 K for total time of 1 ps with the time step of 1fs. During the +simulations the temperature (kinetic energy thermostat) is maintained by the velocity scaling +approach and at every 10 fs, time step, the temperature is checked and corrected. The time +evolution potential energy trajectories and the snapshots are depicted in Figure 9(a) and Figure S3 +(in supporting Information) respectively. The MD simulations at 300 K and 1ps illustrate that +almost all the H2 molecules fly away from the sorption centers, except 1H2 at each center. +Simulations at 500 K shows that all the H2 molecules are desorbed from the host material keeping +the host structure intact. This result suggests that the hydrogen storage in Li functionalized PCP222 +is reversible in process. +For a viable reversible hydrogen storage material, it is important that the host material must not +distorted above the hydrogen desorption temperature. To investigate the solidity of host material +(PCP222-3Li), we performed the MD simulations on the bare host structure (PCP222-3Li) at room +temperature (300 K) and considerably above the H2 desorption temperature (500K) using ADMP. +The molecular dynamics simulations are performed for 1 ps with a time step of 1 fs. The time +evolution trajectory of average distance between Li atom and the carbon atoms of PCP222 benzene +rings is plotted in Figure 9(b). We noticed that the PCP222-3Li structure stays stable at 500 K and +almost no change in C-C and C-H bond distance is observed. The trajectory of average bond length +between the Li atom and C atoms of PCP222 benzene rings seem oscillate but the mean value +(2.25 Å) and the variation is minimal. This validates the structural integrity of the host material +above the H2 desorption temperature. Moreover no Li clustering is also noticed after desorption as +discussed earlier in Section 3.2.1. Thus, we believe that PCP222-3Li can be considered for feasible +reversible hydrogen storage material. + + + +4 Summery and Conclusion +In this study, we investigated the thermodynamical stability and hydrogen storage capacity of Li +functionalized [2,2,2]paracyclophane, using the density functional theory. The Li atoms are found +to bind with the PCP222 via Dewar mechanism and no clustering of Li atoms over PCP222 was +noticed. Each Li atom on PCP222 could adsorb up to 5H2 molecules via charge polarization +mechanism with an average H2 adsorption energy in the range of 0.12 - 0.17 eV/H2, indicating +physisorption type of adsorption. Moreover, the average H-H bond distance got elongated by 0.01 +Å, during the adsorption process, which implied that the adsorbed H2 were in molecular form and +this fact was also confirmed by the charge distribution analysis. When three Li atoms were +functionalized on PCP222, the H2 gravimetric capacity of the system was up to 8.32 wt% which +was fairly above the US-DOE requirements for practical hydrogen applications. During saturation +of H2 adsorption, the host material displayed no significant change in geometry. The +thermodynamic usable hydrogen capacity was found up to ~8.32 wt% at the temperature of 80 K +and pressure of 30-60 bar. On further increase in temperature, up to 180 K under the pressure of +30-60 bar, the PCP222-3Li hydrogen uptake capacity approached 5.5wt% which is the target of +DOE by 2025. At a temperature range of 260 K-300 K and pressure range of 1-10 bar, the PCP222- +3Li system showed 100% desorption of H2. Molecular dynamic simulation confirmed that at 300 +K, almost all the H2 molecules flied away except 1H2 at each center. Simulations at 500 K showed +that all the H2 molecules are desorbed from the host material keeping the structure of the host +structure intact. Since, there is no experimental works reported on Li functionalized PCP222 for +hydrogen storage, we hope our computational work will contribute significantly to the research of +hydrogen storage in macrocyclic compounds and provide supporting reference for the future +experiments. +References +[1] Sachin P. Shet, S. Shanmuga Priya, K. 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CRC press. + + + diff --git a/49E2T4oBgHgl3EQfkAeM/content/tmp_files/load_file.txt b/49E2T4oBgHgl3EQfkAeM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd83b1b66f149e02a63048bc6b363612004966b5 --- /dev/null +++ b/49E2T4oBgHgl3EQfkAeM/content/tmp_files/load_file.txt @@ -0,0 +1,1075 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf,len=1074 +page_content='Hydrogen storage in Li functionalized [2,2,2]paracyclophane at cryogenic to room temperatures: A computational quest Rakesh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Sahoo, Sridhar Sahu Computational Materials Research Lab, Department of Physics, Indian Institute of Technology (Indian School of Mines) Dhanbad, India Abstract In this work, we have studied the hydrogen adsorption-desorption properties, and storage capacities of Li functionalized [2,2,2]paracyclophane (PCP222) using dispersion-corrected density functional theory and molecular dynamic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The Li atom was found bonded strongly with the benzene ring of PCP222 via Dewar interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Subsequently, the calculation of the diffusion energy barrier revealed a significantly high energy barrier of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='38 eV, preventing the Li clustering on PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The host material, PCP222-3Li adsorbed up to 15H2 molecules via charge polarization mechanism with an average adsorption energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='145 eV/5H2, suggesting physisorption type of adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The PCP222 functionalized with three Li atom showed maximum hydrogen uptake capacity up to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='32 wt% which was fairly above the US-DOE criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The practical H2storage estimation revealed that the PCP222-3Li desorbed 100% of adsorbed H2 molecules at the temperature range of 260 K-300 K and pressure range of 1-10 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The maximum H2 desorption temperature estimated by the Vant-Hoff relation was found to be 219 K and 266 K at 1 bar and 5 bar, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The ADMP molecular dynamics simulations assured the reversibility of adsorbed H2 and the structural integrity of the host material at sufficiently above the desorption temperature (300K and 500K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Therefore, the Li-functionalized PCP222 can be considered as a thermodynamically viable and potentially reversible H2 storage material below room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Keywords: Hydrogen storage, DFT, Van’t-Hoff equation, ADMP, [2,2,2]paracyclophane, PCP222, ESP 1 Introduction The excessive consumption of traditional fossil fuels has not only led to the depletion of the energy supplies but also has emerged as the prime cause of environmental pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The global consumption of petroleum and other traditional fossil fuel is anticipated to expand up to 56% by the year 2040 and the crude oil supply is expected to endure until 2060 if the current demand trend continues[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Thus, it is essential to develop alternative energy sources that are free from the drawbacks of traditional fossil fuels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' To meet the world’s energy demand and reduce the pollution caused by fossil fuels, hydrogen has been considered as a plausible alternative due to its natural abundance, environmental friendliness, and regenerative properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' One of the distinctive quality of hydrogen is that it produces a large amount of energy per unit mass (120 MJ/kg) without releasing any pollutant by-products [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Despite these benefits, however, the use of hydrogen in practice is limited due to the obstacle of finding the most appropriate and affordable way to store and deliver hydrogen under normal environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' As per the criteria proposed by the United State department of energy (DOE-US) an effective hydrogen storage material should have a minimum storage capacity of up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 wt% by the year 2025 at moderate thermodynamics[5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' In addition, as reported by many authors, the adsorption energy of hydrogen molecules of an effective storage materials should be in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 eV/H2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='6 eV/H2[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Though, numerous varieties of materials such as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' metal hydrides [7, 8], graphene [9, 10], metal alloys [11, 12], metal-organic frameworks (MOF) [13, 14], covalent-organic frameworks (COF) [15] and carbon nanostructures [16, 17] etc have been investigated both theoretically and experimentally as potential hydrogen storage materials, but there are many drawbacks and unsolved issues to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The metal hydrides and complex hydrides store hydrogen via chemisorption process which is highly irreversible and prevents easy desorption of hydrogen [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' For example, Al(BH4)3, which yields hydrogen uptake capacity of up to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='9 wt%, has high desorption temperature (about 1000 K) that makes the material non-effective practical reversible hydrogen storage applications[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Under ambient conditions, Mg-metal hydrides have a storage capacity up to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='6 wt%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' however, it can only be used for 2-3 cycles[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Tavhare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' studied the hetero atom substituted Ti-benzene and reported an H2 uptake capacity up to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='85 wt%, but at relatively high desorption temperature (1193 K)[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Furthermore, MOF and COF applications are constrained in the practical H2 storage field due to the difficulties of their heavy structure and challenging step-wise production [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The efficient use of carbonaceous materials as hydrogen storage media was initially reported by Dillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Carbonaceous materials are appropriate for H2 storage due to their unique qualities such as, large surface area, high porosity, better stabilities, and low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' However, the early findings have shown that these pure materials are weakly interact with the hydrogen molecules (with BE ~4-5 kJ/mol), thus impractical for realistic hydrogen storage at ambient environment [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Meanwhile, carbon-based pure substrates are excellent materials for hydrogen storage at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' For instance, pure single wall carbon nanotube (SWCNT) can store hydrogen molecules up to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='25 wt%, with a substantially lower desorption temperature of 80 K [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' It has been reported that the H2 interaction strength and the desorption temperature can be tuned by integrating pure carbon substrates with alkali metal (AM)(Li, Na, and K), alkali earth metals (Be, Mg, Ca), and transition metals (TM)(Sc, Ti, V, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' )[27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Numerous theoretical investigations showed that integrating AMs and TMs with the carbon/borane substrates can bind H2 molecules via charge polarization and the Kubas mechanism [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The metallic atom decorated fullerenes were first explored to investigate the impact of metal integration on pure carbon substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' According to studies by Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=', Li decorated fullerene could show a storage capacity of 9 wt%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' however, the hydrogen adsorption energy was estimated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='075 eV/H2, which is much lower than the DOE criterion [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The Li and Na-loaded C60 revealed H2 uptake capacities of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 wt% and 4 wt%, respectively, that were significantly below the target of DoE [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Experimental studies of transition metals like V and Pd decorated CNT reveal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='66 wt% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='69 wt% of hydrogen capacity respectively, while pure CNT has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='53 wt% of storage capacity [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' reported storage of H2 on Li and Sc doped C8N8 cage via Niu-Rao-Jena and Kubas interaction and estimated a desorption temperature of 286 K and 456 K, respectively [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The Li and Na decorated on C24 fullerene could adsorb H2 molecules, with average hydrogen binding energies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='198 eV/H2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='164 eV/H2 and led to storage capacity up to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='7 wt% and 10 wt %, respectively [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Recently, we have investigated the H2 storage on alkali metal decorated C20 fullerene and found the molecular hydrogen are physisorbed on host material via charge polarization mechanism with desorption temperature of 182 -191 K [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Each Li and Na atom on C20 could uptake up to 5H2 molecules with a total gravimetric storage capacity of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='08 wt % and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='82 wt%, respectively, and the H2 binding energies found in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='12 eV—0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='13 eV/H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Other carbonaceous materials such as functionalized organometallic compounds, macrocyclic compounds have also been reported recently as potential candidates for hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' For example, Mahamiya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' revealed the H2 storage capacities of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 9 wt % in K and Ca decorated biphenylene with an average adsorption energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='24-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='33 eV [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Y atom doped zeolite shows high capacity adsorption of H2 with binding energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='35 eV/H2 and the desorption energy of 437K for fuel cells[39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Lithium-doped Calixarenes show an excellent hydrogen storage behaviour but at very low up to 100 K [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Calix[4]arene functionalized with Li metal reveals 10 wt% storage capacity via Kubas—Niu—Rao—Jena interaction, and all most all H2 desorbed at a temperature of 273 K [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Macrocyclic compounds such as, paracyclophane (PCP), a subgroup derivative of cyclophanes, contains aromatic benzene rings, and their nomenclature is established on the arene substitution pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' For a [n,n]paracyclophane, the number of -CH2- moiety connecting the successive benzene rings is indicated by the number in the square bracket [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Due to the existence of aromatic benzene rings in the geometry, PCPs are easy to synthesize experimentally and can be functionalized with metal atoms, making them a viable choice for hydrogen storage candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' A report on Li and Sc functionalized [4,4]paracyclophane revealed the hydrogen uptake capacity up to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='8 wt% and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='7 wt% with an average adsorption energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='08 eV/H2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3 eV/H2 respectively [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Sahoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' recently studied the H2 storage capacity of [1,1]paracyclophane functionalized with Sc and Y metals and found an H2 gravimetric storage capacity of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='22 wt% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='33 wt%, respectively, with an average adsorption energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='36 eV/H2[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' They reported the H2 desorption temperature of 439 K and 412 K for Sc and Y doped PCP11, respectively, at 1 atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The hydrogen molecules are physisorbed on Li, and Sc decorated paracyclophane via Kubas-Niu-Jena interaction and show a storage capacity of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3 wt%, as reported by Sathe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Many more alkali metal-doped macrocyclic compounds have also been investigated for hydrogen storage candidates and found the storage capacity above the DOE target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' however very few reported the practical H2 capacity at various thermodynamic conditions[46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Though few of PCP-based hydrogen storage systems are available in literature, the [2,2,2]paracyclophane (PCP222) which is experimentally synthesized by Tabushi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' [48] is yet to be explored as hydrogen storage material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Because Li, the lightest alkali metal atom and can hold H2 molecules via charge polarization mechanism, it can serve as better sorption center on PCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Therefore, in the current work, we intend to investigate the hydrogen storage properties and potential of Li functionalized [2,2,2]paracyclophane (PCP222).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' We chose the PCP222 for hydrogen storage because it is already experimentally synthesized and can be decorated with metal atoms to form a hydrogen storage media with a high hydrogen uptake capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The Li atoms are functionalized as sorption centers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' this is because the light-weight metal doping method is an effective way to increase the capacity of H2 storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Li being the lightest alkali metal atom, received a lot of attention to for hydrogen sorption application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Though there are few reports available based on hydrogen adsorption mechanism on metal doped macrocyclic organic molecules and other Li decorated nanostructures, our work is the first to reveal the efficiency of Li functionalized PCP222 using the atomistic MD simulation, practical storage capacity and diffusion energy barrier estimation 2 Theory and Computation The theoretical computations are carried out on [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2] paracyclophane (PCP222) and their hydrogenated derivatives within the framework of density functional theory (DFT)[49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The modern range separated hybrid functional wB97Xd is used, and molecular orbitals (MO) are defined as linear combination of atom centered basis functions, with all atoms using the valence diffuse and polarization function 6-311+G(d,p) basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The wB97Xd, a long range separated form of Becke’s 97 functional, also adds Grimme’s D2 dispersion correction[50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' It is worth mentioning that the wB97Xd is a reliable approach to investigate the non-covalent interaction of metal doped organic molecules and their thermochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The harmonic frequencies of all the studied structures are calculated to confirm that they are truly in the ground state on the potential surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Some of the crucial quantitative metrics, including, binding energy of metal atom on host, average H2 adsorption energy and successive H2 desorption energy must be determined in order to analyze the mechanism of hydrogen storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The binding strength of Li atom on the PCP222 is calculated by the following expression[44];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 𝐸𝑏 = 1 𝑚 [𝐸𝑃𝐶𝑃222 + 𝑚𝐸𝐿𝑖 − 𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖] (1) Where 𝐸𝑃𝐶𝑃222, 𝐸𝐿𝑖, and 𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖 are symbolize for the total energy of PCP222, energy of single isolated Li atom and energy of Li-decorated PCP222 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' m denotes for the number of Li atoms used to functionalized the PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The average adsorption energy of H2 molecules with Li functionalized PCP222 is estimated as [52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 𝐸𝑎𝑑𝑠 = 1 𝑛 [𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖 + 𝑛𝐸𝐻2 − 𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖+𝑛𝐻2] (2) Where, EH2, and EPCP222+mLi+nH2 represents the energy of isolated single H2 molecule and hydrogen adsorbed PCP222+mLi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' n is the number of H2 molecules adsorbed in each Li functionalized PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The successive desorption energy of adsorbed H2 molecules is estimated using following equation[52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 𝐸𝑑𝑒𝑠 = 1 𝑛 [𝐸𝐻2 + 𝐸𝐻𝑜𝑠𝑡+(𝑛−1)𝐻2 − 𝐸𝐻𝑜𝑠𝑡+𝑛𝐻2] (3) where 𝐸𝐻𝑜𝑠𝑡+(𝑛−1)𝐻2is the energy of previous H2 molecules adsorbed 𝐸𝐻𝑜𝑠𝑡+𝑛𝐻2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is calculated to ensure the kinetic stability of the Li functionalized PCP222 and their hydrogen derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The Hirshfeld charges and electrostatic potential map (ESP) was used to study electronic charge transfer and interaction mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Further, to understand the metal and hydrogen interaction we have performed the partial density of states (PDOS), and topological using the Bader’s quantum theory of atoms in molecules (QTAIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' To investigate the structural integrity of the host material and H2 reversibility of the system, atomistic molecular dynamic simulations were carried out using the expanded lagrangian approach, atom-centered density matrix propagation (ADMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' To determine the H2 adsorption capacity, gravimetric density (wt%) of hydrogen can be calculated using the following expression [53]: 𝐻2(𝑤𝑡%) = 𝑀𝐻2 𝑀𝐻2+𝑀𝐻𝑜𝑠𝑡 × 100 (4) Here MH2 represent the mass of the total number of H2molecules adsorbed and MHost represent the mass of Li functionalized PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 3 Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 Structural properties of PCP222 Figure 1 depicts the ground state geometrical structure of PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The PCP222 comprises three benzene rings, that are linked via two CH2 moiety as bridge between the adjacent rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The lengths of the nearest CH2-CH2, and the CH2 across the benzene rings are observed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='84 Å , respectively, that agrees with the empirically reported value by Cohen-Addad et al [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' To confirm the aromaticity of the relaxed PCP222, we calculated the Nucleus Independent Chemical Shift (NICS) from center to to 3 Å above the benzene ring by increment of 1 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The NICS(1) is found to have negative maximum (-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 ppm), demonstrating the aromatic character of PCP222[55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' This suggest that the cyclic rings of PCP222 are \uf070-electron rich and most probably can bind the metal atom above (outside of PC222) the benzene rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The Li atom then functionalized above the benzene rings and on every possible site of PCP222 and allowed to relax as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Figure 1: (a) Optimized structure of PCP222 with adsorption site marked with red-colored text, (b) Li functionalized PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2 Functionalization of Li atom on PCP222 To explore the hydrogen adsorption capacity in Li-functionalized PCP222, we must first carefully examine the suitable adsorption site for Li atoms on the PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' In order to do this, we investigated several PCP222 adsorption site, including the C-C bridge of benzene ring (B1), CH2 moiety and benzene bridge (B2), CH2 - CH2 bridge (B3), and above the center of benzene (Rc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' All the possible Li adsorption sites of PCP222 are depicted in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' A single Li atom is placed nearly 2 Å above the several probable adsorption sites of PCP222 and the structure is allowed to get optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' It is observed that functionalization of Li atom over B1 and B2 sites, it migrate towards the Rc site following the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' On optimization of Li atom over B3 site, the it moves away from the PCP222 and does not bind to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' We found that the Li atom is stable on Rc site with binding energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='32 eV that is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 eV higher than that of Li on PCP44, reported by Sathe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The Li atom is supposed to be functionalized on PCP222 via Dewar mechanism, in which is due to the electronic charge transfer between the p-complex and s- orbitals 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='858 B1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='543 (a) (b)of Li atom [43, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' After functionalization of Li, the estimated Hirshfeld charge on benzene ring of PCP222 is increased to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='08 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='u from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='03 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='u (in bare PCP222).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' These charges are transferred from the metal atom, with the Hirshfeld charges on Li atom being +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='35 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='u after functionalization, which make the Li atom ionic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The ionic Li atom is exposed to the guest H2 molecules and can bind them via charge polarization mechanism as proposed by Niu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' No significant change in geometrical bond distances is observed after the functionalization of Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The thermal stability of the structures (host) is discussed in the molecular dynamic simulations section (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' All the hydrogen adsorption/desorption simulations are performed by functionalizing the Li atom above the center of benzene ring of PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 Diffusion energy barrier calculation Figure 2: Diffusion energy barrier plot between energy difference and diffusion coordinates of Li atom on PCP222 The clustering of metal atoms on the substrate can reduce the hydrogen uptake capacity of the system as reported earlier [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The barrier of metal atoms diffusion energy ultimately decides whether or not the clustering will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' With a small rise in temperature, if the Li atom migrated from its adsorption location, the possibility of metal-metal clustering would increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Since, the binding energy of Li atom on the PCP222 is less than the cohesive energy of the isolated Li atom (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='63eV), we calculate if there is an energy barrier for diffusion of Li atom on PCP222 that can avoid the possibility of metal clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' To calculate the energy barrier, we shift the Li atom over its adsorption site (on the benzene ring) by a small distance along the path shown in the Figure 2 and carried out the single point energy calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Then we exhibit the energy difference between initial and current step energy with the diffusion coordinate as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='4- AE=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='38eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='8- ev 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='6- 4 1-3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 - 1 0 1 2 3 4 5 Diffusion coordinatesfigure shows presence of an energy barrier of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='38 eV, that is sufficient to stop the Li atom from diffusing across the PCP222 and thus prevent the metal clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Therefore, our calculated energy barrier for diffusion of Li atom is high enough to prevent metal clustering over the studided PCP222 compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3 Interaction of H2 with PCP222-Li 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 Adsorption Energy Table 1: Average bond distance between carbon bridge (C-C), center of PCP222 benzene ring (Rc) and Lithium atom (Rc-Li), Lithium and hydrogen molecules (Li-H2), and hydrogen Hydrogen (H-H) in Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Average adsorption energy and successive desorption energy of PCP222-Li- nH2 (n=1-5) Name of complex Bridge C-C Rc-Li Li-H H-H Eads (eV) Edes (eV) PCP222-Li 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='542 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='735 PCP222-Li-1H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='542 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='745 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='171 PCP222-Li-2H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='742 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='147 PCP222-Li-3H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='767 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='127 PCP222-Li-4H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='811 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='752 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='089 PCP222-Li-5H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='813 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='030 To explore the storage capacity and characteristics of Li functionalized PCP222, we introduced the H2 molecules in a sequential manner to PCP222-Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Firstly we introduced a single H2 molecule at around 2Å above the Li atom on PCP222 and allowed the structure to get relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' It is observed that, the H2 molecule is adsorbed at a distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='124 Å from the Li atom with an adsorption energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='171 eV and the H-H bond length elongated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='01 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Sathe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' studied the hydrogen storage capacity of Li functionalized PCP11 (PCP22) and reported the adsorption energy of first H2 molecule ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='13 eV (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='11 eV) [46, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Our calculated adsorption energy is slightly higher, which is important in alkali metal doped H2 storage material and leads to the increase in the desorption temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Further, we optimized the structures by adding H2 molecules sequentially onto the PCP222-Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' On addition of second H2 molecule to the system, the average H2 adsorption energy calculated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='159 eV/H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' In this way, adsorption of 3rd, 4th and 5th H2 molecules to PCP222-Li, the average H2 adsorption energy reduces to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='148, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='134 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='113 eV/H2respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' When of more than five H2 molecules are added to the system, they fly away from the Li atom and adsorption energy fall below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' We observed that the average adsorption energy decreases with increase in number of H2 molecules in the system which is due the steric hindrance between the adsorbed H2 crowed around the sorption centers and the increase in Li- H2 distances (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The estimated data of adsorption energy and geometrical parameters of all the bare hydrogenated systems and presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Figure 3: Optimized geometry of hydrogenated Li functionalized PCP222, (a) PCP222-Li- 1H2, (b) PCP222-Li-2H2, (c) PCP222-Li-3H2, (d) PCP222-Li-4H2, (e) PCP222-Li-5H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2 Electrostatics potential and Hirshfeld charges To get a qualitative picture of electronic charge distribution over the surface of Li functionalized PCP222 and their hydrogen adsorbed systems during the hydrogen adsorbed, we generate and plotted the electrostatic potential map (ESP map) on the total electron density as depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The electronic charge distribution is used to identify the active adsorption site, where the hydrogen molecules can be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The red and blue regions in the ESP plot reflects the aggregation and reduction of electronic charge density respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The variation in the charge density is plotted with the sequence of color code as red (highest electron density)> orange > yellow > green > blue (lowest electron density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The ESP map of PCP222-Li shows that the Li atom has the deficiency of electronic charges as marked by the dark blue region over the Li atom, this indicate that the Li atom is somewhat ionic and is prone to bind the guest H2 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' When the first H2 molecule added to the Li atom, the colour of the region over the Li changes from dark (a) b C (d) eblue to light blue, demonstrating the charge transfer from C atom of PCP222 and adsorbed H2 to the Li atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Further sequential adsorption of H2 molecules to PCP222-Li changes the colour of Li region from blue to light blue indicating additional charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The blue region over Li almost disappears on the adsorption of 5th H2 molecules suggesting the saturation of hydrogen uptake and more guest H2 are unlikely to be adsorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The exact charge transfer is determined by calculating the hirshfeld charges as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Figure 4: Electrostatics potential map of (a) PCP222-Li, (b) PCP222-Li-1H2, (c) PCP222-Li- 2H2, (d) PCP222-Li-3H2, (e) PCP222-Li-4H2, (f) PCP222-Li-5H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' We have performed the hirshfeld charge analysis to quantify the charge transfer distributions on the Li functionalized PCP222 and their H2 adsorbed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The computed average Hirshfeld charges on C atoms of benzene ring (Li functionalized site), Li atom, and adsorbed H2 molecules with the number of hydrogen molecules is depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The average charges on C atom of benzene ring is noted to be -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='031 e which raises to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='084 e with the functionalization of Li atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The charge on Li atom of PCP222-Li is noted to be +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='354 e, which illustrate the transfer of charges from benzene ring to Li atom making the sorption center (Li) ionic and more suitable for H2 adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' These results agree well with the aforesaid ESP analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' On adsorption of the first H2 molecule to PCP222-Li, the charge on C atom is reduced by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='38% and at the same time the charge on Li atom is increased by 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='7 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Further addition of hydrogen molecules follows the trend of decrease in charge on benzene ring and increase in charge on Li atom (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' These observations suggest that, the ionic Li atom polarize the guest H2 molecules and the H2 molecules are adsorbed to the sorption center via a charge polarization mechanism due to induced dipole developed in H2 as suggested by the Neu-Rao-Jena [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' It is noted that the electronic charge on 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='000 e-2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='000 e-2 (a) (b) (c) (d) (e) (f) Sideview Top viewLi atom is raised by 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='36 % after the adsorption of the 5th H2 molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The adsorbed H2 molecules are found to have an average charge of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='027e to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='013 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Figure 5: Hirshfeld charges before and after hydrogen adsorption on PCP222-Li 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3 Bader’s topological analysis and PDOS The nature of interaction between the Li functionalized PCP222 and the adsorbed hydrogen molecules is analyzed using the topological Bader’s quantum theory of atoms in molecules (QTAIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The parameters of electron density distribution at the bond critical point (BCP), including the electron density (\uf072BCP), total electron energy density (ℋBCP), and Laplacian (\uf0d12\uf072BCP), are computed and given in Table S1 (in Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The electron density (r) on C-C, and C-Li, of hydrogenated PCP222-Li estimated to be almost equal to that of bare host material, suggesting the post-adsorption chemical stability of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Additionally, the average \uf072BCP values on H-H in PCP222-Li-5H2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='258 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='u which is same as that on isolated bare H2 molecules (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='263).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' This reveal that the adsorbed hydrogens are in molecular form during the adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' According to Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=', the positive value of \uf0d12\uf072BCP indicated an electron density depletion in the region of bonding and implied a close-shell kind of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' We noticed there is no BCP between the Li and H atoms which implies no chemical bond between the Li atom and the adsorbed H2 molecules and the interaction is purely closed-shell type resulting from the charge polarization as proposed by the Neu-Rao-Jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Figure 6 illustrate the density of state plot of Li and adsorbed H atoms of the hydrogenated PCP222-Li including the first and last (5th) H2 molecules adsorbed on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' When one hydrogen molecule is bound to the sorption center (Li), the s-orbital of the H2 molecule appears below the Fermi level (E = 0) and stays unaffected as in the case of bare H2in Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' This 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='6 Ring CbeforeLi decoration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 Ring C after Li decoration Liatom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='4- Hatom Hirshfeld Charges (eu) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='4 0 1 2 3 4 5 Number of H, molecules, nsignifies that there is no hybridization between the Li and adsorbed H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' This implies that the adsorption of H2 molecule is owing to the induced dipole produced by charge polarization in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' With the adsorption of 5H2 molecules on PCP222-Li, the orbital of H atom splits into multiple peaks ranging from -16 eV to -4 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' This implies that the adsorption weakens as the quantity of H2 molecules increases in the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Figure 6: Partial density of state on Li and H atoms of PCP222-Li-1H2 and PCP222-Li-5H2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='4 Thermodynamics and storage capacity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 Storage Capacity Figure 7: Optimized geometry of (a) PCP222-3Li, (b) PCP222-3Li-3H2, (c) PCP222-3Li-6H2, (d) PCP222-3Li-9H2, (e) PCP222-3Li-12H2, (f) PCP222-3Li-15H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 Li PCP222-Li-1H2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 HOMO LUMO 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='97eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='51eV W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 18 16 14 12 10 8 6 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 2 0 2 4 Energy (ev) (a) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 Li PCP222-Li-5H, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 H 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 PDOS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 HOMO LUMO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='91eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='45eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 18 16 14 12 10 8 6 4 T 2 0 2 4 Energy (ev) (b)+3H2 (a) (b) 15H2 +3H2 + 3H,To investigate the optimum hydrogen storage capacity of the studied system, we functionalized the maximum possible number of Li atoms over each benzene ring of PCP222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The geometrical structure of three Li functionalized PCP222 ( PCP222-3Li) is shown in Figure 7 Further, we introduced H2 molecules to each Li atom of PCP222-3Li sequentially as discussed in previous section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The computed average hydrogen adsorption energy and the geometrical parameters of all the hydrogenated systems are provided in the Table S2 (in Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' It is noticed that, the adsorption process of hydrogen molecules on PCP22-3Li is found similar to that of on PCP222-Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' On saturation of H2 adsorption on PCP222-3Li, we found each Li atom can adsorb a maximum of 5H2 molecules resulting in total gravimetric density of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='32 wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The estimated value of hydrogen storage capacity is fairly above the requirement of US-DOE for effective hydrogen storage systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Our results can be compared with earlier reported H2 gravimetric density on metal decorated carbon-based materials for hydrogen storage, for example, Li-decorated C41 allotrope (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='12 wt%) [58], Li doped MOF impregnated with Li-coated fullerenes[59], Li-doped B4C3 monolayer (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='22 wt%) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' To develop a realistically usable hydrogen storage system, a significant quantity of hydrogen molecules must be adsorbed by the host material under achievable storage conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Further the adsorbed hydrogen molecules must also be efficiently desorbed at suitable temperature (T) and pressure (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Thus, we estimated the quantity of adsorbed hydrogen that could be used at a accessible range of temperature (T) and pressure (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' To calculate the number of H2 molecules remain adsorbed on PCP222-3Li (Occupation number) at different T and P, we calculated the empirical value of hydrogen gas chemical potential (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Then the occupation number (N) is estimated by the following expression and plotted with various T and P in Figure 8(b)[60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 𝑁 = ∑ 𝑛𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇] 𝑁𝑚𝑎𝑥 𝑛=0 ∑ 𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇] 𝑛𝑚𝑎𝑥 𝑛=0 (5) Here Nmax is the maximum number of H2 molecules adsorbed at each Li atom on PCP222, n and gn represents the number of H2 molecules adsorbed and configurational degeneracy for a n respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' kB is the Boltzmann constant and -Eads indicates the average adsorption energy of H2 molecules to PCP222-3Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' m is the empirical value of chemical potential of hydrogen gas at specific T and P, and is obtained by using the following expression [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 𝜇 = 𝐻0(𝑇) − 𝐻0(0) − 𝑇𝑆0(𝑇) + 𝐾𝐵𝑇 ln ( 𝑃 𝑃0) (6) Here H0(T), S0(T) are the enthalpy and entropy of H2 at pressure P0 (1 bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' We can see in Figure 8(b) that the PCP222-3Li can adsorbed H2 molecules giving rise to maximum hydrogen uptake capacity of ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='32 wt% up to the temperature of 80 K and pressure of 30-60 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' When the temperature rises beyond 80 K, the H2 molecules begin to desorb from the PCP222-3Li and the gravimetric density closes to ~5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 wt% (target of US-DOE by 2025) when the temperature reaches 180 K under the pressure of 30-60 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Further rise in temperature, the storage capacity of the PCP222-3Li fall below 4 wt% at 220 K and 40-bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' At a temperature range of 260 K-300 K and pressure range of 1-10 bar, the studied system shows a 100% desorption of hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Thus, we can propose the Li functionalized PCP222 as a low-temperature-adsorption and room-temperature-desorption hydrogen storage material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Under the room temperature (300 K), the studied system shows up to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='32 wt % of usable hydrogen storage capacity with 100% reversibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Thus, we believe that, our studied material Li functionalized PCP222 can be used as an efficient hydrogen storage material satisfying the criteria of US-DOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Figure 8: Plot of Van’t-Hoff desorption temperature for Li functionalized PCP222 at different temperature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2 Desorption temperature For a reversible hydrogen storage media, it is crucial to estimate the desorption temperature of hydrogen molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' We have estimated the desorption temperature (T D) of H2 for the Li functionalized PCP222 using the Van’t Hoff equation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 𝑇𝐷 = ( 𝐸𝑎𝑑𝑠 𝐾𝐵 ) ( ∆𝑆 𝑅 − ln 𝑝) −1 (7) 280 81 7,488 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='656 7- 260- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='824 6- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='992 Desorptiontemperature 240 219K 5- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='160 220 G,wt% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='496 200 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='664 182K 2- 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='000 1 160 145K averageT, 140- 一minT, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 Pressure(atm) 300 (a) (b)Where, Eads represents the computed hydrogen adsorption energy, KB, and R denotes for the Boltzmann constant and R the gas constant respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' P represent s the equilibrium pressure (we take a range of 1 to 5 atm with an increment of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 atm) and △S is the entropy change of hydrogen from its gaseous state to liquid state [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Using the highest and lowest adsorption energy of system (with minimum and maximum H2 gravimetric density, respectively), the maximum and minimum desorption temperatures (TDmax∕TDmin) are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' While, TDmin denotes the minimum temperature necessary to initiate the desorption of H2 molecules, the TDmax is the temperature required for complete desorption process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The estimated desorption temperatures along with the equilibrium pressure is depicted in Figure 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The minimum and maximum temperatures for H2 desorption are determined to be 145 K and 219 K, respectively, at 1 atm pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The estimated average TD of Li functionalized PCP222 is 182 K at 1 atm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' This result reveals that, the system can adsorb its full capacity H2 at cryogenic temperature and desorb all the H2 molecules bellow room temperature at 1 atm pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' However, the desorption temperature can be increases by increase in the equilibrium pressure as presented in Figure 8(a) and as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 Molecular dynamics simulations Figure 9: (a) Potential energy trajectories of hydrogenated PCP222-3Li and (b) Time evolution trajectory of average bond length between the Li atom and C atoms of PCP222 at 300K and 500K, 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='88 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='90 300K (Hartree 500K 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='92 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='94 energy 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='96 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='98 969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='00 Potential 969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='02 969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='04 969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='06 969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='08 0 100 200 300 400 500 600 700 800 900 1000 Time (fs) (a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='6 C-Lidistance@300K Average C-Li distance (A) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 C-Lidistance@50oK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='6 300K,1000fs 500K,1000fs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5 0 100 200 300 400 500 600 700 800 006 1000 Time (fs) (b) To validate the reversibility of hydrogen molecules on PCP222-3Li estimated by the DFT computation, we have carried out molecular dynamics (MD) simulations using the atomistic density matrix propagation (ADMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' ADMP is an extended Lagrangian approach to MD, that uses the gaussian basis function and propagates the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The ADMP-MD simulations is performed on system with highest storage capacity (PCP222-3Li-15H2), at two different temperatures of 300K and 500 K for total time of 1 ps with the time step of 1fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' During the simulations the temperature (kinetic energy thermostat) is maintained by the velocity scaling approach and at every 10 fs, time step, the temperature is checked and corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The time evolution potential energy trajectories and the snapshots are depicted in Figure 9(a) and Figure S3 (in supporting Information) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The MD simulations at 300 K and 1ps illustrate that almost all the H2 molecules fly away from the sorption centers, except 1H2 at each center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Simulations at 500 K shows that all the H2 molecules are desorbed from the host material keeping the host structure intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' This result suggests that the hydrogen storage in Li functionalized PCP222 is reversible in process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' For a viable reversible hydrogen storage material, it is important that the host material must not distorted above the hydrogen desorption temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' To investigate the solidity of host material (PCP222-3Li), we performed the MD simulations on the bare host structure (PCP222-3Li) at room temperature (300 K) and considerably above the H2 desorption temperature (500K) using ADMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The molecular dynamics simulations are performed for 1 ps with a time step of 1 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The time evolution trajectory of average distance between Li atom and the carbon atoms of PCP222 benzene rings is plotted in Figure 9(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' We noticed that the PCP222-3Li structure stays stable at 500 K and almost no change in C-C and C-H bond distance is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The trajectory of average bond length between the Li atom and C atoms of PCP222 benzene rings seem oscillate but the mean value (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='25 Å) and the variation is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' This validates the structural integrity of the host material above the H2 desorption temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Moreover no Li clustering is also noticed after desorption as discussed earlier in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Thus, we believe that PCP222-3Li can be considered for feasible reversible hydrogen storage material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' 4 Summery and Conclusion In this study, we investigated the thermodynamical stability and hydrogen storage capacity of Li functionalized [2,2,2]paracyclophane, using the density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The Li atoms are found to bind with the PCP222 via Dewar mechanism and no clustering of Li atoms over PCP222 was noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Each Li atom on PCP222 could adsorb up to 5H2 molecules via charge polarization mechanism with an average H2 adsorption energy in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='12 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='17 eV/H2, indicating physisorption type of adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Moreover, the average H-H bond distance got elongated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='01 Å, during the adsorption process, which implied that the adsorbed H2 were in molecular form and this fact was also confirmed by the charge distribution analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' When three Li atoms were functionalized on PCP222, the H2 gravimetric capacity of the system was up to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='32 wt% which was fairly above the US-DOE requirements for practical hydrogen applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' During saturation of H2 adsorption, the host material displayed no significant change in geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' The thermodynamic usable hydrogen capacity was found up to ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='32 wt% at the temperature of 80 K and pressure of 30-60 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' On further increase in temperature, up to 180 K under the pressure of 30-60 bar, the PCP222-3Li hydrogen uptake capacity approached 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='5wt% which is the target of DOE by 2025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' At a temperature range of 260 K-300 K and pressure range of 1-10 bar, the PCP222- 3Li system showed 100% desorption of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Molecular dynamic simulation confirmed that at 300 K, almost all the H2 molecules flied away except 1H2 at each center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Simulations at 500 K showed that all the H2 molecules are desorbed from the host material keeping the structure of the host structure intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Since, there is no experimental works reported on Li functionalized PCP222 for hydrogen storage, we hope our computational work will contribute significantly to the research of hydrogen storage in macrocyclic compounds and provide supporting reference for the future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' References [1] Sachin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Shet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Shanmuga Priya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' Sudhakar, Muhammad Tahir, A review on current trends in potential use of metal-organic framework for hydrogen storage, International Journal of Hydrogen Energy, 2021, 46, (21), 11782-11803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} +page_content='ijhydene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E2T4oBgHgl3EQfkAeM/content/2301.03974v1.pdf'} 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0000000000000000000000000000000000000000..0e1199a17ae5ac25d12a4bfb31e94596c6cabb0d --- /dev/null +++ b/59E4T4oBgHgl3EQfcAxC/content/tmp_files/2301.05079v1.pdf.txt @@ -0,0 +1,1133 @@ +Deep learning enhanced noise spectroscopy of a spin qubit environment +Stefano Martina,1, 2, ∗ Santiago Hern´andez-G´omez,3, 2, † Stefano +Gherardini,4, 2, ‡ Filippo Caruso,1, 2, § and Nicole Fabbri5, 2, ¶ +1Dipartimento di Fisica e Astronomia, Universit`a di Firenze, I-50019, Sesto Fiorentino, Italy +2European Laboratory for Non-linear Spectroscopy (LENS), +Universit`a di Firenze, I-50019 Sesto Fiorentino, Italy +3Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139 +4Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO), +Area Science Park, Basovizza, I-34149 Trieste, Italy +5Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO), I-50019 Sesto Fiorentino, Italy +(Dated: January 13, 2023) +The undesired interaction of a quantum system with its environment generally leads to a coherence +decay of superposition states in time. A precise knowledge of the spectral content of the noise induced +by the environment is crucial to protect qubit coherence and optimize its employment in quantum +device applications. We experimentally show that the use of neural networks can highly increase +the accuracy of noise spectroscopy, by reconstructing the power spectral density that characterizes +an ensemble of carbon impurities around a nitrogen-vacancy (NV) center in diamond. +Neural +networks are trained over spin coherence functions of the NV center subjected to different Carr- +Purcell sequences, typically used for dynamical decoupling (DD). As a result, we determine that +deep learning models can be more accurate than standard DD noise-spectroscopy techniques, by +requiring at the same time a much smaller number of DD sequences. +I. +INTRODUCTION +Quantum sensing combines theoretical results with ex- +perimental and engineering techniques to carry out infer- +ence of signals with improved accuracy and/or less com- +putation time by making use of quantum physics [1, 2]. +A quantum sensor takes advantage of the fragility of +its quantum properties, such as quantum coherence or +entanglement, to improve the detection of external per- +turbations with higher accuracy compared to any classic +sensor. +However, this same property implies that the +quantum sensor is subjected to detrimental noise stem- +ming from the coupling with its environment. For this +reason, it is desirable to fully characterize the sensor’s +environment, either to filter out its detrimental effect, or +to take it into account when detecting external signals, +for example, in algorithms using quantum optimal con- +trol [3–6]. +Neural networks (NN) [7, 8], i.e., algorithmic models +provided by the interconnection of a group of nodes com- +monly called neurons, could be a powerful tool to infer +the sensor’s environment. In this context, deep learning +has been already proposed theoretically for the classifi- +cation and detection of quantum noise features [9–11], +and employed experimentally for the following tasks. (a) +Estimating the spectra of minuscule amounts of complex +molecules [12] for nano nuclear magnetic resonance; (b) +the sensing of magnetic-field strength at room temper- +∗ stefano.martina@unifi.it; Equal contribution to this work +† shergom@mit.edu; Equal contribution to this work +‡ stefano.gherardini@ino.cnr.it +§ filippo.caruso@unifi.it +¶ fabbri@lens.unifi.it +ature with high precision [13, 14] by using nitrogen va- +cancy (NV) centers; (c) performing error mitigation [15] +and noise learning [16–18]; (d) the tracking of quantum +trajectories [19]; (e) classification of many-body quantum +states [20] in superconducting quantum circuits. How- +ever, to our knowledge, experimental noise spectroscopy +in single color centers in diamond via deep learning is +still missing. +In this paper, we demonstrate that NN can be used +to process the data obtained by a qubit, operating as +a quantum sensor, and then reconstruct the noise spec- +trum that induces dephasing into the qubit itself. In par- +ticular, we focus on a qubit under dynamical decoupling +(DD) control sequences [21, 22] in the presence of classical +random noise with an unknown power density spectrum, +usually denoted as noise spectral density (NSD). Beyond +testing numerically our machine learning models, we use +a single NV center in diamond as a spin qubit sensor and +we perform a spectroscopic reconstruction of the mag- +netic noise of its local environment. The latter comprises +13C nuclear spins randomly distributed in the diamond +lattice [23–25] (see Fig. 1). The dephasing affecting the +qubit sensor is analyzed by applying a set of DD con- +trol pulses that realize filter functions [21, 22, 26, 27] in +the frequency domain. The filter functions are designed +to select specific noise components, without sensing all +other system-bath interactions. A widely used DD con- +trol pulse is the Carr-Purcell (CP) sequence [1, 28] that +is given by N equidistant π pulses, performed between an +initial and a final π/2 pulse. CP sequences act in the fre- +quency domain approximately as Dirac comb filters [29]; +hence, they have been used to perform spectroscopy of in- +tricate signals, e.g., for noise spectroscopy [30, 31]. With +this protocol, the requirement to achieve high values of +the noise reconstruction accuracy is to perform sequences +arXiv:2301.05079v1 [quant-ph] 12 Jan 2023 + +2 +with a high number of pulses meaning N ∈ [30, 120] (as in +Ref. [32]) or higher, so that the Dirac comb filter approx- +imation remains valid (in fact, N determines the filter +width). This usually leads to long experiments to recon- +struct the whole spectrum of the noise. Other techniques +using non-equidistant or even more sophisticated DD se- +quences [4, 33–36] have proved to be effective for noise +sensing, but sometimes at the price of a higher computa- +tional burden. +For our sensing task, NN are designed to solve a re- +gression problem, i.e., the reconstruction of the NSD. +Here, we assume that the NSD of the bath of spins has a +Gaussian profile [32, 37, 38]. The Gaussian NSD is thus +parametrized as a function of key parameters, i.e., the +mean value, variance, offset and noise power that we aim +to reconstruct. Note that our proposal can be adapted to +other parametrized NSD functions. The NN are trained +over a set of synthetic data generated by simulating how +the coherence of the qubit sensor decays over time under +the influence of both the CP control pulses and the NSD. +Moreover, to make the measurement statistics as close as +possible to the ones obtained from the experiments, extra +artificial errors are added. +Our approach using NN entails the following advan- +tages that we have proven experimentally. (i) NN have +the capability to predict never-before-seen experimental +data, and they can work with a better reconstruction +accuracy (even up to 7 times better, as shown in the +section Results below) than standard noise spectroscopy, +as the ´Alvarez-Suter method [31], by making use at the +same time of DD control sequence with a much smaller +number of pulses. (ii) The training dataset, which can +contain both synthetic and experimental data, is gener- +ated just once and then it can be applied several times, +as long as the new collected data reproduce the physical +context under analysis. In connection with (i), we are +going to show that the amount of data used as input to +the NN can be smaller than the one needed to resolve the +NSD by means of standard noise spectroscopy methods. +From our knowledge, this work is the first experimen- +tal proof of enhanced reconstruction performance with +NN for carrying out noise spectroscopy in single color +centers in diamond. We thus expect that the techniques +discussed here could fast become a novel standard spec- +troscopy tool both for such quantum systems and other +quantum platforms in which regression problems have to +be solved. +II. +RESULTS +A. +Generation of training dataset +The training dataset is composed of synthetic data that +are originated by simulating the coherence decay of the +qubit sensor in a noise spectroscopy experiment based on +DD, as the one depicted in Fig 1. This standard sensing +procedure, which stems from Ramsey interferometry [1], +maps information about the quantum coherence of the +sensor into the population in |0⟩ that is then effectively +recorded. After having initialized the qubit sensor in the +ground state |0⟩, a π/2 pulse is applied such that the +qubit state |ψ⟩ is the superposition (|0⟩+|1⟩)/ +√ +2. Then, +we perform a CP control sequence consisting in a train +of π pulses that flips repeatedly the qubit, and finally, a +second π/2 pulse is applied in order to map the phase of +the qubit into its population. The probability that the +state of the quantum sensor is |0⟩, which corresponds to +the observable population, equals to [1, 32] +P = 1 +2 (1 + C(τ, N)) , +(1) +where N is the number of π pulses and τ is the time +between them. The coherence function C(τ, N) is sim- +ulated numerically, for a set of different values of τ and +N, to generate the training dataset. +Let us now introduce the decoherence function that +quantifies how the quantum coherence C(τ, N) is modi- +fied under the action of both the external bath of spins +and a set of CP control pulses. +The control sequence +has the effect to modulate the coherence content of the +qubit sensor, while the interaction with the bath, asso- +ciated to the NSD S(ω), tends on average to destroy +such coherence. Overall, under the joint presence of con- +trol fields and a noise source, the coherence decays as +C(τ, N) ≡ e−χ(τ,N), where χ(τ, N) denotes the decoher- +ence function [27, 39–41]: +χ(τ, N) = +� +dω +πω2 F(ω, τ, N)S(ω) . +(2) +In Eq. (2), the filter function F(ω, τ, N) ≡ |Y (ω, τ, N)|2 +is the square modulus of the Fourier transform of the +so-called modulation function y(t, τ, N). +The latter is +constant piecewise, with values ±1, and switches sign at +the times t = τ/2, 3τ/2, . . . , (N − 1/2)τ where each π +pulse is applied [2]. Notice that we are assuming that +the π pulses are instantaneous, a reasonable assumption +for our experimental setup where a π pulse duration is +∼ 0.1 µs and the time between pulses is τ ∈ [3.3, 6.1] µs. +Let us now recall the expression, in the frequency domain, +of the filter function for a CP sequence with even N: +F(ω, τ, N) = 8 sin2 +�ωτN +2 +� +sec2 �ωτ +2 +� +sin4 �ωτ +4 +� +, (3) +while for odd N, sin2(ωτN/2) has to be replaced with +cos2(ωτN/2) [2, 26]. +In order to generate the training dataset, the NSD +S(ω) is parameterized as +S(ω) = s0 + A exp +� +−(ω − ωc)2 +2σ2 +� +. +(4) +Thus, being a Gaussian distribution, the NSD is fully de- +scribed by the offset s0, amplitude A, width σ and center +ωc. +For the training dataset in the paper, the values + +3 +FIG. 1: NV center and Neural Networks for noise spectroscopy. The NV center is surrounded by an +ensemble of 13C nuclear spins (orange spheres) that collectively induce dephasing to the NV electronic spin (blue +sphere). The NV electronic spin is controlled with a DD sequence (specifically, a Carr-Purcell (CP) sequence) with +the aim to measure its dephasing, and therefore characterize the NSD of the nuclear spin bath, i.e., S(ω; s0, A, σ). +The CP sequence is formed by N equidistant π pulses in between an initial and a final π/2 pulse. The time τ +between the π pulses determines the measurement total time T = Nτ, given that the time between the first π/2 and +the train of π pulse and the time between the last π and π/2 pulses are both equal to τ/2. Then, we measure the +output of this experiment, which is the probability P = 1 +2(1 + C(t)) that the NV center remains in the initial state +|0⟩. The spin coherence function C(t) – evaluated at previously-determined times in the set T ∈ {t1, t2, . . . , tn} (the +tk’s are obtained by changing τ with N fixed) – is the input of the designed Neural Networks (NN). After being +trained, the NN return the estimation of the NSD parameters. +of these parameters are taken from the following inter- +vals: s0 ∈ [4 · 10−4, 4 · 10−3] MHz; A ∈ [0.3, 0.7] MHz; +σ ∈ [2 · 10−3, 9 · 10−3] MHz. +Instead, ωc is kept con- +stant. +This is because in our experimental setup the +NSD stems from the interaction with a large ensemble +of unresolved 13C impurities (nuclear spin bath) around +the NV electronic spin. Therefore, the center of the NSD +corresponds to the Larmor frequency ωc = γB, where +γ = 1.0705 kHz/G is the gyromagnetic ratio of the 13C +nuclear spins, and B is the amplitude of a static mag- +netic field aligned with the NV quantization axis, z. Such +static magnetic field is well known during the experimen- +tal procedure since it determines the NV electronic spin +resonances (B = 403.2 ± 2 G). +The training dataset is generated by uniformly sam- +pling 104 sets of parameters within the chosen intervals. +Hence, overall we consider 104 distinct sequences of NSD +parameters that are used to simulate different coherence +curves C(τ, N). These sequences are taken in the time +intervals τ ∈ [3.3, 3.66] µs and [5.5, 6.1] µs with sampling +time ∆τ = 1 ns (∆τ = 20 ns in the experimental case, see +below), and for N = {1, 8, 16, 24, 32, 40, 48}. These inter- +vals are significant for our study because they include the +values of τ at which the coherence decay curve exhibits +the first and second order collapses induced on the qubit +sensor by the bath of 13C impurities [42]. +Finally, in +order to make the synthetic data used to train the NN +closer to the experimental setting, extra artificial errors +sampled by a normal distribution with standard devia- +tion equal to 0.05 (comparable with the expected error +in our experimental measurements) are added to every +point of the generated coherence decay curves. In this +way, one may mitigate the over-fitting of the employed +machine learning models that are thus expected to better +generalize to unseen data. In general, a model trained on +synthetic data cannot be successfully applied to real data +without fine tuning it. But in our case, it becomes possi- +ble, probably due to the fact that the simulated data of +the coherence decay are quite close to the experimentally +observed decay data induced by the environment. +As final remark, notice that, from the 104 simulated +curves C(τ, N), 6000 are used for the training of the NN +and 2000 for their validation. Instead, the test step is +performed either by using the remaining 2000 simulated +curves, or by using experimental data as described below. +B. +Neural networks working principles +Let us describe the main working features of the NN +employed in this paper to carry out noise spectroscopy. +Specifically, we are going to use the multi-layer percep- +tron (MLP) that is composed of fully-connected layers, +each of them with a variable number of artificial neurons. +A single artificial neuron returns as output the scalar +ˆy ≡ Σ(wT · x + b) +(5) +that, by definition, is provided by applying the non-linear +function Σ : R → R to the weighted sum of the input +vector x ∈ Rk to which the bias term b ∈ R is added. +w ∈ Rk denotes the vector of weights. In our analysis, +the activation function Σ is chosen equal to the rectifier + +4 +Σ(x) ≡ max(0, x) [43, 44]. Thus, a MLP layer composed +of q neurons (each with k inputs) returns the vector +ˆy ≡ Σ(W T x + b), +(6) +where ˆy ∈ Rq, W ∈ Rk×q is the matrix of weights (W +collects all the weight vectors of the single neurons), and +b ∈ Rq is the vector of the biases. Hence, a MLP with L +layers is ruled by the recursion equation +h[ℓ] ≡ Σ +� +W[ℓ]T h[ℓ − 1] + b[ℓ] +� +, +(7) +where ℓ = 1, . . . , L is the index over the number of layers +and h[0] ≡ x. In Eq. (7), W[ℓ] and b[ℓ] are, respectively, +the weights and the biases of the ℓ-th layer. The output +vector of the MLP is ˆy ≡ h[L]. It is worth noting that +the number, dimension and activation functions (they +are usually denoted as the hyperparameters ξ) of the NN +layers are chosen through a single optimization routine +(cfr. Methods). +Let us now introduce the supervised learning process. +Ideally, the purpose of the latter is to find the parameters +θ∗ = argminθRD(θ, ξ) that minimize the theoretical risk +function +RD(θ, ξ) ≡ E(x,y)∼D [L (ˆy, y)] , +(8) +where θ ≡ {W[1], b[1], . . . , W[L], b[L]}, and ˆy are the +estimated values of y. +By definition, RD is the ex- +pected value of the loss function L for (x, y) sampled +from the distribution D that generates the dataset [45]. +The loss function L is a differentiable function that mea- +sures the distance between the prediction ˆy (output of +the MLP) and the desired output y. However, since one +can only dispose of a finite set S = {(x, y)1, . . . , (x, y)m} +of samples to train, validate and test the employed ML +models, the theoretical risk function is approximated by +the empirical risk function. +Considering the partition +{Str, Sva, Ste} of S in training (Str), validation (Sva) and +test (Ste) sets, the empirical risk function is defined by: +RStr(θ, ξ) ≡ +1 +|Str| +� +(x,y)∈Str +L (ˆy, y) , +(9) +where |Str| is the cardinality of the training set. In fact, +RStr is the arithmetic mean of the loss function L eval- +uated on the samples of the training set Str. +In our paper, we take the loss function L equal to the +Mean Squared Error (MSE), also called L2 loss: +L(ˆy, y) = 1 +q +q +� +i=1 +(ˆyi − yi)2 +(10) +for the q outputs of the last layer (in our case three, +corresponding to the noise parameters s0, A, σ). The +MLP is trained by minimizing (step-by-step over time) +the empirical risk function RStr(θ, ξ) with respect to θ +by means of the mini-batch gradient descent method, so +as to obtain the optimal value θ∗ of the NN parameters. +Each gradient descent step is defined by +θt+1 = θt − η∇θ +1 +B +B +� +b=1 +L(ˆyb,t, yb,t), +(11) +where θ0 is a randomly chosen starting point, η is the +learning rate that defines the length of the step and +∇θ 1 +B +�B +b=1 L(ˆyt,b, yt,b) is the gradient of the loss func- +tion. The gradient is calculated for any time t on a batch +of B elements taken from the training set, and the sub- +script θ in ∇θ indicates that the variables of L during +the gradient evaluation are the weights of the NN. In +this paper, RStr is minimized by means of Adam [46] +that is a gradient-based optimization algorithm perform- +ing the adaptive estimation of lower-order moments. The +minimization is stopped when the time-derivative of the +risk function evaluated on the validation set RSva(θ∗, ξ) +becomes positive (early stopping strategy) or after a pre- +defined number of gradient steps using all the data of the +training set (called epochs). Then, we use RSva(θ∗, ξ) to +check if the MLP works also for unseen data and tune +the hyperparameters ξ (cfr. Methods). Finally, the test +set Ste is employed to calculate the metrics (discussed in +detail below) used to generate the figures with the results +that we are going to illustrate. +C. +Training and numerical test of neural networks +We now show the results obtained by using the trained +machine learning models to infer the value of the NSD +parameters {s0, A, σ}. As already mentioned, the NN are +tested with 2000 different NSD parameters. For each of +these sets of parameters, the curves C(τ, N) have been +simulated as described in the previous subsections. +In order to determine the smallest amount of data re- +quired to reconstruct the NSD, we perform the training, +validation and test of the NN with sub-sets of the simu- +lated curves. These sub-sets are defined by introducing +the variable N that denotes the upper bound for the num- +ber of pulses N ≤ N considered during the whole process. +For example, for N = 16 only the curves C(τ, N) with +N ∈ {1, 8, 16} are considered. Note that the sub-sets de- +fined for each value of N contain the curves for all the +different NSD parameters (6000 for training, 2000 for val- +idation, and 2000 for testing), and for all the times τ in +the intervals defined before. +The results of this analysis are shown in Fig. 2 (orange +data), where the MSE (the loss function) between the in- +ferred parameters ( ˆs0, ˆA, ˆσ) and the original parameters +(s0, A, σ) used to generate the dataset is plotted as a +function of N. Remarkably, the MSE seems to achieve +its minimum value after N = 16. This entails that the +NN do not significantly improve their precision on the +reconstruction of the NSD by using more data to train +the NN beyond this point. + +5 +0 +10 +20 +30 +40 +50 +¯N +0.0 +0.1 +0.2 +0.3 +MSE(s0, A, σ) +FIG. 2: Mean-square-errors (MSE) between original +and estimated NSD parameters for a set of 2000 test +cases. Orange bullets with dash-dotted line are the +mean values returned by NN. Blue squares with dotted +line are the mean values provided by the HS method. +Finally, shaded areas denote the standard deviation, +taking into account all the 2000 cases. +To establish how accurately a NN reconstructs the +NSD, we need to compare the corresponding results with +those of a different method. In particular, we concentrate +on the method used in Ref. [32], which is itself based on +Refs. [30, 31]. According to them, the decay of the coher- +ence function C(τ, N) is analyzed as a function of N, for +each fixed value of τi, i.e., for each fixed frequency com- +ponent of the filter functions. In the limit of high N, the +decay of the coherence is exponential, with a rate that is +inversely proportional to the amplitude of the NSD [30]. +In other words, the amplitude of the NSD is directly es- +timated for a discrete set of frequencies (each propor- +tional to 1/τ). In contrast with the original proposals in +Refs. [30, 31], the method in Ref [32] demonstrates that +it is better to use the harmonics of the filter functions +to reconstruct the NSD, in order to avoid extra broad- +ening of the reconstructed spectrum. For this reason, we +denote this method as Harmonics Spectroscopy (HS). +We have analyzed the same 2000 different curves +C(τ, N) (used to test the machine learning models) also +with the HS method. The results are collected and shown +in Fig. 2 (blue data), where the first point is for N = 16. +This is due to the fact that, by definition, the HS method +fits the decay of the coherence as a function of N. This +is possible only for a dataset with at least three points +(in this case N = 1, 8, 16). As one can observe in Fig. 2, +the MSE values for the HS method (blue region) are al- +ways above the MSE values for the NN method (orange +region), especially for lower values of N. These results +demonstrate that the NN method can predict the pa- +rameters of the NSD with an improved accuracy (up to +5 times larger) with respect to the HS method. The test +presented in this subsection have been performed with +simulated data. In the next subsection we are going to +repeat the same test but with experimental data. +D. +Experimental test of neural networks +By this point we know that NN can reliably predict the +NSD from noisy simulated data. In this section, we want +to use the NN (trained and validated with noisy simu- +lated data) to reconstruct the NSD using experimental +data. +As quantum sensor we use a spin qubit encoded in the +electronic spin of the ground state of a single nitrogen- +vacancy (NV) center in a bulk diamond at room temper- +ature. This system has proven as a sensitive quantum +probe of magnetic fields, with outstanding spacial reso- +lution and sensitivity [47, 48]. The diamond sample in +our experiments has a natural abundance of 13C impu- +rities (1.1%) that are randomly distributed in the dia- +mond lattice [23–25]. The 13C nuclear spins constitute +the external environment of the NV center. +They act +as a collective bath of spins that induces dephasing into +the NV electronic spin, limiting the its coherence time +T2 ≈ 100 µs. In the presence of strong bias magnetic +field (≥ 150G) [32, 49], the weak coupling of the NV spin +with these carbon impurities can be modeled as a clas- +sical stochastic field. The latter has a power spectrum +density (here called NSD) that follows a Gaussian dis- +tribution centered at the Larmor frequency of the 13C +nuclear spins. In order to measure the NV spin coher- +ence function C(τ, N), we apply a train of π pulses (in +our case a CP sequence) to the NV spin qubit following +the DD protocol described in Fig 1. For more details on +the experimental implementation and Hamiltonian of the +system see Ref. [32]. We have performed this experiment +for N = {1, 8, 16, 24, 32, 40, 48}, and for τ ∈ [3.3, 3.66] µs +and [5.5, 6.1] µs with sampling time ∆t = 20 ns. The +results are shown in Fig.3(a) (blue bullets). Then, the +collected coherence functions have been processed and +employed to reconstruct the NSD parameters by means +of both the NN (trained with the generated dataset) and +the HS method. In contrast with the test using simu- +lated data in the previous section, in the experimental +case we do not know the exact values of the NSD pa- +rameters. +Therefore, we cannot calculate the MSE to +quantify the accuracy of the reconstructed parameters. +In order to estimate such accuracy we have used the fol- +lowing procedure: from the inferred NSD, the coherence +curves C(τ, N) are simulated and then compared with +the experimental results. An example of this comparison +is shown in Fig.3(a), where C(τ, N) is simulated under +the assumption that the NSD parameters are inferred ei- +ther by the machine learning models (orange) or by the +HS method (red), both for N = 16. Qualitatively it is +clear that the orange curves are much closer to the ex- +perimental data, than the red curves. +There are several options to quantitatively compare +the experimental data and the simulation results. Here +we use both the reduced chi-squared χ2 +ν [50], and the + +6 +FIG. 3: (a) Coherence function C(τ, N). The experimental data (blue bullets) are shown together with the +simulated ones using the NSD predicted respectively by the HS method (red lines) and machine learning models +(orange lines), both for N = 16. (b) Reduced chi-squared χ2 +ν, obtained by comparing simulation and experimental +data, as a function of N. As in panel (a), orange and red curves refer to the NN and HS method, respectively. +Instead, the dashed line denotes the value of the reduced chi-squared for the HS method when we employ additional +measurements for N = 56, 64, 72, 80 in the interval τ ∈ [5.5, 6.1] µs. Inset: Same results but quantified by the +Mean-Absolute-Error (MAE) between the experimental data and the predicted C(τ, N). +Mean-Absolute-Error (MAE) [51] between the exper- +imental data and the predicted coherence functions +C(τ, N) (see Methods for more details). The results of +this comparison are shown in Fig. 3(b), where χ2 +ν and +the MAE are plotted as a function of N. Remarkably, +the NSD reconstructed by the NN for N = 16 behaves +better that any case using the HS method. It is worth +observing that the same experimental data used to infer +the NSD parameters are partially used to estimate the +χ2 +ν and MAE(C(t)). For example, for N = 16, only the +data for N = 1, 8, 16 are used to reconstruct the NSD, +but we employ all the data N = 1, 8, 16, . . . , 48 to ob- +tain the χ2 +ν and MAE(C(t)). Overall, we have observed +enhanced performance in reconstructing the NSD of the +collective bath of spins, with a maximum improvement +(about 7 times higher) for N = 16. In other words, for +N = 16, once we reconstruct the NSD, the quantum sen- +sor dynamics can be predicted with an average square +deviation of ≃ 1.86 experimental error-bars by using the +NN method, or with an average square deviation of ≃ 13 +error-bars if we use the HS method. +III. +DISCUSSION +As shown pictorially in Fig. 1, the NN takes as input +the spin qubit coherence functions (the coherence of the +quantum sensor decays due to the presence of the exter- +nal bath) obtained by using a set of different CP control +sequences. The NN returns as output the parameters of +the unknown NSD in the frequency domain. +One can +thus note that the NN, once validated, acts as a “time- +frequency converter” (making use of a quite complicated +deconvolution) from the measured signals living in the +time domain – the spin coherence functions – to the NSD +defined in the frequency domain. +The results shown in the previous section, and sum- +marized in Figs. 2 and 3(b), demonstrate that NN can +be used to reconstruct the NSD affecting a quantum sen- +sor, achieving higher precision and with considerable less +data than the standard HS method. +Improved values +of the reconstruction accuracy have been obtained with +simulated and experimental data. Both the HS and NN +methods are comparable – in terms of NSD reconstruc- +tion accuracy – for high values of N, but not for small +ones, where NN gives significantly better results. More- +over, the main result of our study is that NN trained +with data obtained for N = 16 reconstruct the NSD +more accurately than the best estimate provided by the +HS method with N = 48. This improvement is remark- +able by itself, but it becomes more significant when we +consider that the time required to complete these exper- +iments has a growth faster than a linear function with +respect to N, following an arithmetic progression. As an +example, the total time to perform all the experiments in +the case of N = 16 and 48 is respectively ≃ 10 minutes +and ≃ 1.2 hours [52]. This is an under-estimation of the +time difference between methods, because we are only +considering the bare measurement time, without taking +into account the time delay between different experi- +ments. Furthermore, it is worth stressing that our results +also show that deep learning has a predictive power since +it can be applied to never-before-seen data. This natu- +rally provides to the employed machine learning models +a connotation of robustness that is crucial in real appli- +cations. + +7 +Let us observe that regression tasks, which are suc- +cessfully solved by multi-layer perceptrons (one of the +easiest form of NN), are less common with respect to the +ones to carry out classification; a review of some exam- +ple datasets and methods for regression is in Ref. [53]. +Hence, we expect that the synthetic data used in this +work could be useful as a test bed also to the audience +of machine learning researchers and developers solving +regression problems in different contexts. With this in +mind, we share the training dataset with synthetic data +and our codes for their generation, as well as the code for +machine learning experiments and NSD reconstruction +[available on the GitHub repository (see Section “Data +and code availability”)]. In this way, we promote the im- +provement of machine learning models for noise sensing +purposes and their use to solve different regression tasks +in the quantum estimation framework. +Conclusions & outlooks +In this paper, we use NN to carry out noise spec- +troscopy with a quantum sensor using dynamical decou- +pling sequences with a much smaller number of π pulses +and, at the same time, achieving a higher reconstruc- +tion accuracy than standard methods (e.g., HS proto- +col). This means that with our proposal the noise spec- +troscopy procedure will take less time and give better +results. More in detail, we experimentally demonstrate +the capability of NN to reconstruct the NSD of the collec- +tive nuclear spin bath that surrounds an electronic spin +qubit, i.e., the ground state of a single nitrogen-vacancy +center in bulk diamond at room temperature. +To conclude, we outline some possible outlooks for our +work. +First of all, one may evaluate the performance +of NN that are trained over input data obtained using +DD control sequences with more degrees of freedom than +the CP ones [54–58]. Secondly, deep learning might be +applied to noise spectroscopy techniques beyond the HS +methods, as for example optimal band-limited control +protocols [34, 35] and even non-Gaussian noise charac- +terization [59–61]. In addition, it might be worth inves- +tigating how deep learning can be integrated to quantum +sensing procedures that rely on the so-called stochastic +quantum Zeno effect [62, 63], whereby the quantum probe +is subjected to a sequence of quantum measurements that +in the ideal case are designed to confine the dynamics of +the probe around the initial (nominal) state [33, 64, 65]. +We are also confident that the extent of our results can +be quite easily replicated in other experimental settings, +as e.g., superconducting flux qubits [66, 67], trapped +ions [68, 69], cold atoms [70, 71], quantum dots [72, 73], +NMR experiments in molecules [31, 74], and nanoelec- +tronic devices [75]. For such a purpose, one might slightly +adapt the deep learning techniques used here to methods +tailored for time series. +IV. +METHODS +A. +Technical details on the training of NN +The NN models are developed using the PyTorch +framework [76] on a machine with 32 CPU cores, 126Gb +of RAM and a GeForce RTX 3090 GPU. The training +time, including the optimization of the hyperparameters, +is around 12 hours for each N . +The hyperparameters optimization is implemented by +means of the Ray Tune library [77]. The Hyperopt pack- +age [78] uses the Tree-structured Parzen Estimators [79] +algorithm as a Bayesian optimization to search for the +best choice of the hyperparameters within a predefined +search space. Hyperopt suggest the likely better configu- +rations of the hyperparameters and the underlying model +is updated after each trial that is run. The ASHA sched- +uler [80] is then used to stop the run of the least promising +trials chosen by the search algorithm, thus speeding up +the hyperparameters optimization process. +The optimized hyperparameters are the following. (1) +The number of hidden layers decides the value of L − +1 in Eq. (7). The hidden layers are between the input +layer h[0] and the output layer h[L]. (2) The dimension +of the hidden layers is the value of q in Eq. (6) that, +for the sake of simplicity, is equal for all the layers in +Eq. (7). Both the number and dimension of the hidden +layers are chosen by sampling log-uniformly an integer +value from the space [1, 32) and [1, 1024), respectively. +(3) The learning rate is responsible for the length of the +gradient descent step and it is optimized with a choice +between 10−2, 10−3 and 10−4. (4) The batch size denotes +the dimension of the batch on which the loss function is +summed for the gradient calculation in a single descent +step. The batch size is chosen between 2, 4, 8, 16, 32. +(5) The dropout is a regularization strategy that aims +to reduce the overfitting by randomly turn off the NN +neurons with a predefined probability. Such probability +is one among 0 (no dropout), 0.2 and 0.5. (6) The weight +decay is another regularization technique that adds to the +loss function the squared weights of the NN multiplied +by a decay value. The latter value is optimized choosing +between 0 (no decay), 10−6, 10−5, 10−4 and 10−3. +B. +Definition of quantifiers for reconstruction +accuracy +The accuracy NN and HS methods can be estimated by +using the reconstructed NSD to simulate the coherence +function C(τ, N), and ‘measuring’ the distance between +the simulated data and the experimental values. To do +so, we use the reduced chi-squared χ2 +ν, and the Mean- +Absolute-Error (MAE(C)): We define Ce ± δCe (Cs) as +the experimental (simulated) values of C(τ, N), where +δCe is the standard deviation of the experimental data. + +8 +Then we can write reduced chi-squared and the MAE as +χ2 +ν ≡ 1 +ν +� +n,N +(Ce(τn, N) − Cs(τn, N))2 +δCe(τn, N)2 +(12) +MAE(C) ≡ 1 +ν +� +n,N +|Ce(τn, N) − Cs(τn, N)| , +(13) +where N = {1, 8, 16, 24, . . . , N}, {τn} are the values of +the time between pulses within the time intervals defined +in main text, and ν is the total number of elements in the +sum. Notice that χ2 +ν takes into account the experimental +precision to scale the difference between experiment and +simulation. The results showing both χ2 +ν and the MAE +are in Fig. 3. +DATA AND CODE AVAILABILITY +The source codes for the generation of the train- +ing dataset and the machine learning experiments +are available on GitHub +at the following address: +https://github.com/trianam/noiseSpectroscopyNV +ACKNOWLEDGEMENTS +This work was supported by the European Com- +mission’s +Horizon +Europe +Framework +Programme +under +the +Research +and +Innovation +Action +GA +n. 101070546–MUQUABIS, and by the European De- +fence Agency under the project Q-LAMPS Contract No +B PRJ-RT-989. +S. H. G. acknowledges support from +CNR-FOE-LENS-2020. F. C. and S. M. acknowledge the +European Union’s Horizon 2020 research and innovation +programme under FET-OPEN GA n. 828946–PATHOS. +[1] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum +sensing, Rev. Mod. Phys. 89, 035002 (2017). +[2] S. Hern´andez-G´omez and N. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Sesto Fiorentino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Italy 2European Laboratory for Non-linear Spectroscopy (LENS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Universit`a di Firenze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' I-50019 Sesto Fiorentino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Italy 3Research Laboratory of Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' MA 02139 4Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Area Science Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Basovizza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' I-34149 Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Italy 5Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' I-50019 Sesto Fiorentino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Italy (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 2023) The undesired interaction of a quantum system with its environment generally leads to a coherence decay of superposition states in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' A precise knowledge of the spectral content of the noise induced by the environment is crucial to protect qubit coherence and optimize its employment in quantum device applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' We experimentally show that the use of neural networks can highly increase the accuracy of noise spectroscopy, by reconstructing the power spectral density that characterizes an ensemble of carbon impurities around a nitrogen-vacancy (NV) center in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Neural networks are trained over spin coherence functions of the NV center subjected to different Carr- Purcell sequences, typically used for dynamical decoupling (DD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' As a result, we determine that deep learning models can be more accurate than standard DD noise-spectroscopy techniques, by requiring at the same time a much smaller number of DD sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' INTRODUCTION Quantum sensing combines theoretical results with ex- perimental and engineering techniques to carry out infer- ence of signals with improved accuracy and/or less com- putation time by making use of quantum physics [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' A quantum sensor takes advantage of the fragility of its quantum properties, such as quantum coherence or entanglement, to improve the detection of external per- turbations with higher accuracy compared to any classic sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' However, this same property implies that the quantum sensor is subjected to detrimental noise stem- ming from the coupling with its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' For this reason, it is desirable to fully characterize the sensor’s environment, either to filter out its detrimental effect, or to take it into account when detecting external signals, for example, in algorithms using quantum optimal con- trol [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Neural networks (NN) [7, 8], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=', algorithmic models provided by the interconnection of a group of nodes com- monly called neurons, could be a powerful tool to infer the sensor’s environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In this context, deep learning has been already proposed theoretically for the classifi- cation and detection of quantum noise features [9–11], and employed experimentally for the following tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (a) Estimating the spectra of minuscule amounts of complex molecules [12] for nano nuclear magnetic resonance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (b) the sensing of magnetic-field strength at room temper- ∗ stefano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='martina@unifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Equal contribution to this work † shergom@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Equal contribution to this work ‡ stefano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='gherardini@ino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='cnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='it § filippo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='caruso@unifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='it ¶ fabbri@lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='unifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='it ature with high precision [13, 14] by using nitrogen va- cancy (NV) centers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (c) performing error mitigation [15] and noise learning [16–18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (d) the tracking of quantum trajectories [19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (e) classification of many-body quantum states [20] in superconducting quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' How- ever, to our knowledge, experimental noise spectroscopy in single color centers in diamond via deep learning is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In this paper, we demonstrate that NN can be used to process the data obtained by a qubit, operating as a quantum sensor, and then reconstruct the noise spec- trum that induces dephasing into the qubit itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In par- ticular, we focus on a qubit under dynamical decoupling (DD) control sequences [21, 22] in the presence of classical random noise with an unknown power density spectrum, usually denoted as noise spectral density (NSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Beyond testing numerically our machine learning models, we use a single NV center in diamond as a spin qubit sensor and we perform a spectroscopic reconstruction of the mag- netic noise of its local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The latter comprises 13C nuclear spins randomly distributed in the diamond lattice [23–25] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The dephasing affecting the qubit sensor is analyzed by applying a set of DD con- trol pulses that realize filter functions [21, 22, 26, 27] in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The filter functions are designed to select specific noise components, without sensing all other system-bath interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' A widely used DD con- trol pulse is the Carr-Purcell (CP) sequence [1, 28] that is given by N equidistant π pulses, performed between an initial and a final π/2 pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' CP sequences act in the fre- quency domain approximately as Dirac comb filters [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' hence, they have been used to perform spectroscopy of in- tricate signals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=', for noise spectroscopy [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' With this protocol, the requirement to achieve high values of the noise reconstruction accuracy is to perform sequences arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='05079v1 [quant-ph] 12 Jan 2023 2 with a high number of pulses meaning N ∈ [30, 120] (as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' [32]) or higher, so that the Dirac comb filter approx- imation remains valid (in fact, N determines the filter width).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This usually leads to long experiments to recon- struct the whole spectrum of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Other techniques using non-equidistant or even more sophisticated DD se- quences [4, 33–36] have proved to be effective for noise sensing, but sometimes at the price of a higher computa- tional burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' For our sensing task, NN are designed to solve a re- gression problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=', the reconstruction of the NSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Here, we assume that the NSD of the bath of spins has a Gaussian profile [32, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The Gaussian NSD is thus parametrized as a function of key parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=', the mean value, variance, offset and noise power that we aim to reconstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Note that our proposal can be adapted to other parametrized NSD functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The NN are trained over a set of synthetic data generated by simulating how the coherence of the qubit sensor decays over time under the influence of both the CP control pulses and the NSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Moreover, to make the measurement statistics as close as possible to the ones obtained from the experiments, extra artificial errors are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Our approach using NN entails the following advan- tages that we have proven experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (i) NN have the capability to predict never-before-seen experimental data, and they can work with a better reconstruction accuracy (even up to 7 times better, as shown in the section Results below) than standard noise spectroscopy, as the ´Alvarez-Suter method [31], by making use at the same time of DD control sequence with a much smaller number of pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (ii) The training dataset, which can contain both synthetic and experimental data, is gener- ated just once and then it can be applied several times, as long as the new collected data reproduce the physical context under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In connection with (i), we are going to show that the amount of data used as input to the NN can be smaller than the one needed to resolve the NSD by means of standard noise spectroscopy methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' From our knowledge, this work is the first experimen- tal proof of enhanced reconstruction performance with NN for carrying out noise spectroscopy in single color centers in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' We thus expect that the techniques discussed here could fast become a novel standard spec- troscopy tool both for such quantum systems and other quantum platforms in which regression problems have to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Generation of training dataset The training dataset is composed of synthetic data that are originated by simulating the coherence decay of the qubit sensor in a noise spectroscopy experiment based on DD, as the one depicted in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This standard sensing procedure, which stems from Ramsey interferometry [1], maps information about the quantum coherence of the sensor into the population in |0⟩ that is then effectively recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' After having initialized the qubit sensor in the ground state |0⟩, a π/2 pulse is applied such that the qubit state |ψ⟩ is the superposition (|0⟩+|1⟩)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Then, we perform a CP control sequence consisting in a train of π pulses that flips repeatedly the qubit, and finally, a second π/2 pulse is applied in order to map the phase of the qubit into its population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The probability that the state of the quantum sensor is |0⟩, which corresponds to the observable population, equals to [1, 32] P = 1 2 (1 + C(τ, N)) , (1) where N is the number of π pulses and τ is the time between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The coherence function C(τ, N) is sim- ulated numerically, for a set of different values of τ and N, to generate the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Let us now introduce the decoherence function that quantifies how the quantum coherence C(τ, N) is modi- fied under the action of both the external bath of spins and a set of CP control pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The control sequence has the effect to modulate the coherence content of the qubit sensor, while the interaction with the bath, asso- ciated to the NSD S(ω), tends on average to destroy such coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Overall, under the joint presence of con- trol fields and a noise source, the coherence decays as C(τ, N) ≡ e−χ(τ,N), where χ(τ, N) denotes the decoher- ence function [27, 39–41]: χ(τ, N) = � dω πω2 F(ω, τ, N)S(ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (2) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (2), the filter function F(ω, τ, N) ≡ |Y (ω, τ, N)|2 is the square modulus of the Fourier transform of the so-called modulation function y(t, τ, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The latter is constant piecewise, with values ±1, and switches sign at the times t = τ/2, 3τ/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' , (N − 1/2)τ where each π pulse is applied [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Notice that we are assuming that the π pulses are instantaneous, a reasonable assumption for our experimental setup where a π pulse duration is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='1 µs and the time between pulses is τ ∈ [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='1] µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Let us now recall the expression, in the frequency domain, of the filter function for a CP sequence with even N: F(ω, τ, N) = 8 sin2 �ωτN 2 � sec2 �ωτ 2 � sin4 �ωτ 4 � , (3) while for odd N, sin2(ωτN/2) has to be replaced with cos2(ωτN/2) [2, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In order to generate the training dataset, the NSD S(ω) is parameterized as S(ω) = s0 + A exp � −(ω − ωc)2 2σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (4) Thus, being a Gaussian distribution, the NSD is fully de- scribed by the offset s0, amplitude A, width σ and center ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' For the training dataset in the paper, the values 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 1: NV center and Neural Networks for noise spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The NV center is surrounded by an ensemble of 13C nuclear spins (orange spheres) that collectively induce dephasing to the NV electronic spin (blue sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The NV electronic spin is controlled with a DD sequence (specifically, a Carr-Purcell (CP) sequence) with the aim to measure its dephasing, and therefore characterize the NSD of the nuclear spin bath, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=', S(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' s0, A, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The CP sequence is formed by N equidistant π pulses in between an initial and a final π/2 pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The time τ between the π pulses determines the measurement total time T = Nτ, given that the time between the first π/2 and the train of π pulse and the time between the last π and π/2 pulses are both equal to τ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Then, we measure the output of this experiment, which is the probability P = 1 2(1 + C(t)) that the NV center remains in the initial state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The spin coherence function C(t) – evaluated at previously-determined times in the set T ∈ {t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' , tn} (the tk’s are obtained by changing τ with N fixed) – is the input of the designed Neural Networks (NN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' After being trained, the NN return the estimation of the NSD parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' of these parameters are taken from the following inter- vals: s0 ∈ [4 · 10−4, 4 · 10−3] MHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' A ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='7] MHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' σ ∈ [2 · 10−3, 9 · 10−3] MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Instead, ωc is kept con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This is because in our experimental setup the NSD stems from the interaction with a large ensemble of unresolved 13C impurities (nuclear spin bath) around the NV electronic spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Therefore, the center of the NSD corresponds to the Larmor frequency ωc = γB, where γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='0705 kHz/G is the gyromagnetic ratio of the 13C nuclear spins, and B is the amplitude of a static mag- netic field aligned with the NV quantization axis, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Such static magnetic field is well known during the experimen- tal procedure since it determines the NV electronic spin resonances (B = 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='2 ± 2 G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The training dataset is generated by uniformly sam- pling 104 sets of parameters within the chosen intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Hence, overall we consider 104 distinct sequences of NSD parameters that are used to simulate different coherence curves C(τ, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' These sequences are taken in the time intervals τ ∈ [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='66] µs and [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='1] µs with sampling time ∆τ = 1 ns (∆τ = 20 ns in the experimental case, see below), and for N = {1, 8, 16, 24, 32, 40, 48}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' These inter- vals are significant for our study because they include the values of τ at which the coherence decay curve exhibits the first and second order collapses induced on the qubit sensor by the bath of 13C impurities [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Finally, in order to make the synthetic data used to train the NN closer to the experimental setting, extra artificial errors sampled by a normal distribution with standard devia- tion equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='05 (comparable with the expected error in our experimental measurements) are added to every point of the generated coherence decay curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In this way, one may mitigate the over-fitting of the employed machine learning models that are thus expected to better generalize to unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In general, a model trained on synthetic data cannot be successfully applied to real data without fine tuning it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' But in our case, it becomes possi- ble, probably due to the fact that the simulated data of the coherence decay are quite close to the experimentally observed decay data induced by the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' As final remark, notice that, from the 104 simulated curves C(τ, N), 6000 are used for the training of the NN and 2000 for their validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Instead, the test step is performed either by using the remaining 2000 simulated curves, or by using experimental data as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Neural networks working principles Let us describe the main working features of the NN employed in this paper to carry out noise spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Specifically, we are going to use the multi-layer percep- tron (MLP) that is composed of fully-connected layers, each of them with a variable number of artificial neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' A single artificial neuron returns as output the scalar ˆy ≡ Σ(wT · x + b) (5) that, by definition, is provided by applying the non-linear function Σ : R → R to the weighted sum of the input vector x ∈ Rk to which the bias term b ∈ R is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' w ∈ Rk denotes the vector of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In our analysis, the activation function Σ is chosen equal to the rectifier 4 Σ(x) ≡ max(0, x) [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Thus, a MLP layer composed of q neurons (each with k inputs) returns the vector ˆy ≡ Σ(W T x + b), (6) where ˆy ∈ Rq, W ∈ Rk×q is the matrix of weights (W collects all the weight vectors of the single neurons), and b ∈ Rq is the vector of the biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Hence, a MLP with L layers is ruled by the recursion equation h[ℓ] ≡ Σ � W[ℓ]T h[ℓ − 1] + b[ℓ] � , (7) where ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' , L is the index over the number of layers and h[0] ≡ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (7), W[ℓ] and b[ℓ] are, respectively, the weights and the biases of the ℓ-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The output vector of the MLP is ˆy ≡ h[L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' It is worth noting that the number, dimension and activation functions (they are usually denoted as the hyperparameters ξ) of the NN layers are chosen through a single optimization routine (cfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Let us now introduce the supervised learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Ideally, the purpose of the latter is to find the parameters θ∗ = argminθRD(θ, ξ) that minimize the theoretical risk function RD(θ, ξ) ≡ E(x,y)∼D [L (ˆy, y)] , (8) where θ ≡ {W[1], b[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' , W[L], b[L]}, and ˆy are the estimated values of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' By definition, RD is the ex- pected value of the loss function L for (x, y) sampled from the distribution D that generates the dataset [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The loss function L is a differentiable function that mea- sures the distance between the prediction ˆy (output of the MLP) and the desired output y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' However, since one can only dispose of a finite set S = {(x, y)1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' , (x, y)m} of samples to train, validate and test the employed ML models, the theoretical risk function is approximated by the empirical risk function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Considering the partition {Str, Sva, Ste} of S in training (Str), validation (Sva) and test (Ste) sets, the empirical risk function is defined by: RStr(θ, ξ) ≡ 1 |Str| � (x,y)∈Str L (ˆy, y) , (9) where |Str| is the cardinality of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In fact, RStr is the arithmetic mean of the loss function L eval- uated on the samples of the training set Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In our paper, we take the loss function L equal to the Mean Squared Error (MSE), also called L2 loss: L(ˆy, y) = 1 q q � i=1 (ˆyi − yi)2 (10) for the q outputs of the last layer (in our case three, corresponding to the noise parameters s0, A, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The MLP is trained by minimizing (step-by-step over time) the empirical risk function RStr(θ, ξ) with respect to θ by means of the mini-batch gradient descent method, so as to obtain the optimal value θ∗ of the NN parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Each gradient descent step is defined by θt+1 = θt − η∇θ 1 B B � b=1 L(ˆyb,t, yb,t), (11) where θ0 is a randomly chosen starting point, η is the learning rate that defines the length of the step and ∇θ 1 B �B b=1 L(ˆyt,b, yt,b) is the gradient of the loss func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The gradient is calculated for any time t on a batch of B elements taken from the training set, and the sub- script θ in ∇θ indicates that the variables of L during the gradient evaluation are the weights of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In this paper, RStr is minimized by means of Adam [46] that is a gradient-based optimization algorithm perform- ing the adaptive estimation of lower-order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The minimization is stopped when the time-derivative of the risk function evaluated on the validation set RSva(θ∗, ξ) becomes positive (early stopping strategy) or after a pre- defined number of gradient steps using all the data of the training set (called epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Then, we use RSva(θ∗, ξ) to check if the MLP works also for unseen data and tune the hyperparameters ξ (cfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Finally, the test set Ste is employed to calculate the metrics (discussed in detail below) used to generate the figures with the results that we are going to illustrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Training and numerical test of neural networks We now show the results obtained by using the trained machine learning models to infer the value of the NSD parameters {s0, A, σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' As already mentioned, the NN are tested with 2000 different NSD parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' For each of these sets of parameters, the curves C(τ, N) have been simulated as described in the previous subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In order to determine the smallest amount of data re- quired to reconstruct the NSD, we perform the training, validation and test of the NN with sub-sets of the simu- lated curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' These sub-sets are defined by introducing the variable N that denotes the upper bound for the num- ber of pulses N ≤ N considered during the whole process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' For example, for N = 16 only the curves C(τ, N) with N ∈ {1, 8, 16} are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Note that the sub-sets de- fined for each value of N contain the curves for all the different NSD parameters (6000 for training, 2000 for val- idation, and 2000 for testing), and for all the times τ in the intervals defined before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The results of this analysis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 2 (orange data), where the MSE (the loss function) between the in- ferred parameters ( ˆs0, ˆA, ˆσ) and the original parameters (s0, A, σ) used to generate the dataset is plotted as a function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Remarkably, the MSE seems to achieve its minimum value after N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This entails that the NN do not significantly improve their precision on the reconstruction of the NSD by using more data to train the NN beyond this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 5 0 10 20 30 40 50 ¯N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='3 MSE(s0, A, σ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 2: Mean-square-errors (MSE) between original and estimated NSD parameters for a set of 2000 test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Orange bullets with dash-dotted line are the mean values returned by NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Blue squares with dotted line are the mean values provided by the HS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Finally, shaded areas denote the standard deviation, taking into account all the 2000 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' To establish how accurately a NN reconstructs the NSD, we need to compare the corresponding results with those of a different method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In particular, we concentrate on the method used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' [32], which is itself based on Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' According to them, the decay of the coher- ence function C(τ, N) is analyzed as a function of N, for each fixed value of τi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=', for each fixed frequency com- ponent of the filter functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In the limit of high N, the decay of the coherence is exponential, with a rate that is inversely proportional to the amplitude of the NSD [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In other words, the amplitude of the NSD is directly es- timated for a discrete set of frequencies (each propor- tional to 1/τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In contrast with the original proposals in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' [30, 31], the method in Ref [32] demonstrates that it is better to use the harmonics of the filter functions to reconstruct the NSD, in order to avoid extra broad- ening of the reconstructed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' For this reason, we denote this method as Harmonics Spectroscopy (HS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' We have analyzed the same 2000 different curves C(τ, N) (used to test the machine learning models) also with the HS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The results are collected and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 2 (blue data), where the first point is for N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This is due to the fact that, by definition, the HS method fits the decay of the coherence as a function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This is possible only for a dataset with at least three points (in this case N = 1, 8, 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' As one can observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 2, the MSE values for the HS method (blue region) are al- ways above the MSE values for the NN method (orange region), especially for lower values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' These results demonstrate that the NN method can predict the pa- rameters of the NSD with an improved accuracy (up to 5 times larger) with respect to the HS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The test presented in this subsection have been performed with simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In the next subsection we are going to repeat the same test but with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Experimental test of neural networks By this point we know that NN can reliably predict the NSD from noisy simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In this section, we want to use the NN (trained and validated with noisy simu- lated data) to reconstruct the NSD using experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' As quantum sensor we use a spin qubit encoded in the electronic spin of the ground state of a single nitrogen- vacancy (NV) center in a bulk diamond at room temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This system has proven as a sensitive quantum probe of magnetic fields, with outstanding spacial reso- lution and sensitivity [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The diamond sample in our experiments has a natural abundance of 13C impu- rities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='1%) that are randomly distributed in the dia- mond lattice [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The 13C nuclear spins constitute the external environment of the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' They act as a collective bath of spins that induces dephasing into the NV electronic spin, limiting the its coherence time T2 ≈ 100 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In the presence of strong bias magnetic field (≥ 150G) [32, 49], the weak coupling of the NV spin with these carbon impurities can be modeled as a clas- sical stochastic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The latter has a power spectrum density (here called NSD) that follows a Gaussian dis- tribution centered at the Larmor frequency of the 13C nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In order to measure the NV spin coher- ence function C(τ, N), we apply a train of π pulses (in our case a CP sequence) to the NV spin qubit following the DD protocol described in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' For more details on the experimental implementation and Hamiltonian of the system see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' We have performed this experiment for N = {1, 8, 16, 24, 32, 40, 48}, and for τ ∈ [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='66] µs and [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='1] µs with sampling time ∆t = 20 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='3(a) (blue bullets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Then, the collected coherence functions have been processed and employed to reconstruct the NSD parameters by means of both the NN (trained with the generated dataset) and the HS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In contrast with the test using simu- lated data in the previous section, in the experimental case we do not know the exact values of the NSD pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Therefore, we cannot calculate the MSE to quantify the accuracy of the reconstructed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In order to estimate such accuracy we have used the fol- lowing procedure: from the inferred NSD, the coherence curves C(τ, N) are simulated and then compared with the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' An example of this comparison is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='3(a), where C(τ, N) is simulated under the assumption that the NSD parameters are inferred ei- ther by the machine learning models (orange) or by the HS method (red), both for N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Qualitatively it is clear that the orange curves are much closer to the ex- perimental data, than the red curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' There are several options to quantitatively compare the experimental data and the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Here we use both the reduced chi-squared χ2 ν [50], and the 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 3: (a) Coherence function C(τ, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The experimental data (blue bullets) are shown together with the simulated ones using the NSD predicted respectively by the HS method (red lines) and machine learning models (orange lines), both for N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (b) Reduced chi-squared χ2 ν, obtained by comparing simulation and experimental data, as a function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' As in panel (a), orange and red curves refer to the NN and HS method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Instead, the dashed line denotes the value of the reduced chi-squared for the HS method when we employ additional measurements for N = 56, 64, 72, 80 in the interval τ ∈ [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='1] µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Inset: Same results but quantified by the Mean-Absolute-Error (MAE) between the experimental data and the predicted C(τ, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Mean-Absolute-Error (MAE) [51] between the exper- imental data and the predicted coherence functions C(τ, N) (see Methods for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The results of this comparison are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 3(b), where χ2 ν and the MAE are plotted as a function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Remarkably, the NSD reconstructed by the NN for N = 16 behaves better that any case using the HS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' It is worth observing that the same experimental data used to infer the NSD parameters are partially used to estimate the χ2 ν and MAE(C(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' For example, for N = 16, only the data for N = 1, 8, 16 are used to reconstruct the NSD, but we employ all the data N = 1, 8, 16, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' , 48 to ob- tain the χ2 ν and MAE(C(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Overall, we have observed enhanced performance in reconstructing the NSD of the collective bath of spins, with a maximum improvement (about 7 times higher) for N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In other words, for N = 16, once we reconstruct the NSD, the quantum sen- sor dynamics can be predicted with an average square deviation of ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='86 experimental error-bars by using the NN method, or with an average square deviation of ≃ 13 error-bars if we use the HS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' DISCUSSION As shown pictorially in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 1, the NN takes as input the spin qubit coherence functions (the coherence of the quantum sensor decays due to the presence of the exter- nal bath) obtained by using a set of different CP control sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The NN returns as output the parameters of the unknown NSD in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' One can thus note that the NN, once validated, acts as a “time- frequency converter” (making use of a quite complicated deconvolution) from the measured signals living in the time domain – the spin coherence functions – to the NSD defined in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The results shown in the previous section, and sum- marized in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 2 and 3(b), demonstrate that NN can be used to reconstruct the NSD affecting a quantum sen- sor, achieving higher precision and with considerable less data than the standard HS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Improved values of the reconstruction accuracy have been obtained with simulated and experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Both the HS and NN methods are comparable – in terms of NSD reconstruc- tion accuracy – for high values of N, but not for small ones, where NN gives significantly better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' More- over, the main result of our study is that NN trained with data obtained for N = 16 reconstruct the NSD more accurately than the best estimate provided by the HS method with N = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This improvement is remark- able by itself, but it becomes more significant when we consider that the time required to complete these exper- iments has a growth faster than a linear function with respect to N, following an arithmetic progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' As an example, the total time to perform all the experiments in the case of N = 16 and 48 is respectively ≃ 10 minutes and ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='2 hours [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This is an under-estimation of the time difference between methods, because we are only considering the bare measurement time, without taking into account the time delay between different experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Furthermore, it is worth stressing that our results also show that deep learning has a predictive power since it can be applied to never-before-seen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This natu- rally provides to the employed machine learning models a connotation of robustness that is crucial in real appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 7 Let us observe that regression tasks, which are suc- cessfully solved by multi-layer perceptrons (one of the easiest form of NN), are less common with respect to the ones to carry out classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' a review of some exam- ple datasets and methods for regression is in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Hence, we expect that the synthetic data used in this work could be useful as a test bed also to the audience of machine learning researchers and developers solving regression problems in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' With this in mind, we share the training dataset with synthetic data and our codes for their generation, as well as the code for machine learning experiments and NSD reconstruction [available on the GitHub repository (see Section “Data and code availability”)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In this way, we promote the im- provement of machine learning models for noise sensing purposes and their use to solve different regression tasks in the quantum estimation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Conclusions & outlooks In this paper, we use NN to carry out noise spec- troscopy with a quantum sensor using dynamical decou- pling sequences with a much smaller number of π pulses and, at the same time, achieving a higher reconstruc- tion accuracy than standard methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=', HS proto- col).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' This means that with our proposal the noise spec- troscopy procedure will take less time and give better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' More in detail, we experimentally demonstrate the capability of NN to reconstruct the NSD of the collec- tive nuclear spin bath that surrounds an electronic spin qubit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=', the ground state of a single nitrogen-vacancy center in bulk diamond at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' To conclude, we outline some possible outlooks for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' First of all, one may evaluate the performance of NN that are trained over input data obtained using DD control sequences with more degrees of freedom than the CP ones [54–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Secondly, deep learning might be applied to noise spectroscopy techniques beyond the HS methods, as for example optimal band-limited control protocols [34, 35] and even non-Gaussian noise charac- terization [59–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' In addition, it might be worth inves- tigating how deep learning can be integrated to quantum sensing procedures that rely on the so-called stochastic quantum Zeno effect [62, 63], whereby the quantum probe is subjected to a sequence of quantum measurements that in the ideal case are designed to confine the dynamics of the probe around the initial (nominal) state [33, 64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' We are also confident that the extent of our results can be quite easily replicated in other experimental settings, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=', superconducting flux qubits [66, 67], trapped ions [68, 69], cold atoms [70, 71], quantum dots [72, 73], NMR experiments in molecules [31, 74], and nanoelec- tronic devices [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' For such a purpose, one might slightly adapt the deep learning techniques used here to methods tailored for time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Technical details on the training of NN The NN models are developed using the PyTorch framework [76] on a machine with 32 CPU cores, 126Gb of RAM and a GeForce RTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The training time, including the optimization of the hyperparameters, is around 12 hours for each N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The hyperparameters optimization is implemented by means of the Ray Tune library [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The Hyperopt pack- age [78] uses the Tree-structured Parzen Estimators [79] algorithm as a Bayesian optimization to search for the best choice of the hyperparameters within a predefined search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Hyperopt suggest the likely better configu- rations of the hyperparameters and the underlying model is updated after each trial that is run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The ASHA sched- uler [80] is then used to stop the run of the least promising trials chosen by the search algorithm, thus speeding up the hyperparameters optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The optimized hyperparameters are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (1) The number of hidden layers decides the value of L − 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The hidden layers are between the input layer h[0] and the output layer h[L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (2) The dimension of the hidden layers is the value of q in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (6) that, for the sake of simplicity, is equal for all the layers in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Both the number and dimension of the hidden layers are chosen by sampling log-uniformly an integer value from the space [1, 32) and [1, 1024), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (3) The learning rate is responsible for the length of the gradient descent step and it is optimized with a choice between 10−2, 10−3 and 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (4) The batch size denotes the dimension of the batch on which the loss function is summed for the gradient calculation in a single descent step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The batch size is chosen between 2, 4, 8, 16, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (5) The dropout is a regularization strategy that aims to reduce the overfitting by randomly turn off the NN neurons with a predefined probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Such probability is one among 0 (no dropout), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' (6) The weight decay is another regularization technique that adds to the loss function the squared weights of the NN multiplied by a decay value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The latter value is optimized choosing between 0 (no decay), 10−6, 10−5, 10−4 and 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Definition of quantifiers for reconstruction accuracy The accuracy NN and HS methods can be estimated by using the reconstructed NSD to simulate the coherence function C(τ, N), and ‘measuring’ the distance between the simulated data and the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' To do so, we use the reduced chi-squared χ2 ν, and the Mean- Absolute-Error (MAE(C)): We define Ce ± δCe (Cs) as the experimental (simulated) values of C(τ, N), where δCe is the standard deviation of the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 8 Then we can write reduced chi-squared and the MAE as χ2 ν ≡ 1 ν � n,N (Ce(τn, N) − Cs(τn, N))2 δCe(τn, N)2 (12) MAE(C) ≡ 1 ν � n,N |Ce(τn, N) − Cs(τn, N)| , (13) where N = {1, 8, 16, 24, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' , N}, {τn} are the values of the time between pulses within the time intervals defined in main text, and ν is the total number of elements in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' Notice that χ2 ν takes into account the experimental precision to scale the difference between experiment and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' The results showing both χ2 ν and the MAE are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' DATA AND CODE AVAILABILITY The source codes for the generation of the train- ing dataset and the machine learning experiments are available on GitHub at the following address: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content='com/trianam/noiseSpectroscopyNV ACKNOWLEDGEMENTS This work was supported by the European Com- mission’s Horizon Europe Framework Programme under the Research and Innovation Action GA n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' 101070546–MUQUABIS, and by the European De- fence Agency under the project Q-LAMPS Contract No B PRJ-RT-989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' acknowledges support from CNR-FOE-LENS-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E4T4oBgHgl3EQfcAxC/content/2301.05079v1.pdf'} +page_content=' acknowledge the European Union’s Horizon 2020 research and innovation programme under FET-OPEN GA n.' metadata={'source': 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Natural Language +processing (NLP) has been used extensively to process this data. +Deep Learning based NLP especially Large Language Models +(LLMs) such as BERT have found broad acceptance and are +used extensively for many applications. A Language Model is a +probability distribution over a word sequence. Self-supervised +Learning on a large corpus of data automatically generates deep +learning-based language models. BioBERT and Med-BERT are +language models pre-trained for the healthcare domain. +Healthcare uses typical NLP tasks such as question answering, +information extraction, named entity recognition, and search to +simplify and improve processes. However, to ensure robust +application of the results, NLP practitioners need to normalize +and standardize them. One of the main ways of achieving +normalization and standardization is the use of Knowledge +Graphs. A Knowledge Graph captures concepts and their +relationships for a specific domain, but their creation is time- +consuming and requires manual intervention from domain +experts, +which +can +prove +expensive. +SNOMED +CT +(Systematized Nomenclature of Medicine - Clinical Terms), +Unified Medical Language System (UMLS), and Gene Ontology +(GO) are popular ontologies from the healthcare domain. +SNOMED CT and UMLS capture concepts such as disease, +symptoms and diagnosis and GO is the world's largest source of +information on the functions of genes. Healthcare has been +dealing with an explosion in information about different types +of drugs, diseases, and procedures. This paper argues that using +Knowledge Graphs is not the best solution for solving problems +in this domain. We present experiments using LLMs for the +healthcare domain to demonstrate that language models +provide the same functionality as knowledge graphs, thereby +making knowledge graphs redundant. +Keywords—Medical +data, +Language +Models, +Natural +Language Processing, Knowledge Graphs, Deep Learning +I. INTRODUCTION +Knowledge graphs (KG) are knowledge bases that capture +concepts and their relationships for a specific domain using a +graph-structured data model. Systematized Nomenclature of +Medicine – Clinical Terms (SNOMED CT) (SNOMED), +Unified Medical Language Systems(UMLS) [Bodenreider O. +2004], etc., are some of the popular KG in the healthcare +domain. Fig. 1 shows a sample from a representative medical +entity, KG. On the other hand, a language model is a +probability distribution over a word sequence and is the +backbone of modern natural language processing (NLP). +Language models try to capture any language's linguistic +intuition and writing, and large language models like BERT +[Devlin et al., 2019] and GPT-2 [Radford et al., 2019] have +shown remarkable performance. The paper presents a study +demonstrating that language models' ability to learn +relationships among different entities makes knowledge +graphs redundant for many applications. + + This paper uses similar terms from SNOMED-CT KG and +passes them through a language model for the healthcare +domain BioRedditBERT to get a 768-dimensional dense +vector representation. The paper presents the results for +analyzing these embeddings. The experiments presented in +the paper validate that similar terms cluster together. The +paper uses simple heuristics to assign names to clusters. The +results show that the cluster names match the names in the +KG. Finally, the experiments demonstrate that the cosine +similarity of vector representation of similar terms is high and +vice versa. + + Our contributions include: (i) We propose a study to +demonstrate the value and application of Large Language +Models (LLMs) in comparison to Knowledge Graph-based +approaches for the task of synonym extraction. (ii) We +extensively evaluate our approach on a standard, widely +accepted dataset, and the results are encouraging. + + + Fig 1. Medical entity Knowledge Graph Representation + The rest of the paper is organized as follows: Section II +presents the background required to understand the work +presented in this paper. Section III presents a literature survey +of related work on knowledge graphs and language models. +Section IV presents our understanding of how current days +language models are making knowledge graphs redundant. +Section V describes our proposed approach. Section VI +describes the experiments conducted and the results obtained. +Finally, section VII summarizes our work and discusses +possible directions for future study. +II. BACKGROUND +This section defines and describes Language Models and +Knowledge Graphs as used in this paper: + + +Medicine +Fever +Allergy +Dolo +ClaritinA. Language Models + +A Language Model predicts the probability of a sequence of +words in a human language such as English. In the equation +below P(w1,…wm) is the probability of the word sequence +S, where S = (w1, w2, …, wm) and wi is the ith word in the +sequence. + + + + Large Language Models (LLMs) are language models +trained on large general corpora that learn associations and +relationships +among +different +word +entities +in +an +unsupervised manner. Large Language Models (LLMs) are +considered universal language learners. LLMs such as BERT +and GPTare deep neural networks based on transformer +architecture. One of many reasons for the immense popularity +of LLMs is that these models are pre-trained self-supervised +models and can be adapted or fine-tuned to cater to a wide +range of NLP tasks. Few-shot learning has enabled these +LLMs to be adapted to a given NLP task using fewer training +samples. + + Another reason for the immense popularity of LLMs is that +a single language model is applicable for multiple +downstream applications such as Token classification, Text +classification, and Question answering. LLMs generate +embeddings or word vectors for words, and these embeddings +capture the context of the word in the corpus. This ability of +LLMs to generate embeddings based on the corpus makes +them ubiquitous in almost NLP tasks. + + In this paper, we use BioRedditBERT [Basaldella et al., +2020], a variant of BERT trained for the healthcare domain. +It is a domain-specific language representation model trained +on large-scale biomedical corpora from Reddit. + +B. Knowledge Graphs + +Knowledge Graphs (KGs) organize data and capture +relationships between different entities for a domain. Domain +experts create KGs to map domain-based relations between +various entities. + + Knowledge graphs are Graph data structures with nodes +and edges. Nodes or vertices represent entities of interest, and +edges represent relations between them, as shown in Fig 1. +KGs can map and model direct and latent relationships +between entities of interest. Typically, KGs are used to model +and map information from model sources. Once KGs are +designed, typically, NLP is used to populate & create the +knowledge base from unstructured text corpora. + + Knowledge graphs play a crucial role in healthcare +knowledge representation. There are many widely used +knowledge graphs like SNOMED and UMLS etc. In +healthcare, KGs are used for drug discovery drugs, +identifying tertiary symptoms for diseases and augmented +decision-making, etc. + + COMETA: A Corpus for Medical Entity Linking in social +media [Basaldella et al., 2020] – a corpus containing four +years of content in 68 health-themed subreddits and +annotating the most frequent with their corresponding +SNOMED-CT entities. In this paper, we have used COMETA +to obtain synonyms from SNOMED-CT. +III. RELATED WORK +In 2019, Jawahar et al. performed experiments to understand +the underlying language structure learned by a language +model like BERT [Ganesh Jawahar et al. 2019]. The authors +show that BERT captures the semantic information from the +language hierarchically through experiments. BERT captures +surface features in the bottom layer, syntactic elements in the +middle and semantic features in the top layer. The work +presented in this paper treats the BERT model as a black box +and demonstrates that BERT can learn the information in a +knowledge graph through experiments on real-life healthcare +use cases. + + There have been studies to generate a knowledge graph +directly from the output of LLMs. [Wang C et al., 2020; +Wang X et al. 2022] proposes a mechanism to create a KG +directly from LLMs. This mechanism talks about a two-step +mechanism to generate a KG from LLM. In the first step, +different candidate triplets are created from the text corpus. +Attention weights from a pre-trained LLM are used to get the +best-matched candidate triplets and then validated through a +beam search. In the second stage, the matched candidate +triplets are mapped to a pre-defined KG for validation, and +the unmatched candidates are used to create an open +knowledge graph. The work demonstrates the feasibility of +the idea presented in this paper that LLM can be used as a +substitute for knowledge graphs, especially since they +contain the information in the KG. + + There is a body of research on integrating Knowledge +graphs and LLMs. Structured knowledge from Knowledge +Graphs is effectively integrated into Language models to +enhance the pre-trained language models [Lei He et al., +2021]. However, these approaches have found limited +success, thereby strengthening the position in this paper that +LLMs contain information from KGs. +IV. LANGUAGE MODELS FOR KNOWLEDGE GRAPHS +Language Models can find associations between different +words based on the attention weight matrix. The +methodology to use attention weights as a measure of +relationship among the entities indicates that Knowledge +graphs are getting replaced by LLMs as they learn more +generic relationships in an unsupervised way. The proposed +methodology in this paper is built on this idea to demonstrate +that Knowledge graphs are increasingly getting redundant for +many NLP tasks. +V. PROPOSED APPROACH +The paper demonstrates that language models' ability to learn +relationships among different entities makes knowledge +graphs redundant for many applications. To illustrate this, we +have used word embeddings for all the synonyms of a set of +medical terms from a large language model. This work uses + +m +P(w1,..., Wm) =|[P(wi I Wi,..., Wi-1) +i=1COMETA data to obtain synonyms for a set of medical terms. +In COMETA data, the work focuses on the following +columns: a) Example column, which contains the sentences +from health-themed forums on Reddit, b) Term column +contains the medical terms present in the Example column, c) +General SNOMED Label column; contains the literal +meaning of the Term column from the SNOMED Knowledge +Graphs. To obtain synonyms, we use the different values +from the Terms column for a specific value of the General +SNOMED Label column. For example, for Abdominal Wind +Pain General SNOMED label, we have the following three +synonyms that we can obtain from the Terms column: gas +pains, painful gas, and gas pain. + + To calculate the word embeddings of every synonym term, +we +use +the +word_vector +function +from +the +biobert_embeddings python module [Jitendra Jangid, 2020]. +Since the original code was incompatible with the current +version of Pytorch [Paszke, A. et al., 2019] and Huggingface +[Wolf et al., 2020], we modified it just enough to satisfy the +current version requirements – the core logic remains the +same. We tokenize every Term using HuggingFace +tokenizers +and +pass +the +tokenized +Term +through +BioRedditBERT model. The previous step gives us +embedding for the Term (or sub-terms if the model didn't see +the Term before). If the model has not seen the Term before, +then we sum up the embedding of all the subterms). We then +store all the embeddings for the next steps. + + We perform the following two experiments after +generating the word embeddings for the synonyms of a set of +medical terms. In the first experiment, we cluster the word +embeddings for the synonyms of a set of medical terms and +assign names to clusters. The word embeddings are passed +into UMAP to generate a 2-dimensional representation. We +plot the 2-dimensional representation to examine how the +term cluster visually. UMAP is used as the dimensionality +reduction technique over PCA because it is a non-linear +dimensionality reduction technique and does very well to +preserve the local and global structure of the data as +compared to PCA. However, unlike PCA [Karl Pearson +F.R.S. , 1901], UMAP is very sensitive to hyperparameters +that we chose, so we visualize the embeddings for several +values of number of neighbours (n_neighbors) and minimum +distance (min_dist). This step will help us visually validate +that a fine-tuned LLM indeed groups together similar terms +while ensuring different terms are further apart. + + After identifying clusters from the above step, we use +Humans in the Loop approach to identify all terms that belong +together and run KMeans Clustering Algorithm [Lloyd, +Stuart P., 1982] on them. We identify the term closest to the +cluster's centroid, which becomes the Parent Node – one of +the core uses of Knowledge Graphs. + + In the second experiment, we analyze the similarity +between the word embeddings of the synonyms of the set of +medical terms. In this step, we compute the cosine similarity +between all the word embeddings and then we examine the +similarity to demonstrate that the synonyms for the same term +are similar with a small cosine distance between them. +VI. EXPERIMENTS AND RESULTS +We use Term and General SNOMED Label columns from +COMETA dataset for our experiments. To calculate the +embeddings of every term, we use word_vector function from +biobert_embeddings package [Jitendra Jangid, 2020]. Since +the original code was incompatible with current version of +Pytorch [Paszke, A. et al., 2019] and Huggingface [Wolf et al., +2020], we modified it just enough to satisfy the current version +requirements – the core logic remains the same. + To test the rich representation of language models for our +use case, we perform 2 experiments, (1) Cluster the word +embeddings for the synonyms of a set of medical terms and +assign names to clusters (2) Analyze the similarity between +the word embeddings of the synonyms of the set of medical +terms. + For the reasons discussed in Sec. III, we use UMAP as our +choice of dimensionality reduction. For experiment (1), Fig. 2 +shows that entities having similar nature are grouped together +and dissimilar entities are further apart which proves utility of +a Fine-tuned Language Models. + +Fig 2. Clusters resulting from UMAP dimensionality reduction + Next we perform KMeans clustering on mentions +belonging to same group using cosine similarity. The centroid +of each clusters were then used to identify concepts by finding +terms that were closest to the centers by cosine similarity. We +found the following terms for the concepts visible in Table. 1. + +Concept (General SNOMED +Label) +Term (closest to the cluster) +Oral contraception +hormonal BC pills +Crohn's disease +crohns disease +Diabetes mellitus type 2 +T2 diabetes +Analgesic +Pain Medication +Diabetes mellitus type 1 +T1 diabetic +Autoimmune disease +autoimmune disease +Hypoglycemia +low blood sugars +Headache +head pain +Tachycardia +heart racing + +10 +general_snomed_label +Oral contraception +Crohn's disease +Diabetes mellitus type 2 +Analgesic +Diabetes mellitus type 1 +Autoimmunedisease +5 +Hypoglycemia +E +Headache +Tachycardia +Tired +4 +Itching +0 +2 +4 +6 +8 +10 +dim_0Tired +feel tired +Itching +itching +Table 1. Terms closest to the cluster center of each Concept + While Fig. 2 illustrates global and local structure among +different mentions of a concept, as a part of experiment (2), +we also analyze distribution of similarity scores (which are +calculated by using cosine similarity) to visualize distribution +of cosine similarity among terms belonging to same concept +(Fig. 3 and 4) and terms belonging to different concepts (Fig. +5). We can see that distribution of mentions belonging to same +concept are closer to each other on average as compared to +mentions from different concepts. This point again validates +the utility of Language Model in finding different mentions of +a concept in multiple documents. + +Fig 3. Cosine similarity between mentions from Oral Contraception + +Fig 4. Cosine similarity between mentions from Cron’s disease + +In addition to these plots, we also analyze similarity +between unrelated terms, and we see the following trend – + +Fig 5. Cosine similarity between mentions from different concepts +VII. CONCLUSION AND FUTURE WORK +In this paper we have empirically shown how Language +Models fine-tuned on domain specific data can be used to +replace Knowledge Graphs for tasks where identifying +synonyms is involved. + + Language Models do a very good job in calculating +embeddings which contains semantic information about +terms that can be used to identify if two terms are close to +each other or not. This information is used in this paper to +identify terms which are closer to each other, and which are +not. Once groups of similar terms have been identifying using +non-linear dimensionality techniques, using Humans in the +Loop approach we can annotate such groups. After +annotating the groups, we use KMeans to identify centroids +of each cluster which are then used the identify terms with +the closest cosine distance from them. These terms can then +be used as parent nodes for their respective clusters. The +primary way in which our algorithm improves over current +Knowledge Graph based approaches is that unlike KGs which +are created by subject matter experts, our algorithm doesn’t +require subject matter experts for annotation. + + Our current algorithm handles synonym mapping quite +well, but it requires human intervention and for next steps, we +would be exploring ways in which we can extract Knowledge +Graphs from Language Models themselves. This would be +required to remove the human intervention in the current +process and handling cases where hypernyms are involved. +REFERENCES + +[1] Bodenreider O. 2004. The Unified Medical Language System (UMLS): +integrating biomedical terminology. Nucleic Acids Res. 2004 Jan +1;32(Database issue):D267-70. +[2] SNOMED. URL: http://www.snomed.org/ +[3] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. +2019. BERT: Pre-training of Deep Bidirectional Transformers for +Language Understanding. 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KLMo: +Knowledge Graph Enhanced Pretrained Language Model with Fine- +Grained +Relationships. +In Findings +of +the +Association +for +Computational Linguistics: EMNLP 2021, pages 4536–4542, Punta +Cana, +Dominican +Republic. +Association +for +Computational +Linguistics. + + + + + diff --git a/6dE2T4oBgHgl3EQfkgfV/content/tmp_files/load_file.txt b/6dE2T4oBgHgl3EQfkgfV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b099d0ea81af9291a773318fc986b9e8a2ab07b --- /dev/null +++ b/6dE2T4oBgHgl3EQfkgfV/content/tmp_files/load_file.txt @@ -0,0 +1,324 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf,len=323 +page_content='XXX-X-XXXX-XXXX-X/XX/$XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='00 ©20XX IEEE Language Models sounds the Death Knell of Knowledge Graphs Kunal Suri Optum, India kunal_suri@optum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='com Swapna Sourav Rout Optum, India rout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='swapnasourav@optum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='com Atul Singh Optum, India atul_singh18@optum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='com Prakhar Mishra Optum, India prakhar_mishra29@optum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='com Rajesh Sabapathy Optum,India rajesh_sabapathy@uhc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='com Abstract—Healthcare domain generates a lot of unstructured and semi-structured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Natural Language processing (NLP) has been used extensively to process this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Deep Learning based NLP especially Large Language Models (LLMs) such as BERT have found broad acceptance and are used extensively for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' A Language Model is a probability distribution over a word sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Self-supervised Learning on a large corpus of data automatically generates deep learning-based language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' BioBERT and Med-BERT are language models pre-trained for the healthcare domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Healthcare uses typical NLP tasks such as question answering, information extraction, named entity recognition, and search to simplify and improve processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' However, to ensure robust application of the results, NLP practitioners need to normalize and standardize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' One of the main ways of achieving normalization and standardization is the use of Knowledge Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' A Knowledge Graph captures concepts and their relationships for a specific domain, but their creation is time- consuming and requires manual intervention from domain experts, which can prove expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' SNOMED CT (Systematized Nomenclature of Medicine - Clinical Terms), Unified Medical Language System (UMLS), and Gene Ontology (GO) are popular ontologies from the healthcare domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=" SNOMED CT and UMLS capture concepts such as disease, symptoms and diagnosis and GO is the world's largest source of information on the functions of genes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Healthcare has been dealing with an explosion in information about different types of drugs, diseases, and procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' This paper argues that using Knowledge Graphs is not the best solution for solving problems in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' We present experiments using LLMs for the healthcare domain to demonstrate that language models provide the same functionality as knowledge graphs, thereby making knowledge graphs redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Keywords—Medical data, Language Models, Natural Language Processing, Knowledge Graphs, Deep Learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' INTRODUCTION Knowledge graphs (KG) are knowledge bases that capture concepts and their relationships for a specific domain using a graph-structured data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Systematized Nomenclature of Medicine – Clinical Terms (SNOMED CT) (SNOMED), Unified Medical Language Systems(UMLS) [Bodenreider O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 2004], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', are some of the popular KG in the healthcare domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 1 shows a sample from a representative medical entity, KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' On the other hand, a language model is a probability distribution over a word sequence and is the backbone of modern natural language processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=" Language models try to capture any language's linguistic intuition and writing, and large language models like BERT [Devlin et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2019] and GPT-2 [Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2019] have shown remarkable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=" The paper presents a study demonstrating that language models' ability to learn relationships among different entities makes knowledge graphs redundant for many applications." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' This paper uses similar terms from SNOMED-CT KG and passes them through a language model for the healthcare domain BioRedditBERT to get a 768-dimensional dense vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The paper presents the results for analyzing these embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The experiments presented in the paper validate that similar terms cluster together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The paper uses simple heuristics to assign names to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The results show that the cluster names match the names in the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Finally, the experiments demonstrate that the cosine similarity of vector representation of similar terms is high and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Our contributions include: (i) We propose a study to demonstrate the value and application of Large Language Models (LLMs) in comparison to Knowledge Graph-based approaches for the task of synonym extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' (ii) We extensively evaluate our approach on a standard, widely accepted dataset, and the results are encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Medical entity Knowledge Graph Representation The rest of the paper is organized as follows: Section II presents the background required to understand the work presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Section III presents a literature survey of related work on knowledge graphs and language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Section IV presents our understanding of how current days language models are making knowledge graphs redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Section V describes our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Section VI describes the experiments conducted and the results obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Finally, section VII summarizes our work and discusses possible directions for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' BACKGROUND This section defines and describes Language Models and Knowledge Graphs as used in this paper: Medicine Fever Allergy Dolo ClaritinA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Language Models A Language Model predicts the probability of a sequence of words in a human language such as English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In the equation below P(w1,…wm) is the probability of the word sequence S, where S = (w1, w2, …, wm) and wi is the ith word in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Large Language Models (LLMs) are language models trained on large general corpora that learn associations and relationships among different word entities in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Large Language Models (LLMs) are considered universal language learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' LLMs such as BERT and GPTare deep neural networks based on transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' One of many reasons for the immense popularity of LLMs is that these models are pre-trained self-supervised models and can be adapted or fine-tuned to cater to a wide range of NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Few-shot learning has enabled these LLMs to be adapted to a given NLP task using fewer training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Another reason for the immense popularity of LLMs is that a single language model is applicable for multiple downstream applications such as Token classification, Text classification, and Question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' LLMs generate embeddings or word vectors for words, and these embeddings capture the context of the word in the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' This ability of LLMs to generate embeddings based on the corpus makes them ubiquitous in almost NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In this paper, we use BioRedditBERT [Basaldella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2020], a variant of BERT trained for the healthcare domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' It is a domain-specific language representation model trained on large-scale biomedical corpora from Reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Knowledge Graphs Knowledge Graphs (KGs) organize data and capture relationships between different entities for a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Domain experts create KGs to map domain-based relations between various entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Knowledge graphs are Graph data structures with nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Nodes or vertices represent entities of interest, and edges represent relations between them, as shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' KGs can map and model direct and latent relationships between entities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Typically, KGs are used to model and map information from model sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Once KGs are designed, typically, NLP is used to populate & create the knowledge base from unstructured text corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Knowledge graphs play a crucial role in healthcare knowledge representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' There are many widely used knowledge graphs like SNOMED and UMLS etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In healthcare, KGs are used for drug discovery drugs, identifying tertiary symptoms for diseases and augmented decision-making, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' COMETA: A Corpus for Medical Entity Linking in social media [Basaldella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2020] – a corpus containing four years of content in 68 health-themed subreddits and annotating the most frequent with their corresponding SNOMED-CT entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In this paper, we have used COMETA to obtain synonyms from SNOMED-CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' RELATED WORK In 2019, Jawahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' performed experiments to understand the underlying language structure learned by a language model like BERT [Ganesh Jawahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The authors show that BERT captures the semantic information from the language hierarchically through experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' BERT captures surface features in the bottom layer, syntactic elements in the middle and semantic features in the top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The work presented in this paper treats the BERT model as a black box and demonstrates that BERT can learn the information in a knowledge graph through experiments on real-life healthcare use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' There have been studies to generate a knowledge graph directly from the output of LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' [Wang C et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Wang X et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 2022] proposes a mechanism to create a KG directly from LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' This mechanism talks about a two-step mechanism to generate a KG from LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In the first step, different candidate triplets are created from the text corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Attention weights from a pre-trained LLM are used to get the best-matched candidate triplets and then validated through a beam search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In the second stage, the matched candidate triplets are mapped to a pre-defined KG for validation, and the unmatched candidates are used to create an open knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The work demonstrates the feasibility of the idea presented in this paper that LLM can be used as a substitute for knowledge graphs, especially since they contain the information in the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' There is a body of research on integrating Knowledge graphs and LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Structured knowledge from Knowledge Graphs is effectively integrated into Language models to enhance the pre-trained language models [Lei He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' However, these approaches have found limited success, thereby strengthening the position in this paper that LLMs contain information from KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' LANGUAGE MODELS FOR KNOWLEDGE GRAPHS Language Models can find associations between different words based on the attention weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The methodology to use attention weights as a measure of relationship among the entities indicates that Knowledge graphs are getting replaced by LLMs as they learn more generic relationships in an unsupervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The proposed methodology in this paper is built on this idea to demonstrate that Knowledge graphs are increasingly getting redundant for many NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=" PROPOSED APPROACH The paper demonstrates that language models' ability to learn relationships among different entities makes knowledge graphs redundant for many applications." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' To illustrate this, we have used word embeddings for all the synonyms of a set of medical terms from a large language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' This work uses m P(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', Wm) =|[P(wi I Wi,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', Wi-1) i=1COMETA data to obtain synonyms for a set of medical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In COMETA data, the work focuses on the following columns: a) Example column, which contains the sentences from health-themed forums on Reddit, b) Term column contains the medical terms present in the Example column, c) General SNOMED Label column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' contains the literal meaning of the Term column from the SNOMED Knowledge Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' To obtain synonyms, we use the different values from the Terms column for a specific value of the General SNOMED Label column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' For example, for Abdominal Wind Pain General SNOMED label, we have the following three synonyms that we can obtain from the Terms column: gas pains, painful gas, and gas pain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' To calculate the word embeddings of every synonym term, we use the word_vector function from the biobert_embeddings python module [Jitendra Jangid, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Since the original code was incompatible with the current version of Pytorch [Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2019] and Huggingface [Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2020], we modified it just enough to satisfy the current version requirements – the core logic remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' We tokenize every Term using HuggingFace tokenizers and pass the tokenized Term through BioRedditBERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=" The previous step gives us embedding for the Term (or sub-terms if the model didn't see the Term before)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' If the model has not seen the Term before, then we sum up the embedding of all the subterms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' We then store all the embeddings for the next steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' We perform the following two experiments after generating the word embeddings for the synonyms of a set of medical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In the first experiment, we cluster the word embeddings for the synonyms of a set of medical terms and assign names to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The word embeddings are passed into UMAP to generate a 2-dimensional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' We plot the 2-dimensional representation to examine how the term cluster visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' UMAP is used as the dimensionality reduction technique over PCA because it is a non-linear dimensionality reduction technique and does very well to preserve the local and global structure of the data as compared to PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' However, unlike PCA [Karl Pearson F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' , 1901], UMAP is very sensitive to hyperparameters that we chose, so we visualize the embeddings for several values of number of neighbours (n_neighbors) and minimum distance (min_dist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' This step will help us visually validate that a fine-tuned LLM indeed groups together similar terms while ensuring different terms are further apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' After identifying clusters from the above step, we use Humans in the Loop approach to identify all terms that belong together and run KMeans Clustering Algorithm [Lloyd, Stuart P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 1982] on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=" We identify the term closest to the cluster's centroid, which becomes the Parent Node – one of the core uses of Knowledge Graphs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In the second experiment, we analyze the similarity between the word embeddings of the synonyms of the set of medical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In this step, we compute the cosine similarity between all the word embeddings and then we examine the similarity to demonstrate that the synonyms for the same term are similar with a small cosine distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' EXPERIMENTS AND RESULTS We use Term and General SNOMED Label columns from COMETA dataset for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' To calculate the embeddings of every term, we use word_vector function from biobert_embeddings package [Jitendra Jangid, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Since the original code was incompatible with current version of Pytorch [Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2019] and Huggingface [Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', 2020], we modified it just enough to satisfy the current version requirements – the core logic remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' To test the rich representation of language models for our use case, we perform 2 experiments, (1) Cluster the word embeddings for the synonyms of a set of medical terms and assign names to clusters (2) Analyze the similarity between the word embeddings of the synonyms of the set of medical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' For the reasons discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' III, we use UMAP as our choice of dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' For experiment (1), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 2 shows that entities having similar nature are grouped together and dissimilar entities are further apart which proves utility of a Fine-tuned Language Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Clusters resulting from UMAP dimensionality reduction Next we perform KMeans clustering on mentions belonging to same group using cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The centroid of each clusters were then used to identify concepts by finding terms that were closest to the centers by cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' We found the following terms for the concepts visible in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Concept (General SNOMED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Label) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Term (closest to the cluster) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Autoimmunedisease ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Hypoglycemia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Headache ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Tachycardia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Tired ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Itching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='dim_0Tired ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='feel tired ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Itching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='itching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Terms closest to the cluster center of each Concept While Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 2 illustrates global and local structure among different mentions of a concept, as a part of experiment (2), we also analyze distribution of similarity scores (which are calculated by using cosine similarity) to visualize distribution of cosine similarity among terms belonging to same concept (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 3 and 4) and terms belonging to different concepts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' We can see that distribution of mentions belonging to same concept are closer to each other on average as compared to mentions from different concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' This point again validates the utility of Language Model in finding different mentions of a concept in multiple documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Cosine similarity between mentions from Oral Contraception Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Cosine similarity between mentions from Cron’s disease In addition to these plots, we also analyze similarity between unrelated terms, and we see the following trend – Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Cosine similarity between mentions from different concepts VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK In this paper we have empirically shown how Language Models fine-tuned on domain specific data can be used to replace Knowledge Graphs for tasks where identifying synonyms is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Language Models do a very good job in calculating embeddings which contains semantic information about terms that can be used to identify if two terms are close to each other or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' This information is used in this paper to identify terms which are closer to each other, and which are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Once groups of similar terms have been identifying using non-linear dimensionality techniques, using Humans in the Loop approach we can annotate such groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' After annotating the groups, we use KMeans to identify centroids of each cluster which are then used the identify terms with the closest cosine distance from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' These terms can then be used as parent nodes for their respective clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' The primary way in which our algorithm improves over current Knowledge Graph based approaches is that unlike KGs which are created by subject matter experts, our algorithm doesn’t require subject matter experts for annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Our current algorithm handles synonym mapping quite well, but it requires human intervention and for next steps, we would be exploring ways in which we can extract Knowledge Graphs from Language Models themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' This would be required to remove the human intervention in the current process and handling cases where hypernyms are involved.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='95 Crohn disease Crohn et *$ Crohn at\' " s disease 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='9 Crohnie Crohnies crohns crohns colitis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content="85 crohns disease Crohn ' s disease CrohnsDisease 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='8 crohns flare 3 olitlis diseas eDissimilarityMatrix hormonal μC pills crohns disease T2 diabetes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='95 Pain Medicatior T1 diabetic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='9 autoimmune disease low blood sugars head pair 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='85 heart racing feel tired 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='8 itching[12] Radford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', Child, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', Luan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', Amodei, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', & Sutskever, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Language Models are Unsupervised Multitask Learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' [13] Ganesh Jawahar etal, What does BERT learn about the structure of language?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 3651–3657, 2019 [14] Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', & Song, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Language Models are Open Knowledge Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' ArXiv, abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content='11967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' [15] Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', He, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', Liang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=', & Xiao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Language Models as Knowledge Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence (IJCAI-22) [16] Lei He, Suncong Zheng, Tao Yang, and Feng Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' KLMo: Knowledge Graph Enhanced Pretrained Language Model with Fine- Grained Relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: EMNLP 2021, pages 4536–4542, Punta Cana, Dominican Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'} diff --git a/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf b/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..49d9e1b1a548b9a0f46f1f7d14fa785f8ad4e7fe --- /dev/null +++ b/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1cfff4374d9cdaf117b05aad9fb5bcf8b085a14eec1bcb89685a78849129bec8 +size 150273 diff --git a/79AzT4oBgHgl3EQfSPsz/vector_store/index.faiss b/79AzT4oBgHgl3EQfSPsz/vector_store/index.faiss new file mode 100644 index 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b/7NE1T4oBgHgl3EQfnATd/content/tmp_files/2301.03305v1.pdf.txt @@ -0,0 +1,1613 @@ +Topological charge quantization on localized imperfections in crystalline insulators +and the nearsightedness principle of Kohn +Kiryl Piasotski1, 2, ∗ +1Institut f¨ur Theorie der Kondensierten Materie, +Karlsruher Institut f¨ur Technologie, 76131 Karlsruhe, Germany +2Institut f¨ur QuantenMaterialien und Technologien, +Karlsruher Institut f¨ur Technologie, 76021 Karlsruhe, Germany† +(Dated: January 10, 2023) +We study the quantization of the excess charge on N localized (ultra-screened) impurities in d- +dimensional crystalline insulating systems. Solving Dyson’s equation, we demonstrate that such +charges are topological, by expressing them as winding numbers of appropriate functionals of bulk +position space Green’s functions. We discuss the ties of our topological invariant with the nearsight- +edness principle of W. Kohn, stating that the electronic charge density at fixed chemical potential +depends on the external field only locally, meaning that localized perturbations by external fields +may only result in localized charge redistributions. We arrive at the same conclusion by demonstrat- +ing that an adiabatic perturbation comprised of a variation of impurities’ positions and/or strengths +may only result in the change in the occupancy of impurity-localized bound states sitting, energy- +wise, close to the Fermi level. Finally, we conclude by discussing the relations of the nearsightedness +principle with the topological invariants characterizing the boundary charge. +I. +INTRODUCTION +With the discovery of the quantum Hall effect [1] and +its topological origins [2, 3] the study of the topological +structures in condensed matter systems became a style +rather than a fashion. The possibly best-known contem- +porary example of that is the field of topological insu- +lators, where the boundary-localized edge states, with +topologically granted existence and robustness, are be- +ing studied (popular reviews are [4, 5]). Despite being a +well-defined endeavor with its very own periodic table [6], +this discipline leaves a number of questions open. One +of these regards the direct experimental accessibility of +these topological surface states. In particular, aside from +the subspace of such states, their basis is incomplete for a +description of the physical system accommodating them +– a topological insulator, making it highly questionable +whether an actual physical observable may be expanded +into a basis of intra- and inter-surface state transition +operators. This is, for example, not true of the excess +charge density that these insulators accumulate at their +boundaries as it, as such, also features the exponentially +localized contributions of all of the occupied extended +states. Despite that, it is clear that as the surface states +also contribute to such an observable, the change in their +occupancy has to have an observable effect. +In a series of recent works [7, 8, 9, 10], the topologi- +cal properties of the boundary-localized electronic excess +charges (the boundary charges) in unidimensional crys- +tals were examined. In particular, a pair of topological +invariants characterizing the boundary charge upon two +bulk energy spectrum-preserving transformations of crys- +tal’s potential, translations and local inversions, were de- +vised. Specifically, it was demonstrated that upon local +inversion (inversion of coordinates within the unit cell), +the boundary charge maps to its negative, up to an inte- +gral topological quantum number known as the interface +invariant. Likewise, upon the lattice translation by xϕ, +the boundary charge was shown to grow linearly with +the shift variable xϕ (with the slope being the unit cell- +averaged average charge density in the bulk ¯ρ = ν +L, with +ν – filling factor and L-system’s period), whilst perform- +ing discontinuous downward jumps by a unit of the elec- +tron charge, as quantified by another topological quan- +tum number – the boundary invariant. These topologi- +cal invariants were shown to be generated by the spectral +flow of the energies corresponding to the edge states in- +side the energy gap that hosts the chemical potential, in +complete analogy with the integer quantum Hall effect +[3]. As opposed to the edge states in topological insu- +lators, the quantization of these invariants does not rely +on the internal symmetries of the bulk Bloch’s Hamilto- +nian (such as particle-hole or time-reversal symmetries) +and is instead guaranteed by a number of fundamental +physical principles, such as charge conservation, Pauli +principle, and the nearsightedness principle of W. Kohn +[11, 12, 13, 14] (to be discussed further on). Moreover, +these invariants are directly linked with the properties of +an experimental observable, a privilege shared by both +the quantum Hall effect and the topological defects (see +Ref. [15] for a review), while not being entirely clear in +the domain of the topological insulators. +Further, in a different paper [16], rational quantization +of boundary and interface charges was discussed. Partic- +ularly, with the aid of the aforementioned physical princi- +ples, a general framework for studying quantized charges +in one dimension was laid down, allowing us to quantify +all possible quantization patterns of the boundary charge +in terms of the non-symmorphic symmetries of the crys- +tal. The charges on the interfaces between pairs of in- +sulators sharing their bulk properties were demonstrated +to follow a lattice version of the Goldstone-Wilczek for- +mula [17], relating the interface charge to the sum of the +arXiv:2301.03305v1 [cond-mat.mes-hall] 9 Jan 2023 + +2 +boundary charges right and left to their septum, mod- +ulo an unknown integer generated by the local coupling +between the two subsystems. +The key feature of the method developed in Ref. [16] +is this “modulo an unknown integer” paradigm, arising +from the nearsightedness principle of the electronic mat- +ter. As such, the nearsightedness principle tells us that +(see Ref. [13, 14]), in insulators, localized perturbations +by external fields may result in localized charge redistri- +butions only. To be more specific, the corrections beyond +the characteristic length scale ξg = vF +Eg (where vF and Eg +are the Fermi velocity and the gap opening up at the +Fermi level, see Ref. [18] for example) are exponentially +suppressed. +An even further refinement of this state- +ment would be that such perturbations may only remove +or add an additional number of bound states whose wave +functions are localized around the corresponding pertur- +bations. One of the key purposes of the present paper +is to substantiate this claim mathematically, which turns +out to be possible in pretty general d-dimensional models. +To be more specific, this paper concerns the topolog- +ical properties of the electronic excess charges accumu- +lated around point-like defects in d-dimensional insula- +tors. Although we purposefully specify the Hamiltonian +of the crystal under consideration to make our exposi- +tion more transparent, the derivations presented in this +manuscript are shown to be independent of its choice. +What indeed matters is that the spectrum of the clean +system consists of the energy bands occasionally sepa- +rated by the energy gaps, that is, there exists at least +one bulk energy gap into which we can put the chemical +potential to promote the resulting statistical system into +an insulator. +Furthermore, neither we specify the internal structure +of the impurity vertices, nor do we assume any particu- +lar arrangement of them, making our analysis applicable +to a wide range of experimental setups. In particular, +quite conventionally, we may assume that a number of +randomly located point-like impurities exerting an ultra- +screened electrostatic force on the system’s electrons are +scattered through the charge sampling region of a crystal +under consideration. A slightly less familiar situation is +inspired by the work of Nomura and Nagaosa [19] and +may be formulated as follows. Assuming that a crystal is +further magnetic, we know that, in an insulating regime, +its ground state may accurately be described by a Heisen- +berg model that, by itself, features topological defects. +A familiar example of such a defect would be a magnetic +skyrmion or a hedgehog texture. Assuming that the total +spin of atoms comprising our crystal is large, these tex- +tures may be seen as an arrangement of classical magnetic +moments nailed down to the atomic positions. Their in- +teraction with the electron’s spin degree of freedom may +then be written as a sum of the Zeeman-like terms, each +weighted with the Dirac δ-function centered at the posi- +tion of the corresponding atom. +Quite generically, we show that the total electronic +excess charge accumulated around these defects is an +integer-valued topological invariant, which we express +as a contour integral winding number of an appropriate +functional of bulk position space Green’s functions. Fur- +ther analysis of this topological quantum number reveals +that upon an adiabatic modification of positions and/or +vertex functions of the localized scattering centers, the +value of the invariant may only be affected by the change +in the occupancy of the imperfection-localized bound +states in the process of the spectral flow of their eigenen- +ergies inside the chemical potential-accommodating en- +ergy gap. This observation allows for an immediate in- +terpretation in terms of the nearsightedness principle, +as well as for a direct read-off of the central memo of +Ref. +[16]: “localized perturbations in insulators result +in localized charge redistribution, leading to an addi- +tion/removal of the corresponding perturbation-localized +bound states to/from the occupied spectral region”. We +conclude our analysis by commenting on the relation be- +tween the nearsightedness principle and the topological +invariants characterizing the boundary charge. +In what follows, we set the reduced Plank’s constant ̵h +and the electron charge e equal to unity ̵h = e = 1. +II. +ADIABATIC RESPONSE OF THE EXCESS +CHARGE TO LOCALIZED PERTURBATIONS IN +AN INSULATING STATE +A. +A translationally invariant model +In the following, we shall specifically refer to an elec- +tronic system governed by the following Hamiltonian +H(0) +x += p2 +2m + 1 +2m +d +∑ +j=1 +{ ˜Aj(x),pj} + V (x), +(1) +with V (x) and ˜Aj(x), j = 1, ..., d being the lattice +periodic Nc × Nc Hermitian matrices. More specifically, +{V (x) +˜A(x)} = {V (x + Rm) +˜A(x + Rm)}, +∀m ∈ Zd, +(2) +where Rm = ∑d +j=1 mjaj is a lattice vector characterized +by a d-dimensional vector of integers m = (m1 ⋯ md) +T , +specifying its components in the basis of primitive vec- +tors {aj}j spanning the unit cell of a Bravais lattice. +Furthermore, p and x are vectorial momentum and po- +sition operators comprised of the individual components +pj = −i ∂ +∂xj and xj. +This model naturally generalizes the one recently stud- +ied in Ref. [10] in connection with the universal prop- +erties of one-dimensional boundary charge, to higher +dimensions. +We remark that other models of multi- +dimensional periodic structures [20] are expected to share +the same physics, as the effects we are about to describe +are rather generic to an insulating state. +Translationally invariant systems are characterized by +their band structure, comprised of the individual energy + +3 +bands dispersing as ϵα,k, α = 1, 2, ..., as a function +of the vectorial quasimomentum variable k, confined to +the first Brillouin zone of the reciprocal space. +The +eigenstates of the Hamiltonian to which ϵα,k are the +corresponding eigenvalues are known as Bloch functions +ψα,k(x), and may be generically expressed as +ψα,k(x) = eik⋅xuα,k(x), +(3) +where uα,k(x) in the Nc-component object and is lattice +periodic in the same sense as vector and scalar potentials +are uα,k(x) = uα,k(x+Rm), ∀m ∈ Zd. The completeness +and identity resolution relations may be written as +VUC +(2π)d ∫Rd d(d)xψ† +α,k(x)ψα′,k′(x) = δα,α′δ(d)(k − k′), +(4) +VUC +(2π)d +∞ +∑ +α=1∫BZ d(d)kψα,k(x)ψ† +α,k(x′) = 1Ncδ(d)(x − x′), +(5) +where VUC is the volume of the unit cell, defined via +VUC = ∫UC d(d)x = det(a1∣ ⋯ ∣ad). +(6) +When studying charge, it is more convenient to intro- +duce the retarded single-particle Green’s function, con- +taining the information on both the eigenstates and the +energy spectrum. +In thermodynamic equilibrium, the +Laplace image of the latter is defined as the resolvent +of the single-particle Hamiltonian (1) +[z − H(0) +x ]G(0)(x,x′) = 1Ncδ(d)(x − x′), +(7) +where z is the complex energy variable, defined in terms +of the physical frequency variable ω as z = ω + iη, where +η → 0+. Owing to the identity resolution relation (5) we +can establish the conventional Lehmann representation +G(0)(x,x′) = VUC +(2π)d +∞ +∑ +α=1∫BZ d(d)k +ψα,k(x)ψ† +α,k(x′) +z − ϵα,k +. (8) +Further, using the completeness of the basis (4), in Ap- +pendix A, we establish the following important fusion +rule for the bare propagators +∫Rd d(d)x′G(0)(x,x′)G(0)(x′,x′′) = − ∂ +∂ω G(0)(x,x′′). +(9) +As it is shown in Appendix A, this relation holds pretty +generally, without any reference to the Hamiltonian (1). +B. +Localized perturbations and Dyson’s equation +Now we perturb the translationally invariant (on the +scale of the unit cell) system by a finite number of point- +like impurities +˜V (x) = +N +∑ +n=1 +˜V (n) +0 +δ(d)(x − xn), +(10) +where ˜V (n) +0 +are Nc × Nc matrices describing the action +of the nth impurity on the channel space. This action is +further assumed to be local as prescribed by Dirac delta- +function δ(d)(x − xn) centered at the impurity position +xn. +Let us remark that the problem of a Dirac delta- +function potential is well-known to be ill-defined in spa- +tial dimensions higher than d = 1. +In our analysis, +this is manifested in the ill-definiteness of the bulk po- +sition space Green’s function at equal spatial arguments +G(0)(x,x) due to the divergence of the defining integrals +(8) in the ultraviolet. +Such a divergence is not physi- +cal and has to be circumvented by an appropriate reg- +ularization scheme. +In particular, in the metallic case +˜A(x) = 0, V (x) = 0, in d > 1 the problem of the delta- +potential has been extensively studied in both physical +[21, 22, 23] and mathematical [24] literature and several +meaningful regularization techniques were proposed and +shown to produce physically sensible results. Since the +presence of the energy gaps is of no importance in the +deep ultraviolet regime, the same methods may be ap- +plied in our case. +The Dyson’s equation for the full Green’s function of +the system is given by +G(x,x′) =G(0)(x,x′) ++ +N +∑ +n=1 +G(0)(x,xn) ˜V (n) +0 +G(xn,x′). +(11) +First we want to consistently solve for the functions +G(xn,x′), n = 1, ..., N. This problem is brought to +the solution of the following matrix equation +M(z)D(x′) = D(0)(x′), +(12) +where M(z) is the Nc ⋅ N × Nc ⋅ N block matrix defined +by +M(z) =1Nc⋅N − G(0)(z)˜V0, +(13) +(G(0)(z))n,n′ =G(0)(xn,xn′), (˜V0)n,n′ = δn,n′ ˜V (n) +0 +. (14) +Likewise, D(x′) and D(0)(x′) are the Nc ⋅ N × Nc matri- +ces comprised of the full G(xn,x′) and bare G(0)(xn,x′) +propagators, respectively. With these notations we ob- +tain +G(x,x′) =G(0)(x,x′) + D(0)†(x)˜V0D(x′) +=G(0)(x,x′) + D(0)†(x)˜V0M−1(z)D(0)(x′), +(15) +where in our definition the Hermitian conjugate does not +affect the z-variable, i.e. +(G(0)(x,x′))† = G(0)(x′,x). +(16) + +4 +C. +Measuring the excess charge +We define the excess charge density operator in the +following manner: +δ̂ρ(x) = ̂ρ(x) − ¯ρ, +(17) +where +̂ρ(x) = ̂ψ†(x)̂ψ(x), +(18) +is the density operator, expressed in terms of the Nc- +component fermionic field operators ̂ψ(x) and ̂ψ†(x). +The field operators ̂ψ(x) and ̂ψ†(x) are further assumed +to destroy/create excitations of the full Hamiltonian in- +cluding the effect of localized scattering centers in Eq. +(10). The constant contribution ¯ρ describes the unit cell- +averaged average charge density in the bulk: +¯ρ = 1 +VUC ∫VUC +d(d)xρ(0)(x), +(19) +ρ(0)(x) =⟨̂ψ(0)†(x)̂ψ(0)(x)⟩ += − 1 +π Im∫ +µ +−∞ dωtr{G(0)(x,x)}, +(20) +where the field operators ̂ψ(0)(x) and ̂ψ(0)†(x) describe +the excitations of the translationally invariant system, µ +denotes the chemical potential, and G(0)(x, x′) is the +bare Green’s function defined by Eqs. (7) and (8). +We measure the excess charge with the help of the clas- +sical device, described by the envelope function f(x) (see +Refs. [7, 8, 9, 10] and Ref. [25] for similar definitions). +To be more specific, we define the excess charge operator +as +δ ̂Q = ∫Rd d(d)xf(x)δ̂ρ(x). +(21) +It is sensible to define the function f(x) relative to a +certain point xp, to which the charge probe is applied, +and further assume that the charge is sampled equiva- +lently in all directions f(x) = f(∣x − xp∣). Additionally, +we assume that all of the charge f(∣x − xp∣) ≈ 1 is sam- +pled in sufficiently large vicinity of the sampling point +xp, while the envelope function smoothly decays to zero +f(∣x − xp∣) → 0 far away from xp. For that matter, it is +convenient to choose +f(∣x − xp∣) = 1 − Θlp(∣x − xp∣ − Lp), +(22) +where Θlp(∣x − xp∣ − Lp) is some representation of the +Heaviside function broadened by lp. The length scales +characteristic of the charge probe are assumed to satisfy +Lp ≫ lp ≫ ξg, +(23) +where ξg ≃ vF +Eg is the charge localization length in an insu- +lator (also it is the charge correlation length, defining the +exponential decay length of the density-density correla- +tion function, see Ref. [18]), roughly defined as the ratio +between the Fermi velocity vF and size of the energy gap +at the Fermi level Eg. +D. +Topological invariant characterizing the excess +charge +Let us assume that N impurities, as characterized by +the potential (10), are placed in a region of a crystal +falling into the sampling district of the envelope function +∣x∣ ≲ Lp. We define the total excess charge as the zero +temperature expectation value of the excess charge op- +erator in the grandcanonical equilibrium density matrix, +so that +δQ =⟨δ ̂Q⟩ = ∫Rd d(d)xf(x)(ρ(x) − ¯ρ), +(24) +ρ(x) = − 1 +π Im∫ +µ +−∞ dωtr{G(x,x)}. +(25) +With the help of the representation (15), we obtain +δQ = Q′ + QP , +(26) +where Q′ contains the Friedel charge as well as the charge +due to the impurity-localized bound states +Q′ = ∫Rd d(d)xf(x)ρ′(x), +(27) +ρ′(x) = − 1 +π Im∫ +µ +−∞ dωtr{D(0)†(x)˜V0M−1(z)D(0)(x)}, +(28) +while QP is the so-called polarization charge given by +QP = ∫Rd d(d)xf(x)(ρ(0)(x) − ¯ρ), +(29) +and, with the help of the properties of the envelope func- +tion, is shown to be zero QP = 0 in Appendix B. It hence +follows that +δQ =Q′ = − 1 +π Im∫Rd d(d)xf(x) +× ∫ +µ +−∞ dωtr{D(0)†(x)˜V0M−1(z)D(0)(x)}. +(30) +Due to the branch cuts and poles of the T-matrix +T(x,x′) = ∑n,n′[˜V0M−1(z)]n,n′δ(x − xn)δ(x′ − xn′), the +integrand of the outer integral is exponentially sup- +pressed ∼ e−∣x∣/ξg at large x, allowing us to set f(x) = 1. +Interchanging the order of the integrals, we consider +∫Rd d(d)xtr{D(0)†(x)˜V0M−1(z)D(0)(x)} += − +N +∑ +n,n′=1 +tr{[M−1(z)]n,n′ ∂ +∂ω G(0)(xn′,xn) ˜V (n) +0 +} += +N +∑ +n,n′=1 +tr{[M−1(z)]n,n′ ∂ +∂ω [M(z)]n′,n} += +N +∑ +n=1 +tr{[M−1(z) ∂ +∂ω M(z)] +n,n +} += ∂ +∂ω tr{log M(z)} = ∂ +∂ω log det{M(z)}, +(31) + +5 +where, in the last line, trace and determinant of the full +Nc ⋅N ×Nc ⋅N block matrix M(z) are understood. Using +the result in Eq. (31), we arrive at the following compact +formula for the total excess charge +δQ = − 1 +π Im∫ +µ +−∞ dω ∂ +∂ω log det{M(z)}. +(32) +To see why the integral in Eq. (32) may take on in- +tegral values only, in Appendix C we find an alternative +contour integral representation +δQ = − ∮C +dz +2πi +∂ +∂z log det{M(z)}, +(33) +where C is an arbitrary non-self-intersecting curve that +crosses the real axis at two points only, below the low- +est eigenvalue of the full Hamiltonian and at the chemi- +cal potential µ, and the direction of C is assumed to be +clockwise. +In the representation (33), the excess charge δQ is nec- +essarily an integer as it is expressed as a contour integral +winding number and the chemical potential is by def- +inition inside one of the energy gaps (we focus on the +insulating systems solely). In other words, the integral +in Eq. (33) measures the degree of the mapping S1 → S1 +and is thus a member of the only non-trivial homotopy +group of the unit circle π1(S1) = Z. +In particular, the integral in Eq. +(33), is a sum +of two distinct contributions: +the contribution of the +branch cuts corresponding to the extended or scatter- +ing states, and the contribution of poles corresponding +to the imperfection-localized bound states. +The bands in multidimensional (d > 1) and/or mul- +tichannel (Nc > 1) systems are typically composite, i.e. +overlapping with one another along the frequency axis. +For that matter, it is convenient to choose the branch +cuts to connect the bottom of the lowest sub-band with +the top of the highest one, within every patch of the en- +ergy bands surrounded by a pair of energy gaps. +The bound state poles, determined as a solution of +det{M(z)}∣z∈R = 0, are located on the complement of +the bare Hamiltonian’s spectrum, i.e. inside the energy +gaps and, in some cases (e.g. an attractive scalar impu- +rity), below the bottom of the lowest energy band of the +unperturbed Hamiltonian. +III. +RELATION WITH THE +NEARSIGHTEDNESS PRINCIPLE +A. +Discussion +Now we would like to discuss the topological invari- +ant (33) in greater detail. In what follows, we specify +the contour C as a rectangle of length µ − B in the real +direction and width 2η in the imaginary one. +Here B +is by definition an energy lying below the lowest eigen- +value of the total Hamiltonian Hx = H(0) +x ++ ˜V (x) (i.e. +c) +b) +a) +Im{z} +Re{z} +C +FIG. 1. A schematic illustration of how the spectral flow of the +energies of the imperfection-circumscribing bound states sit- +ting inside the gap that accommodates the Fermi level affects +the total excess charge. The spectrum of the system is visu- +alized through the local spectral density as looked down on +the complex frequency plane. The occupied part of the spec- +trum is demonstrated in blue, while the yellow color marks its +complement (the states of the system that are unoccupied). +Panel a) shows a rectangular contour C encircling the occu- +pied spectral region. Panels b) and c) show the zoomed-in +vicinity of the chemical potential before and after the per- +turbation. As is demonstrated in panel c), the spectral flow +results in the removal of a single bound state, carrying away +a unity of the electron charge from the system (an inverse +process is of course also possible). +B ∈ (−∞,min{spec{Hx}})), and η is not necessarily an +infinitesimal positive but is rather a finite positive num- +ber (which is allowed as the integral is invariant under +such contour deformations (see Appendix C)). Further- +more, we assume that the chemical potential is located +above the νth bulk energy band. +Let us now consider making an adiabatic perturba- +tion to the system that is comprised of the change in the +positions {xn}n and/or vertex functions { ˜V (n) +0 +}n of the +impurities. As the span of the extended states’ energy +bands is unaffected by such adiabatic perturbations, the +branch cut contribution to the winding number remains +invariant (up to the cases when the bound state merges +with the band, as discussed below). This remark is essen- +tially true as such deformations of the parameter space +do not change the analytical structure of G(0)(xn, xn′), +through the functionals of which alone our topological +invariant is expressed. +We hence conclude that such +changes may only unleash themselves in the spectral flow +of the bound state energies. +As was anticipated in Section II D, the bound state +energies are energy-wise located inside the energy gaps +of the bulk system. +This assertion also regards the +energy gap below the bottom of the lowest band ω ∈ +(−∞, mink ϵ1,k], which can accommodate the bound +states in the case of attractive impurities, for example. + +6 +The energies of the bound states ϵbs inside the energy +gaps [maxk ϵα,k, mink ϵα+1,k] surrounded by a pair of +bands α, α + 1, (α = 1, ..., ν − 1), are solely character- +ized by their location within the gap. The same holds +true for the infinite gap below the bottom of the lowest +bulk energy band, with ϵbs now being energy-wise located +in (−∞, mink ϵ1,k]. This implies that the spectral flow +of these energies is constituted in the motion of ϵbs in be- +tween the top of ϵα,k and the bottom of ϵα+1,k, or between +the negative infinity and mink ϵ1,k shall some states be +also found in there. When merging with one of the energy +bands (either ϵα,k or ϵα+1,k, and ϵ1,k solely when consid- +ering the gap preceding the entire band structure), the +value of the contour integral winding number (33) relat- +ing to that band gets modified by unity26. It follows that +the motion of the bound state poles, inside such energy +gaps below the one hosting the chemical potential, has +absolutely no effect on the topological invariant (33) (one +may see this result as a form of charge conservation), as +B, by definition, resides below the lowest pole (effectively +meaning that none of the states are allowed to escape the +occupied spectral region from below). +The flow of the energies of the impurity-localized +bound states residing inside the gap separating the con- +duction and the valence bands apart (the gap where the +chemical potential is located), on the other hand, affects +the winding number in Eq. (33). When a bound state +crosses the chemical potential from above or below, the +number of poles encompassed by the integration contour +increases or decreases correspondingly. That means that +the unit of the electron charge gets either pumped in or +out of the system, modifying the topological invariant by +±1. This discussion is summarized in Fig. 1. +The elaboration above allows us to draw the following +physical conclusion: +Localized adiabatic perturbations in insulators, may only +result in the localized charge redistributions, owing to +the change in the occupancy of the perturbation-localized +bound states at the Fermi level. +This intuitive result is nothing but a direct consequence +of the universal nearsightedness principle of W. Kohn +[12, 13, 14] stating that, at fixed chemical potential, the +electronic charge density depends on the external field +(in our case being an assembly of localized scattering +centers) only at nearby points. +Another conclusion drawn by E. Prodan and W. Kohn +in Ref. +[13] (see also Ref. +[14] for the fine details in +d = 1) is that the adiabatic perturbations to the exter- +nal potential, no matter how strong, have a negligible +effect on the local charge density beyond a certain char- +acteristic length scale, which, in the insulating regime, +is naturally provided by the charge correlation length +ξg. From the viewpoint of our topological invariant (33), +this means that in the case of well-separated impurities +∣xn −xn′∣/ξg ≫ 1, the topological invariant is expected to +approach a sum of the individual single-impurity invari- +ants, as distant impurities are not supposed to be able +to “talk” with one another on such scales. +Indeed, in +an insulating state, it is well-known, that the two-point +correlation functions G(0)(xn, xn′) decay exponentially +at large distances ∼ e−∣Rmn−Rmn′ ∣/ξg (where mn labels +the unit cell accommodating the nth scattering center), +meaning that we can approximate +(G(0)(z))n,n′ ≃δn,n′G(0)(xn,xn), +(34) +implying that +M(z) ≃ +N +⊕ +n=1 +(1Nc − G(0)(xn,xn)V (n) +0 +), +(35) +and +δQ ≃ − +N +∑ +n=1∮C +dz +2πi +∂ +∂z log det{1Nc − G(0)(xn,xn)V (n) +0 +}. +(36) +This result may be seen as a form of the conven- +tional Born approximation of the linear transport theory, +whereby, to the lowest order in the impurity density, one +considers impurities as independent. +B. +An illustration: A pair of magnetic impurities +in an illuminated quantum wire +To illustrate some of the points highlighted in the +above discussion, we here consider a simple model of a +spin-orbit-interacting ballistic quantum wire, submersed +into the background of the spatially oscillating electro- +magnetic field. The bulk Hamiltonian assumes the form +of the Pauli Hamiltonian with an extra Rashba-like term: +H(0) +x += +(p + e +cAx(x)) +2 +2m ++ kR ⋅ σ +m +(p + e +cAx(x)) ++ µBge +2 +σ ⋅ B(x). +(37) +Above, kR = (kR,x, kR,y, kR,z) is the Rashba spin-orbit +vector, σ = (σx, σy, σz) is the vector of the Pauli spin +matrices, µB = +e +2mc is the Bohr magneton, c is the speed +of light in vacuum, ge is the electron’s Land´e g-factor, +and +B(x) =∇ × A(x)∣ +x=xˆex, +Ax(x) = ˆex ⋅ A(x)∣ +x=xˆex, (38) +with ˆex being the ort in the x-direction, and A(x) be- +ing the electromagnetic vector potential of the monochro- +matic plane-wave form +A(x) = A0 cos(q ⋅ x + ϕ), +(39) +in the Coulomb gauge +∇ ⋅ A(x) = 0 ⇐⇒ q ⋅ A0 = 0. +(40) + +7 +° º +L +° º +2L +0 +º +2L +º +L +k +°2.00 +°1.75 +°1.50 +°1.25 +°1.00 +°0.75 +°0.50 +≤k +x +y +z +kR +m(1) +eff +m(2) +eff +R +q +B +E +λ +a) +b) +FIG. 2. Panel a): A schematic illustration of a ballistic quantum wire featuring Rashba-style spin-orbit coupling (defined by a +spin-orbit vector kR) and submersed into a spatially-periodic arrangement of electric E and magnetic B fields of wavelength λ. +The two impurity atoms, separated by distance R and carrying an effective magnetic moment of m(j) +eff , j = 1, 2, are schematically +shown by atomic symbols pierced with the magnetic moment-symbolizing arrows. Panel b): The bulk energy spectrum of the +two-impurity problem. The energy bands are shown in dark blue, the chemical potential located inside the second spectral gap +(above the fourth energy band) is depicted in orange, and the relevant spectral region is highlighted in light blue. +The wave vector of the background electromagnetic field +defines the fictitious lattice spacing +L = +2π +ˆex ⋅ q, +(41) +where we have excluded the uninteresting case of the or- +thogonally propagating wave ˆex ⋅ q = 0. +We note that the Hamiltonian in Eq. (37) falls into +the class of systems defined by the Hamiltonian (1), with +d = 1 and +V (x) =µBge +2 +σ ⋅ B(x) + e2A2 +x(x) +2mc2 ++ ekR ⋅ σAx(x) +mc +, (42) +˜Ax(x) =e +cAx(x) + kR ⋅ σ. +(43) +As this demonstration is assumed to be interpretative, +it suffices to consider the case of a pair of impurities, +which we assume to be separated by distance R: +˜V (x) = ˜V (1) +0 +δ(x) + ˜V (2) +0 +δ(x − R). +(44) +Note that we can place the first impurity at x = 0 with- +out loss of generality, as its other positions inside the wire +may be achieved by appropriate tuning of the modula- +tion’s phase ϕ. Furthermore, we assume the impurities +to exert both the electrostatic and the exchange “force” +on the wire’s electrons, which we encode in the following +form of the impurities’ vertex functions +˜V (j) +0 += Ujσ0 + µBge +2 +σ ⋅ B(j) +eff , +(45) +where B(j) +eff is the effective (also appropriately screened to +have a short-ranged effect only) magnetic field, produced +by the effective magnetic moment of the impurity atom +m(j) +eff = +qjgj +2MjcS(j), with qj, gj, and Mj being the charge, +g-factor, and mass of the jth impurity. Furthermore, Uj +denotes the strength of the electrostatic potential, defin- +ing the corresponding force exerted by the impurity on +the electrons. Not going into much of the microscopic +details, in the following, we treat Uj and B(j) +eff as some +constant parameters. +The resulting setup is schemati- +cally illustrated in panel a) of Fig. 2. +Further, to illustrate our point, we assume that the +associated impurity parameters {R, {Uj}j, {B(j) +eff }j} +evolve with a fictitious “adiabatic time” τ ∈ [0,T], in +such a manner that their temporal derivatives remain +much smaller than the Fermi energy ϵF times their value, +for all τ ∈ [0,T]. +The particular form of the pumping protocol used to +produce the numerical data and the concrete numerical +values of the free model parameters are provided in Ap- +pendix D. The resulting bulk energy spectrum is demon- +strated in panel b) of Fig. 2. +The +numerical +data +for +the +excess +charge- +characterizing topological invariant, +as well as the +spectral flow of bound state energies inside the chem- +ical potential-accommodating spectral gap, is shown +in Fig. +3. +In particular, using the parametrization +R(τ) = (nR − 1)L + ¯R(τ), suggested in the Appendix +D, we present the data for five different values of +nR ∈ {1,...,5}, as shown in five different columns of +the corresponding figure, with upper and lower rows +corresponding to the spectral flow and the topolog- +ical invariant, respectively. +Solid black and dashed +burgundy lines mark the cases of independent and +“interacting” +impurities, +correspondingly. +By +in- +dependent impurities, we here understand that the +separation between them is effectively infinite, so that +the off-diagonal blocks of the M(z) matrix (see Eq. +(14) for the definition) may be completely ignored. +This means that the bound state spectrum of the +independent impurities is provided by the solution of +det(12 − G(0)(0,0) ˜V (1) +0 +)det(12 − G(0)(R,R) ˜V (2) +0 +)∣z∈R = + +8 +0 +T/4 +T/2 +3T/4 +T +τ +max ϵk,4 +min ϵk,5 +nR = 1 +0 +T/4 +T/2 +3T/4 +T +τ +nR = 2 +0 +T/4 +T/2 +3T/4 +T +τ +nR = 3 +0 +T/4 +T/2 +3T/4 +T +τ +nR = 4 +0 +T/4 +T/2 +3T/4 +T +τ +nR = 5 +0 +T/4 +T/2 +3T/4 +T +τ +−2 +−1 +0 +1 +2 +δQ +0 +T/4 +T/2 +3T/4 +T +τ +0 +T/4 +T/2 +3T/4 +T +τ +0 +T/4 +T/2 +3T/4 +T +τ +0 +T/4 +T/2 +3T/4 +T +τ +FIG. 3. The figure demonstrates the adiabatic flows of both the bound state energy spectrum and the excess charge invariant +in the toy model proposed in Section III B. Specifically, the spectral flow of the impurity-localized bound state energies is +shown in the upper row, while the second row is dedicated to the invariant itself. As is explained in Appendix D, the position +of the second impurity is parametrized as R(τ) = (nR − 1)L + ¯R(τ), where nR denotes the number of the unit cell hosting +the second imperfection, and ¯R(τ) ∈ [0, L] describes its location within the unit cell. The five distinct columns in the above +figure correspond to five choices of nR = 1, 2, 3, 4, 5. In all of the panels, red dashed lines correspond to the actual solution, +while black solid lines relate to the case of two independent impurities (see the approximate formula (36)). As the separation +between the impurities becomes of the order of the charge localization length ξg = O(L) (see Appendix D), both adiabatic flows +approach the limit of two independent impurities. +0, while the topological invariant is given by the ap- +proximation (36). +By the “interacting” impurities, on +the other hand, we understand that the exact relations +were used to produce the numerical data. The numerical +technique for evaluation of bulk position space Green’s +functions, as well as the topological indices of the form +(33), is outlined in Ref. +[10]. +In our calculations, +the values of the contour parameters were chosen as +η = 1, B = −30 (such a choice of B is motivated by +the presence of the bound states below the lowest band +ω ∈ (−∞, mink ϵk,1] in our model (37)). +The central purpose of our demonstration is to show +that upon the increase in the impurity’s separation be- +yond the charge localization length ξg = O(L) (see Ap- +pendix D), both the topological invariant and the bound +state spectrum approach that of a pair of independent +impurities. +This effect is a direct consequence of the +nearsightedness principle, telling us that a localized cause +leads to a localized effect. Furthermore, as one may an- +ticipate, the discontinuous jumps of the excess charge in- +variant occur precisely at the points where bound states +enter/leave the occupied part of the energy spectrum, as +is explained in Section III A. Another interesting obser- +vation is the non-zero value of the topological invariant +at the beginning of the adiabatic evolution in τ, where +the strengths of the electrostatic repulsion are the small- +est 0 < Uj ≪ 1 (see Appendix D). This feature is a conse- +quence of the presence of impurity-localized bound states +below the bottom of the lowest energy band. Such an +effect is well-known in the case of attractive scalar impu- +rities, whereas here, it is generated by the non-Abelian +structure of the model, and, to the best of our knowledge, +was not reported previously in the literature. +C. +Topological invariants characterizing the +boundary charge in unidimensional crystals +In this section, we would like to comment on the topo- +logical invariants characterizing boundary charges in uni- +dimensional crystals, extensively discussed in Refs. [7, +8, 9, 10]. +In particular, let us consider a d = 1 semi- +infinite system described by the Hamiltonian (1), with +the boundary placed at x = xb. An appropriate restric- +tion of x defines the respective right and left subsystems: +x ∈ [xb,∞), +right sub-system, +(46) +x ∈ (−∞,xb], +left sub-system. +(47) +In our definition, the primitive unit cell is defined as the +one starting at the boundary of the right semi-infinite +system UC = [xb, xb+L], with L being the lattice period. +In this definition, the left half-system is always obtained +from the right one by a local inversion operation, which +acts by the inversion of local coordinates within each unit +cell. +Now we define the boundary charge operators corre- +sponding to right and left semi-infinite systems as the +envelope-weighted integrals of the expectation values of +the appropriate excess charge density operators: +Q(R) +B +=∫ +∞ +xb +dxf(x)⟨δ̂ρR(x)⟩, +(48) +Q(L) +B +=∫ +xb +−∞ dxf(x)⟨δ̂ρL(x)⟩, +(49) +where, in analogy with Eq. +(17), δ̂ρS(x) = ̂ρS(x) − ¯ρ, +and ̂ρS(x) is the density operator referring to the system +S = R, L. Furthermore, the envelope function f(x) is +chosen in accordance with Eq. (22), with xp = xb, and + +9 +the range of x being restricted according to Eqs. (46) +and (47). +Let us now consider measuring the total excess charge +δQ accumulated around x = xb in a translationally in- +variant system x ∈ (−∞, ∞). By the polarization charge +neutrality condition QP = 0, demonstrated in Appendix +B, the total excess charge also vanishes δQ = 0. +On +the other hand, we may consider a translationally invari- +ant system as a sum of right and left semi-infinite sys- +tems with a coupling corresponding to the bulk Hamilto- +nian switched in between them. This coupling manifests +itself as a local perturbation and, by the nearsighted- +ness principle of Kohn, is capable of affecting the total +charge locally by at most introducing or removing a num- +ber of additional bound states, resulting in an integer +contribution QI. In this connection, we conclude that +δQ = Q(R) +B ++ Q(L) +B +− QI = 0, where QI is known as the in- +terface invariant. One of the central results of Ref. [10], +was to demonstrate that +QI =Q(R) +B ++ Q(L) +B += −∮C +dz +2πi +∂ +∂z log det{G(0)(xb,xb)}. +(50) +That is, the interface invariant, characterizing the bound- +ary charge upon local inversions, is a topological quan- +tum number given by the winding of the determinant +of bulk position space Green’s function evaluated at the +location of the boundary. +Now let us proceed with the transformations of the +boundary charge under translations. First, we consider +the right boundary charge of the so-called reference sys- +tem, starting at xb = 0: +Q(R) +B (0) = ∫ +∞ +0 +dxf(x)(ρ(x) − ¯ρ), +(51) +and we would like to analyze the changes in this quantity +upon the translation of the boundary by xϕ ∈ [0, L]. In- +stead of shifting the boundary, we consider adding the fol- +lowing potential ˆV (x) = ˆV0Θ(x)Θ(xϕ − x), ˆV0 → ∞. By +the Pauli principle, the charge density becomes zero for +x ∈ [0,xϕ] as these states sit at infinite energy above the +chemical potential µ. From the definition of the bound- +ary charge, we are left with the following contribution: +δQ(R) +B (xϕ) = ∫ +xϕ +0 +dxf(x)(0 − ¯ρ) +mod 1 += −¯ρxϕ +mod 1, +(52) +where +mod 1 contribution again comes from the near- +sightedness principle. This analysis allows us to conclude +that: +Q(R) +B (xϕ) − Q(R) +B (0) = ¯ρxϕ + I(xϕ), +(53) +where I(xϕ) is known as the boundary invariant. An- +other important result of Ref. [10] was to show that +I(xϕ) = −∮C +dz +2πi +∂ +∂z lndetU(xϕ), +(54) +where U(xϕ) is defined via the path-ordered exponential +U(xϕ) =Pexp{∫ +x +0 +dx′L(x′)}, +(55) +L(x) =[G(0)(x,x)]−1G(0) +2 (x,x+) − iA(x), +(56) +and G(0) +2 (x,x′) = ∂x′G(0)(x,x′). +In other words, the +boundary invariant is also a topological quantum number +expressed as a winding of the appropriate functional of +bulk position space Green’s functions. +In this way, we see that the quantization of the topo- +logical invariants characterizing the boundary charge in +one-dimensional insulators is a direct consequence of the +nearsightedness principle. As this intuitive physical prin- +ciple holds beyond the single spatial dimension, one ex- +pects the excess charges accumulated on inhomogeneities +of various spatial co-dimensions in d-dimensional crystals +to possess similar topological characterization schemes. +Indeed, linear scaling of the boundary charge, along with +its discontinuous jumps by a unit of the electron charge +at the bound state escape/entrance spectral points, was +recently demonstrated in a two-dimensional system [27]. +IV. +CONCLUSIONS AND OUTLOOK +In this paper, the quantization of the excess charges on +localized scattering centers in d-dimensional insulators +was discussed. Our analysis reveals that an assembly of +such imperfections accumulates an integral excess charge, +given by a winding number expression. We find that an +adiabatic perturbation (no matter how strong) comprised +of either relocation of the impurities or a modification of +their vertex functions (or both at the same time) results +in the change of the total charge by an integer, deter- +mined by the saldo of the imperfection-localized bound +states that entered or escaped the occupied spectral re- +gion, inside the chemical potential-hosting bulk spectral +gap. The quantization of this topological invariant was +shown to be a direct consequence of the nearsightedness +principle of the electronic matter, limiting the range of +the effect of a localized cause. Additionally, this local +behavior of the electronic matter in the insulating state +was shown to be responsible for the quantization of the +topological invariants characterizing the unidimensional +boundary charge studied in [7, 8, 9, 10]. Furthermore, +our study confirms the central paradigm of Ref. +[16], +namely that localized perturbations in insulators specifi- +cally lead to the change in occupancy of the correspond- +ing perturbation-localized bound states, modifying the +total charge, defined as the macroscopic average on the +scales significantly exceeding both the unit cell size L and +the charge correlation length ξg, by at most an integer. +As is now obvious, the present paper is of conceptual +value only as the evaluation of the suggested topologi- +cal invariant (33) for a specific multi-impurity (N ≫ 1) +system poses a challenge on its own. In particular, this +concerns questions regarding the regularization schemes + +10 +for the higher-dimensional equal-argument Green’s func- +tions, as well as the basic questions regarding the numer- +ical feasibility of the problem. Furthermore, it would be +of future interest to study the expansion of the topolog- +ical invariant in the interaction between the individual +impurities, as generated by the off-diagonal blocks of the +M(z) matrix, and analyze its ties with the conventional +Born series for the impurity-dressed T-matrix. As it is +suggested in the present study, in the insulating state, +the impurity density ρI has to be always contrasted with +the inverse charge localization length ξg, in such a man- +ner that the condition 1 ≫ ρIξd +g implies the validity of the +Born approximation, treating impurities as independent. +V. +ACKNOWLEDGMENTS +The author gratefully acknowledges the durable ex- +change of ideas with M. Pletyukhov and H. Schoeller. +Further, the author generously thanks S. Miles and M. +Pletyukhov for their valuable comments. +Most of the present work was done at the Institut f¨ur +Theorie der Statistischen Physik of RWTH Aachen and +was financially supported by the Deutsche Forschungsge- +meinschaft via RTG 1995. +Appendix A: Contraction of two Green’s functions +Quite generically we may represent +G = ⨋s +∣s⟩⟨s∣ +z − ϵs +, +(A1) +where the meta-index s labels the eigenstates ∣s⟩ and +eigenenergies ϵs of the Hamiltonian. +Considering the +product of the Green’s function with itself +GG = ⨋s ⨋s′ +∣s⟩⟨s′∣ +(z − ϵs)(z − ϵs′) ⟨s∣s′⟩ +� +δ(s,s′) += ⨋s +∣s⟩⟨s∣ +(z − ϵs)2 += − ∂ +∂ω ⨋s +∣s⟩⟨s∣ +z − ϵs += − ∂ +∂ω G. +(A2) +Taking the position space matrix elements +⟨x∣GG∣x′′⟩ = − ∂ +∂ω G(x,x′′), +and inserting +1 = ∫Rd d(d)x∣x⟩⟨x∣, +(A3) +we obtain the desired identity +∫Rd d(d)x′G(x,x′)G(x′,x′′) = − ∂ +∂ω G(x,x′′). +(A4) +Appendix B: Polarization charge +We consider +∫Rd d(d)xf(x)(ρ(0)(x) − ¯ρ) += ∑ +m ∫UC d(d)¯xf(¯x + Rm)(ρ(0)(¯x) − ¯ρ). +(B1) +Above we parametrized the position space variable x as +x = Rm+¯x, for some vector of integers m, and ¯x is the lo- +cal coordinate within the unit cell ¯x ∈ UC. Furthermore, +we used the periodicity property of ρ(0)(x), implied by +the periodicity of the equal-argument Green’s function +G(0)(x, x) = G(0)(x + Rm, x + Rm), +∀m ∈ Zd. (B2) +The envelope function varies significantly only in the +crossover region ∣Rm∣ = O(Lp), allowing us to approxi- +mate +f(¯x + Rm) ≈ f(Rm) + ¯x ⋅ ∇f(Rm), +(B3) +leading to +∫Rd d(d)xf(x)(ρ(0)(x) − ¯ρ) += ∫UC d(d)¯x∑ +m +(¯x ⋅ ∇f(Rm))(ρ(0)(¯x) − ¯ρ). +(B4) +Now approximating +∑ +m +(¯x ⋅ ∇f(Rm)) ≈ +1 +VUC ∫Rd d(d)y(¯x ⋅ ∇yf(y)) += +1 +VUC ∫Rd d(d)y∇y ⋅ (¯xf(y)) = 0, +(B5) +where in the last step we used Gauss’ divergence theorem. +Appendix C: Contour integral representation +First, we rewrite Eq. (32) as +δQ = − 1 +2πi ∫ +µ +−∞ dω ∂ +∂ω log det{M(z)} ++ 1 +2πi ∫ +µ +−∞ dω ∂ +∂ω (log det{M(z)})∗ . +(C1) +Now we remind ourselves that +(log f(z))∗ = log (f(z))∗ ≡ log f ∗(z). +(C2) +Furthermore, one has +(det{M(z)})∗ = det{M†(z∗)}, +(C3) +where, as before, the Hermitian conjugate does not affect +the z-variable. Now we have +det{M†(z∗)} = det{(1 − G(0)(z∗)˜V(0) )†} += det{1 − ˜V(0)G(0)(z∗)} += det{M(z∗)}, +(C4) + +11 +where to get from the pre-last to the last lines we em- +ployed the Weinstein–Aronszajn identity. +It hence follows that +δQ = − 1 +2πi ∫ +µ +−∞ dω ∂ +∂ω log det{M(ω + iη)} +− 1 +2πi ∫ +−∞ +µ +dω ∂ +∂ω log det{M(ω − iη)} += − ∮C +dz +2πi +∂ +∂z log det{M(z)}. +(C5) +Above, C is the counterclockwise rectangular contour de- +fined as a union of four segments: +C =[B + iη,µ + iη) ∪ [µ + iη,µ − iη) ∪ [µ − iη,B − iη) +∪ [B − iη,B + iη), +B → −∞, +η → 0+. +(C6) +We note that the integral in (C5) remains unaffected +under continuous contour deformations, so long as the +analytic structure of the integrand within the patch of +the complex plane enclosed by contour C remains intact. +In this connection, we may replace C with an arbitrary +non-self-intersecting curve crossing the real axis at two +points only, at any energy below the lowest eigenvalue of +the full Hamiltonian, and at the chemical potential. +Appendix D: Parameters and protocols +In the numerical example provided in Section III B, the +parameters of the model were chosen according to +q = 2π(ex + κey) +λ +√ +1 + κ2 +, +A0 = A0(κex − ey) +√ +1 + κ2 +, +m = 1, (D1) +κ = 1 + +√ +5 +2 +, +λ = 4, +e +cA0 = 1.17, +kR = +⎛ +⎜ +⎝ +0.32 +1.39 +1.24 +⎞ +⎟ +⎠ +. (D2) +Note that as we have set the electron’s mass m = 1 to +unity (in addition to the electric charge e = 1 and reduced +Plank’s constant ̵h = 1), we work in Hartree’s atomic +units. +In this way, the electromagnetic wave is propagating +in the x − y plane, with the corresponding magnetic field +being +B(x) = 2πA0 +λ +ez sin(q ⋅ x + ϕ). +(D3) +By definition, the corresponding lattice period is given +by +L = λ +√ +1 + κ2 = 2 +√ +2(5 + +√ +5). +(D4) +To produce the data, we used the following pumping +protocol for the impurities’ separation +R(τ) = (nR − 1)L + ¯R(τ), +¯R(τ) = L +T τ, +L +T ≪ vF , +(D5) +where vF is the Fermi velocity and nR is an integer spec- +ifying the number of the unit cell hosting the second +impurity. For the impurities’ vertex functions, we fur- +ther make an assumption of the equivalent impurities: +U (1)(τ) = U (2)(τ) =∶ U(τ) and ∣B(1) +eff (τ)∣ = ∣B(2) +eff (τ)∣ =∶ +BI(τ). The direction of the magnetic moments, on the +other hand, is allowed to be different in two scattering +centers and is parametrized in spherical polar coordinates +B(j) +eff (τ) +BI(τ) = +⎛ +⎜ +⎝ +cosφ(j)(τ)sinθ(j)(τ) +sinφ(j)(τ)sinθ(j)(τ) +cosθ(j)(τ) +⎞ +⎟ +⎠ +. +(D6) +In the following, we assume that, as is the case with the +location of the second impurity within the unit cell num- +ber nR, the impurity strength also grows linearly with +τ +U(τ) = U0 + δU τ +T , +δU +T ≪ ϵ2 +F . +(D7) +On the other hand, we assume the effective magnetic field +of the impurity to oscillate as +BI(τ) = B0 + δB sin(6πτ +T ). +(D8) +The direction of the spins is prescribed by +φ(1)(τ) = φ(2)(τ) = 2π sin(8πτ +T ), +(D9) +θ(j)(τ) = π +2 (1 + (−1)j τ +T ). +(D10) +The rest of the parameters are chosen as +U0 = 0, +δU = 10, +e +cB0 = 3, +e +cδB = 1.5. +(D11) +Now let us estimate the charge localization length ξg +for the second bulk spectral gap, where the chemical po- +tential µ is assumed to be placed. According to Ref. [10], +the Fermi velocity may roughly be estimated as vF ≈ +kF +m ≈ +2π +mL ≈ 0.825816. The energy gap at the Fermi level +was numerically computed to be roughly Eg ≈ 0.271394, +leading to the following estimate ξg ≈ 3 = O(L). + +12 +∗ Email: kiryl.piasotski@kit.edu +† On the leave from Institut f¨ur Theorie der Statistischen +Physik, RWTH Aachen, 52056 Aachen, Germany +1 K. von Klitzing, G. Dorda, and M. Pepper, “New method +for high-accuracy determination of the fine-structure con- +stant based on quantized Hall resistance”, Phys. Rev. Lett. +45, 494 (1980). +2 D. J. Thouless, M. Kohmoto, M. P. Nightingale, and +M. den Nijs, “Quantized Hall conductance in a two- +dimensional periodic potential”, Phys. Rev. Lett. 49, 405 +(1982). +3 Y. Hatsugai, “Chern number and edge states in the integer +quantum Hall effect”, Phys. Rev. Lett. 71, 3697 (1993); Y. +Hatsugai, “Edge states in the integer quantum Hall effect +and the Riemann surface of the Bloch function”, Phys. +Rev. B 48, 11851, (1993). +4 M. Z. Hasan and C. L. Kane, “Colloquium: topological +insulators”, Rev. Mod. Phys. 82, 3045 (2010). +5 X.-L. Qi and S.-C. Zhang, “Topological insulators and su- +perconductors”, Rev. Mod. Phys. 83, 1057 (2011). +6 A. Kitaev, “Periodic table for topological insulators and +superconductors”, AIP Conf. Proc. 1134, 22 (2009). +7 M. Pletyukhov, D. M. Kennes, J. Klinovaja, D. Loss, +and H. Schoeller, “Surface charge theorem and topologi- +cal constraints for edge states: An analytical study of one- +dimensional nearest-neighbor tight-binding models” Phys. +Rev. B 101, 165304 (2020); M. Pletyukhov, D. M. Kennes, +J. Klinovaja, D. Loss, and H. Schoeller, “Topological in- +variants to characterize universality of boundary charge in +one-dimensional insulators beyond symmetry constraints”, +Phys. Rev. B 101, 161106(R) (2020). +8 N. M¨uller, K. Piasotski, D. M. Kennes, H. Schoeller, and +M. Pletyukhov, “Universal properties of boundary and in- +terface charges in multichannel one-dimensional models +without symmetry constraints”, Phys. Rev. B 104, 125447 +(2021). +9 S. Miles, D. M. Kennes, H. Schoeller, and M. Pletyukhov, +“Universal properties of boundary and interface charges in +continuum models of one-dimensional insulators”, Phys. +Rev. B 104, 155409 (2021). +10 K. Piasotski, N. M¨uller, D. M. Kennes, H. Schoeller, and +M. Pletyukhov, “Universal properties of boundary and in- +terface charges in multichannel one-dimensional continuum +models”, Phys. Rev. B 106, 165405 (2022). +11 W. Kohn and A. Yaniv, “Locality principle in wave me- +chanics”, PNAS 75(11), 5270 (1978). +12 W. +Kohn, +“Density +Functional +and +Density +Matrix +Method Scaling Linearly with the Number of Atoms”, +Phys. Rev. Lett. 76, 3168 (1996). +13 E. Prodan and W. Kohn, “Nearsightedness of electronic +matter”, PNAS 102, 11635 (2005). +14 E. Prodan, “Nearsightedness of electronic matter in one +dimension”, Phys. Rev. B 73, 085108 (2006). +15 H.-R. Trebin, “The topology of non-uniform media in con- +densed matter physics”, Adv. Phys. 31, 195 (1982). +16 M. Pletyukhov, D. M. Kennes, K. Piasotski, J. Klinovaja, +D. Loss, and H. Schoeller, “Rational boundary charge in +one-dimensional systems with interaction and disorder”, +Phys. Rev. Res. 2, 033345 (2020). +17 J. Goldstone and F. Wilczek, “Fractional Quantum Num- +bers on Solitons”, Phys. Rev. Lett. 47, 986 (1981). +18 C. S. Weber, K. Piasotski, M. Pletyukhov, J. Klinovaja, +D. Loss, H. Schoeller, and D. M. Kennes, “Universality +of Boundary Charge Fluctuations”, Phys. Rev. Lett. 126, +016803 (2021). +19 K. Nomura and N. Nagaosa, “Electric Charging of Mag- +netic Textures on the Surface of a Topological Insulator”, +Phys. Rev. B 82, 161401(R) (2010). +20 This includes models featuring a momentum-cubic spin- +orbit interaction, or even tight-binding models, describing +crystals in terms of incomplete basis sets of localized Wan- +nier orbitals. +21 D. K. Park, “Green’s-function approach to two- and three- +dimensional delta-function potentials and application to +the spin-1/2 Aharonov–Bohm problem”, J. Math. Phys. +36, 5453 (1995). +22 D. A. Atkinson, H. W. Crater, “An exact treatment of +the Dirac delta function potential in the Schr¨odinger equa- +tion”, Amer. Jour. Phys. 43, 301 (1975). +23 R. Jackiw, “Delta-function potentials in two- and three- +dimensional quantum mechanics” MAB B´eg memorial vol- +ume (1991). +24 C. N. Friedman, “Perturbations of the Schr¨odinger equa- +tion by potentials with small support”, J. Funct. Anal. 10, +346 (1972). +25 S. Kivelson and J. R. Schrieffer, “Fractional charge, a sharp +quantum observable”, Phys. Rev. B 25, 6447 (1982). +26 In the case of one-dimensional (d = 1) single-channel +(Nc = 1) systems, it is not possible for the bound state +poles to coexist with the continuum (or extended) states. +As a result, the effect of the bound state pole merging with +the band is to increase the order of the branching pole at +the band edge. In multichannel (Nc > 1) and/or multidi- +mensional (d > 1) systems, on the other hand, the poles +may, in principle, coexist with the band continuum. In ei- +ther case, as a result, the band contribution to the winding +number gets modified by unity. +27 Z. Hou, C. S. Weber, D. M. Kennes, D. Loss, H. Schoeller, +J. Klinovaja, M. Pletyukhov, “Realization of a three- +dimensional quantum Hall effect in a Zeeman-induced sec- +ond order topological insulator on a torus”, arXiv preprint +arXiv:2212.09053, (2022) + diff --git a/7NE1T4oBgHgl3EQfnATd/content/tmp_files/load_file.txt b/7NE1T4oBgHgl3EQfnATd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..20104a88a61acf02a12a3679d53250e1ebfc4444 --- /dev/null +++ b/7NE1T4oBgHgl3EQfnATd/content/tmp_files/load_file.txt @@ -0,0 +1,555 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf,len=554 +page_content='Topological charge quantization on localized imperfections in crystalline insulators and the nearsightedness principle of Kohn Kiryl Piasotski1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' ∗ 1Institut f¨ur Theorie der Kondensierten Materie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Karlsruher Institut f¨ur Technologie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 76131 Karlsruhe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Germany 2Institut f¨ur QuantenMaterialien und Technologien,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Karlsruher Institut f¨ur Technologie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 76021 Karlsruhe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Germany† (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 2023) We study the quantization of the excess charge on N localized (ultra-screened) impurities in d- dimensional crystalline insulating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Solving Dyson’s equation, we demonstrate that such charges are topological, by expressing them as winding numbers of appropriate functionals of bulk position space Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' We discuss the ties of our topological invariant with the nearsight- edness principle of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Kohn, stating that the electronic charge density at fixed chemical potential depends on the external field only locally, meaning that localized perturbations by external fields may only result in localized charge redistributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' We arrive at the same conclusion by demonstrat- ing that an adiabatic perturbation comprised of a variation of impurities’ positions and/or strengths may only result in the change in the occupancy of impurity-localized bound states sitting, energy- wise, close to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Finally, we conclude by discussing the relations of the nearsightedness principle with the topological invariants characterizing the boundary charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' INTRODUCTION With the discovery of the quantum Hall effect [1] and its topological origins [2, 3] the study of the topological structures in condensed matter systems became a style rather than a fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The possibly best-known contem- porary example of that is the field of topological insu- lators, where the boundary-localized edge states, with topologically granted existence and robustness, are be- ing studied (popular reviews are [4, 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Despite being a well-defined endeavor with its very own periodic table [6], this discipline leaves a number of questions open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' One of these regards the direct experimental accessibility of these topological surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In particular, aside from the subspace of such states, their basis is incomplete for a description of the physical system accommodating them – a topological insulator, making it highly questionable whether an actual physical observable may be expanded into a basis of intra- and inter-surface state transition operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This is, for example, not true of the excess charge density that these insulators accumulate at their boundaries as it, as such, also features the exponentially localized contributions of all of the occupied extended states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Despite that, it is clear that as the surface states also contribute to such an observable, the change in their occupancy has to have an observable effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In a series of recent works [7, 8, 9, 10], the topologi- cal properties of the boundary-localized electronic excess charges (the boundary charges) in unidimensional crys- tals were examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In particular, a pair of topological invariants characterizing the boundary charge upon two bulk energy spectrum-preserving transformations of crys- tal’s potential, translations and local inversions, were de- vised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Specifically, it was demonstrated that upon local inversion (inversion of coordinates within the unit cell), the boundary charge maps to its negative, up to an inte- gral topological quantum number known as the interface invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Likewise, upon the lattice translation by xϕ, the boundary charge was shown to grow linearly with the shift variable xϕ (with the slope being the unit cell- averaged average charge density in the bulk ¯ρ = ν L, with ν – filling factor and L-system’s period), whilst perform- ing discontinuous downward jumps by a unit of the elec- tron charge, as quantified by another topological quan- tum number – the boundary invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' These topologi- cal invariants were shown to be generated by the spectral flow of the energies corresponding to the edge states in- side the energy gap that hosts the chemical potential, in complete analogy with the integer quantum Hall effect [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As opposed to the edge states in topological insu- lators, the quantization of these invariants does not rely on the internal symmetries of the bulk Bloch’s Hamilto- nian (such as particle-hole or time-reversal symmetries) and is instead guaranteed by a number of fundamental physical principles, such as charge conservation, Pauli principle, and the nearsightedness principle of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Kohn [11, 12, 13, 14] (to be discussed further on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Moreover, these invariants are directly linked with the properties of an experimental observable, a privilege shared by both the quantum Hall effect and the topological defects (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [15] for a review), while not being entirely clear in the domain of the topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Further, in a different paper [16], rational quantization of boundary and interface charges was discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Partic- ularly, with the aid of the aforementioned physical princi- ples, a general framework for studying quantized charges in one dimension was laid down, allowing us to quantify all possible quantization patterns of the boundary charge in terms of the non-symmorphic symmetries of the crys- tal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The charges on the interfaces between pairs of in- sulators sharing their bulk properties were demonstrated to follow a lattice version of the Goldstone-Wilczek for- mula [17], relating the interface charge to the sum of the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='03305v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='mes-hall] 9 Jan 2023 2 boundary charges right and left to their septum, mod- ulo an unknown integer generated by the local coupling between the two subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The key feature of the method developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [16] is this “modulo an unknown integer” paradigm, arising from the nearsightedness principle of the electronic mat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As such, the nearsightedness principle tells us that (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [13, 14]), in insulators, localized perturbations by external fields may result in localized charge redistri- butions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' To be more specific, the corrections beyond the characteristic length scale ξg = vF Eg (where vF and Eg are the Fermi velocity and the gap opening up at the Fermi level, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [18] for example) are exponentially suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' An even further refinement of this state- ment would be that such perturbations may only remove or add an additional number of bound states whose wave functions are localized around the corresponding pertur- bations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' One of the key purposes of the present paper is to substantiate this claim mathematically, which turns out to be possible in pretty general d-dimensional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' To be more specific, this paper concerns the topolog- ical properties of the electronic excess charges accumu- lated around point-like defects in d-dimensional insula- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Although we purposefully specify the Hamiltonian of the crystal under consideration to make our exposi- tion more transparent, the derivations presented in this manuscript are shown to be independent of its choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' What indeed matters is that the spectrum of the clean system consists of the energy bands occasionally sepa- rated by the energy gaps, that is, there exists at least one bulk energy gap into which we can put the chemical potential to promote the resulting statistical system into an insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Furthermore, neither we specify the internal structure of the impurity vertices, nor do we assume any particu- lar arrangement of them, making our analysis applicable to a wide range of experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In particular, quite conventionally, we may assume that a number of randomly located point-like impurities exerting an ultra- screened electrostatic force on the system’s electrons are scattered through the charge sampling region of a crystal under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' A slightly less familiar situation is inspired by the work of Nomura and Nagaosa [19] and may be formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Assuming that a crystal is further magnetic, we know that, in an insulating regime, its ground state may accurately be described by a Heisen- berg model that, by itself, features topological defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' A familiar example of such a defect would be a magnetic skyrmion or a hedgehog texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Assuming that the total spin of atoms comprising our crystal is large, these tex- tures may be seen as an arrangement of classical magnetic moments nailed down to the atomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Their in- teraction with the electron’s spin degree of freedom may then be written as a sum of the Zeeman-like terms, each weighted with the Dirac δ-function centered at the posi- tion of the corresponding atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Quite generically, we show that the total electronic excess charge accumulated around these defects is an integer-valued topological invariant, which we express as a contour integral winding number of an appropriate functional of bulk position space Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Fur- ther analysis of this topological quantum number reveals that upon an adiabatic modification of positions and/or vertex functions of the localized scattering centers, the value of the invariant may only be affected by the change in the occupancy of the imperfection-localized bound states in the process of the spectral flow of their eigenen- ergies inside the chemical potential-accommodating en- ergy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This observation allows for an immediate in- terpretation in terms of the nearsightedness principle, as well as for a direct read-off of the central memo of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [16]: “localized perturbations in insulators result in localized charge redistribution, leading to an addi- tion/removal of the corresponding perturbation-localized bound states to/from the occupied spectral region”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' We conclude our analysis by commenting on the relation be- tween the nearsightedness principle and the topological invariants characterizing the boundary charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In what follows, we set the reduced Plank’s constant ̵h and the electron charge e equal to unity ̵h = e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' ADIABATIC RESPONSE OF THE EXCESS CHARGE TO LOCALIZED PERTURBATIONS IN AN INSULATING STATE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' A translationally invariant model In the following, we shall specifically refer to an elec- tronic system governed by the following Hamiltonian H(0) x = p2 2m + 1 2m d ∑ j=1 { ˜Aj(x),pj} + V (x), (1) with V (x) and ˜Aj(x), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=', d being the lattice periodic Nc × Nc Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' More specifically, {V (x) ˜A(x)} = {V (x + Rm) ˜A(x + Rm)}, ∀m ∈ Zd, (2) where Rm = ∑d j=1 mjaj is a lattice vector characterized by a d-dimensional vector of integers m = (m1 ⋯ md) T , specifying its components in the basis of primitive vec- tors {aj}j spanning the unit cell of a Bravais lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Furthermore, p and x are vectorial momentum and po- sition operators comprised of the individual components pj = −i ∂ ∂xj and xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This model naturally generalizes the one recently stud- ied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [10] in connection with the universal prop- erties of one-dimensional boundary charge, to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' We remark that other models of multi- dimensional periodic structures [20] are expected to share the same physics, as the effects we are about to describe are rather generic to an insulating state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Translationally invariant systems are characterized by their band structure, comprised of the individual energy 3 bands dispersing as ϵα,k, α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=', as a function of the vectorial quasimomentum variable k, confined to the first Brillouin zone of the reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The eigenstates of the Hamiltonian to which ϵα,k are the corresponding eigenvalues are known as Bloch functions ψα,k(x), and may be generically expressed as ψα,k(x) = eik⋅xuα,k(x), (3) where uα,k(x) in the Nc-component object and is lattice periodic in the same sense as vector and scalar potentials are uα,k(x) = uα,k(x+Rm), ∀m ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The completeness and identity resolution relations may be written as VUC (2π)d ∫Rd d(d)xψ† α,k(x)ψα′,k′(x) = δα,α′δ(d)(k − k′), (4) VUC (2π)d ∞ ∑ α=1∫BZ d(d)kψα,k(x)ψ† α,k(x′) = 1Ncδ(d)(x − x′), (5) where VUC is the volume of the unit cell, defined via VUC = ∫UC d(d)x = det(a1∣ ⋯ ∣ad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (6) When studying charge, it is more convenient to intro- duce the retarded single-particle Green’s function, con- taining the information on both the eigenstates and the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In thermodynamic equilibrium, the Laplace image of the latter is defined as the resolvent of the single-particle Hamiltonian (1) [z − H(0) x ]G(0)(x,x′) = 1Ncδ(d)(x − x′), (7) where z is the complex energy variable, defined in terms of the physical frequency variable ω as z = ω + iη, where η → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Owing to the identity resolution relation (5) we can establish the conventional Lehmann representation G(0)(x,x′) = VUC (2π)d ∞ ∑ α=1∫BZ d(d)k ψα,k(x)ψ† α,k(x′) z − ϵα,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (8) Further, using the completeness of the basis (4), in Ap- pendix A, we establish the following important fusion rule for the bare propagators ∫Rd d(d)x′G(0)(x,x′)G(0)(x′,x′′) = − ∂ ∂ω G(0)(x,x′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (9) As it is shown in Appendix A, this relation holds pretty generally, without any reference to the Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Localized perturbations and Dyson’s equation Now we perturb the translationally invariant (on the scale of the unit cell) system by a finite number of point- like impurities ˜V (x) = N ∑ n=1 ˜V (n) 0 δ(d)(x − xn), (10) where ˜V (n) 0 are Nc × Nc matrices describing the action of the nth impurity on the channel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This action is further assumed to be local as prescribed by Dirac delta- function δ(d)(x − xn) centered at the impurity position xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Let us remark that the problem of a Dirac delta- function potential is well-known to be ill-defined in spa- tial dimensions higher than d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In our analysis, this is manifested in the ill-definiteness of the bulk po- sition space Green’s function at equal spatial arguments G(0)(x,x) due to the divergence of the defining integrals (8) in the ultraviolet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Such a divergence is not physi- cal and has to be circumvented by an appropriate reg- ularization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In particular, in the metallic case ˜A(x) = 0, V (x) = 0, in d > 1 the problem of the delta- potential has been extensively studied in both physical [21, 22, 23] and mathematical [24] literature and several meaningful regularization techniques were proposed and shown to produce physically sensible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Since the presence of the energy gaps is of no importance in the deep ultraviolet regime, the same methods may be ap- plied in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The Dyson’s equation for the full Green’s function of the system is given by G(x,x′) =G(0)(x,x′) + N ∑ n=1 G(0)(x,xn) ˜V (n) 0 G(xn,x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (11) First we want to consistently solve for the functions G(xn,x′), n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This problem is brought to the solution of the following matrix equation M(z)D(x′) = D(0)(x′), (12) where M(z) is the Nc ⋅ N × Nc ⋅ N block matrix defined by M(z) =1Nc⋅N − G(0)(z)˜V0, (13) (G(0)(z))n,n′ =G(0)(xn,xn′), (˜V0)n,n′ = δn,n′ ˜V (n) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (14) Likewise, D(x′) and D(0)(x′) are the Nc ⋅ N × Nc matri- ces comprised of the full G(xn,x′) and bare G(0)(xn,x′) propagators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' With these notations we ob- tain G(x,x′) =G(0)(x,x′) + D(0)†(x)˜V0D(x′) =G(0)(x,x′) + D(0)†(x)˜V0M−1(z)D(0)(x′), (15) where in our definition the Hermitian conjugate does not affect the z-variable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (G(0)(x,x′))† = G(0)(x′,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (16) 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Measuring the excess charge We define the excess charge density operator in the following manner: δ̂ρ(x) = ̂ρ(x) − ¯ρ, (17) where ̂ρ(x) = ̂ψ†(x)̂ψ(x), (18) is the density operator, expressed in terms of the Nc- component fermionic field operators ̂ψ(x) and ̂ψ†(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The field operators ̂ψ(x) and ̂ψ†(x) are further assumed to destroy/create excitations of the full Hamiltonian in- cluding the effect of localized scattering centers in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The constant contribution ¯ρ describes the unit cell- averaged average charge density in the bulk: ¯ρ = 1 VUC ∫VUC d(d)xρ(0)(x), (19) ρ(0)(x) =⟨̂ψ(0)†(x)̂ψ(0)(x)⟩ = − 1 π Im∫ µ −∞ dωtr{G(0)(x,x)}, (20) where the field operators ̂ψ(0)(x) and ̂ψ(0)†(x) describe the excitations of the translationally invariant system, µ denotes the chemical potential, and G(0)(x, x′) is the bare Green’s function defined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (7) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' We measure the excess charge with the help of the clas- sical device, described by the envelope function f(x) (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [7, 8, 9, 10] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [25] for similar definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' To be more specific, we define the excess charge operator as δ ̂Q = ∫Rd d(d)xf(x)δ̂ρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (21) It is sensible to define the function f(x) relative to a certain point xp, to which the charge probe is applied, and further assume that the charge is sampled equiva- lently in all directions f(x) = f(∣x − xp∣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Additionally, we assume that all of the charge f(∣x − xp∣) ≈ 1 is sam- pled in sufficiently large vicinity of the sampling point xp, while the envelope function smoothly decays to zero f(∣x − xp∣) → 0 far away from xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' For that matter, it is convenient to choose f(∣x − xp∣) = 1 − Θlp(∣x − xp∣ − Lp), (22) where Θlp(∣x − xp∣ − Lp) is some representation of the Heaviside function broadened by lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The length scales characteristic of the charge probe are assumed to satisfy Lp ≫ lp ≫ ξg, (23) where ξg ≃ vF Eg is the charge localization length in an insu- lator (also it is the charge correlation length, defining the exponential decay length of the density-density correla- tion function, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [18]), roughly defined as the ratio between the Fermi velocity vF and size of the energy gap at the Fermi level Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Topological invariant characterizing the excess charge Let us assume that N impurities, as characterized by the potential (10), are placed in a region of a crystal falling into the sampling district of the envelope function ∣x∣ ≲ Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' We define the total excess charge as the zero temperature expectation value of the excess charge op- erator in the grandcanonical equilibrium density matrix, so that δQ =⟨δ ̂Q⟩ = ∫Rd d(d)xf(x)(ρ(x) − ¯ρ), (24) ρ(x) = − 1 π Im∫ µ −∞ dωtr{G(x,x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (25) With the help of the representation (15), we obtain δQ = Q′ + QP , (26) where Q′ contains the Friedel charge as well as the charge due to the impurity-localized bound states Q′ = ∫Rd d(d)xf(x)ρ′(x), (27) ρ′(x) = − 1 π Im∫ µ −∞ dωtr{D(0)†(x)˜V0M−1(z)D(0)(x)}, (28) while QP is the so-called polarization charge given by QP = ∫Rd d(d)xf(x)(ρ(0)(x) − ¯ρ), (29) and, with the help of the properties of the envelope func- tion, is shown to be zero QP = 0 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' It hence follows that δQ =Q′ = − 1 π Im∫Rd d(d)xf(x) × ∫ µ −∞ dωtr{D(0)†(x)˜V0M−1(z)D(0)(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (30) Due to the branch cuts and poles of the T-matrix T(x,x′) = ∑n,n′[˜V0M−1(z)]n,n′δ(x − xn)δ(x′ − xn′), the integrand of the outer integral is exponentially sup- pressed ∼ e−∣x∣/ξg at large x, allowing us to set f(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Interchanging the order of the integrals, we consider ∫Rd d(d)xtr{D(0)†(x)˜V0M−1(z)D(0)(x)} = − N ∑ n,n′=1 tr{[M−1(z)]n,n′ ∂ ∂ω G(0)(xn′,xn) ˜V (n) 0 } = N ∑ n,n′=1 tr{[M−1(z)]n,n′ ∂ ∂ω [M(z)]n′,n} = N ∑ n=1 tr{[M−1(z) ∂ ∂ω M(z)] n,n } = ∂ ∂ω tr{log M(z)} = ∂ ∂ω log det{M(z)}, (31) 5 where, in the last line, trace and determinant of the full Nc ⋅N ×Nc ⋅N block matrix M(z) are understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Using the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (31), we arrive at the following compact formula for the total excess charge δQ = − 1 π Im∫ µ −∞ dω ∂ ∂ω log det{M(z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (32) To see why the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (32) may take on in- tegral values only, in Appendix C we find an alternative contour integral representation δQ = − ∮C dz 2πi ∂ ∂z log det{M(z)}, (33) where C is an arbitrary non-self-intersecting curve that crosses the real axis at two points only, below the low- est eigenvalue of the full Hamiltonian and at the chemi- cal potential µ, and the direction of C is assumed to be clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In the representation (33), the excess charge δQ is nec- essarily an integer as it is expressed as a contour integral winding number and the chemical potential is by def- inition inside one of the energy gaps (we focus on the insulating systems solely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In other words, the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (33) measures the degree of the mapping S1 → S1 and is thus a member of the only non-trivial homotopy group of the unit circle π1(S1) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In particular, the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (33), is a sum of two distinct contributions: the contribution of the branch cuts corresponding to the extended or scatter- ing states, and the contribution of poles corresponding to the imperfection-localized bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The bands in multidimensional (d > 1) and/or mul- tichannel (Nc > 1) systems are typically composite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' overlapping with one another along the frequency axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' For that matter, it is convenient to choose the branch cuts to connect the bottom of the lowest sub-band with the top of the highest one, within every patch of the en- ergy bands surrounded by a pair of energy gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The bound state poles, determined as a solution of det{M(z)}∣z∈R = 0, are located on the complement of the bare Hamiltonian’s spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' inside the energy gaps and, in some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' an attractive scalar impu- rity), below the bottom of the lowest energy band of the unperturbed Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' RELATION WITH THE NEARSIGHTEDNESS PRINCIPLE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Discussion Now we would like to discuss the topological invari- ant (33) in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In what follows, we specify the contour C as a rectangle of length µ − B in the real direction and width 2η in the imaginary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Here B is by definition an energy lying below the lowest eigen- value of the total Hamiltonian Hx = H(0) x + ˜V (x) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' c) b) a) Im{z} Re{z} C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' A schematic illustration of how the spectral flow of the energies of the imperfection-circumscribing bound states sit- ting inside the gap that accommodates the Fermi level affects the total excess charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The spectrum of the system is visu- alized through the local spectral density as looked down on the complex frequency plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The occupied part of the spec- trum is demonstrated in blue, while the yellow color marks its complement (the states of the system that are unoccupied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Panel a) shows a rectangular contour C encircling the occu- pied spectral region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Panels b) and c) show the zoomed-in vicinity of the chemical potential before and after the per- turbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As is demonstrated in panel c), the spectral flow results in the removal of a single bound state, carrying away a unity of the electron charge from the system (an inverse process is of course also possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' B ∈ (−∞,min{spec{Hx}})), and η is not necessarily an infinitesimal positive but is rather a finite positive num- ber (which is allowed as the integral is invariant under such contour deformations (see Appendix C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Further- more, we assume that the chemical potential is located above the νth bulk energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Let us now consider making an adiabatic perturba- tion to the system that is comprised of the change in the positions {xn}n and/or vertex functions { ˜V (n) 0 }n of the impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As the span of the extended states’ energy bands is unaffected by such adiabatic perturbations, the branch cut contribution to the winding number remains invariant (up to the cases when the bound state merges with the band, as discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This remark is essen- tially true as such deformations of the parameter space do not change the analytical structure of G(0)(xn, xn′), through the functionals of which alone our topological invariant is expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' We hence conclude that such changes may only unleash themselves in the spectral flow of the bound state energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As was anticipated in Section II D, the bound state energies are energy-wise located inside the energy gaps of the bulk system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This assertion also regards the energy gap below the bottom of the lowest band ω ∈ (−∞, mink ϵ1,k], which can accommodate the bound states in the case of attractive impurities, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 6 The energies of the bound states ϵbs inside the energy gaps [maxk ϵα,k, mink ϵα+1,k] surrounded by a pair of bands α, α + 1, (α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=', ν − 1), are solely character- ized by their location within the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The same holds true for the infinite gap below the bottom of the lowest bulk energy band, with ϵbs now being energy-wise located in (−∞, mink ϵ1,k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This implies that the spectral flow of these energies is constituted in the motion of ϵbs in be- tween the top of ϵα,k and the bottom of ϵα+1,k, or between the negative infinity and mink ϵ1,k shall some states be also found in there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' When merging with one of the energy bands (either ϵα,k or ϵα+1,k, and ϵ1,k solely when consid- ering the gap preceding the entire band structure), the value of the contour integral winding number (33) relat- ing to that band gets modified by unity26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' It follows that the motion of the bound state poles, inside such energy gaps below the one hosting the chemical potential, has absolutely no effect on the topological invariant (33) (one may see this result as a form of charge conservation), as B, by definition, resides below the lowest pole (effectively meaning that none of the states are allowed to escape the occupied spectral region from below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The flow of the energies of the impurity-localized bound states residing inside the gap separating the con- duction and the valence bands apart (the gap where the chemical potential is located), on the other hand, affects the winding number in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' When a bound state crosses the chemical potential from above or below, the number of poles encompassed by the integration contour increases or decreases correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' That means that the unit of the electron charge gets either pumped in or out of the system, modifying the topological invariant by ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This discussion is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The elaboration above allows us to draw the following physical conclusion: Localized adiabatic perturbations in insulators, may only result in the localized charge redistributions, owing to the change in the occupancy of the perturbation-localized bound states at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This intuitive result is nothing but a direct consequence of the universal nearsightedness principle of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Kohn [12, 13, 14] stating that, at fixed chemical potential, the electronic charge density depends on the external field (in our case being an assembly of localized scattering centers) only at nearby points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Another conclusion drawn by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Prodan and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Kohn in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [13] (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [14] for the fine details in d = 1) is that the adiabatic perturbations to the exter- nal potential, no matter how strong, have a negligible effect on the local charge density beyond a certain char- acteristic length scale, which, in the insulating regime, is naturally provided by the charge correlation length ξg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' From the viewpoint of our topological invariant (33), this means that in the case of well-separated impurities ∣xn −xn′∣/ξg ≫ 1, the topological invariant is expected to approach a sum of the individual single-impurity invari- ants, as distant impurities are not supposed to be able to “talk” with one another on such scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Indeed, in an insulating state, it is well-known, that the two-point correlation functions G(0)(xn, xn′) decay exponentially at large distances ∼ e−∣Rmn−Rmn′ ∣/ξg (where mn labels the unit cell accommodating the nth scattering center), meaning that we can approximate (G(0)(z))n,n′ ≃δn,n′G(0)(xn,xn), (34) implying that M(z) ≃ N ⊕ n=1 (1Nc − G(0)(xn,xn)V (n) 0 ), (35) and δQ ≃ − N ∑ n=1∮C dz 2πi ∂ ∂z log det{1Nc − G(0)(xn,xn)V (n) 0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (36) This result may be seen as a form of the conven- tional Born approximation of the linear transport theory, whereby, to the lowest order in the impurity density, one considers impurities as independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' An illustration: A pair of magnetic impurities in an illuminated quantum wire To illustrate some of the points highlighted in the above discussion, we here consider a simple model of a spin-orbit-interacting ballistic quantum wire, submersed into the background of the spatially oscillating electro- magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The bulk Hamiltonian assumes the form of the Pauli Hamiltonian with an extra Rashba-like term: H(0) x = (p + e cAx(x)) 2 2m + kR ⋅ σ m (p + e cAx(x)) + µBge 2 σ ⋅ B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (37) Above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' kR = (kR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' kR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' kR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='z) is the Rashba spin-orbit vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' σ = (σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' σy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' σz) is the vector of the Pauli spin matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' µB = e 2mc is the Bohr magneton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' c is the speed of light in vacuum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' ge is the electron’s Land´e g-factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' and B(x) =∇ × A(x)∣ x=xˆex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Ax(x) = ˆex ⋅ A(x)∣ x=xˆex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (38) with ˆex being the ort in the x-direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' and A(x) be- ing the electromagnetic vector potential of the monochro- matic plane-wave form A(x) = A0 cos(q ⋅ x + ϕ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (39) in the Coulomb gauge ∇ ⋅ A(x) = 0 ⇐⇒ q ⋅ A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (40) 7 ° º L ° º 2L 0 º 2L º L k °2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='00 °1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='75 °1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='50 °1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='25 °1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='00 °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='75 °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='50 ≤k x y z kR m(1) eff m(2) eff R q B E λ a) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Panel a): A schematic illustration of a ballistic quantum wire featuring Rashba-style spin-orbit coupling (defined by a spin-orbit vector kR) and submersed into a spatially-periodic arrangement of electric E and magnetic B fields of wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The two impurity atoms, separated by distance R and carrying an effective magnetic moment of m(j) eff , j = 1, 2, are schematically shown by atomic symbols pierced with the magnetic moment-symbolizing arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Panel b): The bulk energy spectrum of the two-impurity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The energy bands are shown in dark blue, the chemical potential located inside the second spectral gap (above the fourth energy band) is depicted in orange, and the relevant spectral region is highlighted in light blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The wave vector of the background electromagnetic field defines the fictitious lattice spacing L = 2π ˆex ⋅ q, (41) where we have excluded the uninteresting case of the or- thogonally propagating wave ˆex ⋅ q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' We note that the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (37) falls into the class of systems defined by the Hamiltonian (1), with d = 1 and V (x) =µBge 2 σ ⋅ B(x) + e2A2 x(x) 2mc2 + ekR ⋅ σAx(x) mc , (42) ˜Ax(x) =e cAx(x) + kR ⋅ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (43) As this demonstration is assumed to be interpretative, it suffices to consider the case of a pair of impurities, which we assume to be separated by distance R: ˜V (x) = ˜V (1) 0 δ(x) + ˜V (2) 0 δ(x − R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (44) Note that we can place the first impurity at x = 0 with- out loss of generality, as its other positions inside the wire may be achieved by appropriate tuning of the modula- tion’s phase ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' we assume the impurities to exert both the electrostatic and the exchange “force” on the wire’s electrons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' which we encode in the following form of the impurities’ vertex functions ˜V (j) 0 = Ujσ0 + µBge 2 σ ⋅ B(j) eff ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (45) where B(j) eff is the effective (also appropriately screened to have a short-ranged effect only) magnetic field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' produced by the effective magnetic moment of the impurity atom m(j) eff = qjgj 2MjcS(j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' with qj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' and Mj being the charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' g-factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' and mass of the jth impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Furthermore, Uj denotes the strength of the electrostatic potential, defin- ing the corresponding force exerted by the impurity on the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Not going into much of the microscopic details, in the following, we treat Uj and B(j) eff as some constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The resulting setup is schemati- cally illustrated in panel a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Further, to illustrate our point, we assume that the associated impurity parameters {R, {Uj}j, {B(j) eff }j} evolve with a fictitious “adiabatic time” τ ∈ [0,T], in such a manner that their temporal derivatives remain much smaller than the Fermi energy ϵF times their value, for all τ ∈ [0,T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The particular form of the pumping protocol used to produce the numerical data and the concrete numerical values of the free model parameters are provided in Ap- pendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The resulting bulk energy spectrum is demon- strated in panel b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The numerical data for the excess charge- characterizing topological invariant, as well as the spectral flow of bound state energies inside the chem- ical potential-accommodating spectral gap, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In particular, using the parametrization R(τ) = (nR − 1)L + ¯R(τ), suggested in the Appendix D, we present the data for five different values of nR ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=',5}, as shown in five different columns of the corresponding figure, with upper and lower rows corresponding to the spectral flow and the topolog- ical invariant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Solid black and dashed burgundy lines mark the cases of independent and “interacting” impurities, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' By in- dependent impurities, we here understand that the separation between them is effectively infinite, so that the off-diagonal blocks of the M(z) matrix (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (14) for the definition) may be completely ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This means that the bound state spectrum of the independent impurities is provided by the solution of det(12 − G(0)(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='0) ˜V (1) 0 )det(12 − G(0)(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='R) ˜V (2) 0 )∣z∈R = 8 0 T/4 T/2 3T/4 T τ max ϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='4 min ϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='5 nR = 1 0 T/4 T/2 3T/4 T τ nR = 2 0 T/4 T/2 3T/4 T τ nR = 3 0 T/4 T/2 3T/4 T τ nR = 4 0 T/4 T/2 3T/4 T τ nR = 5 0 T/4 T/2 3T/4 T τ −2 −1 0 1 2 δQ 0 T/4 T/2 3T/4 T τ 0 T/4 T/2 3T/4 T τ 0 T/4 T/2 3T/4 T τ 0 T/4 T/2 3T/4 T τ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The figure demonstrates the adiabatic flows of both the bound state energy spectrum and the excess charge invariant in the toy model proposed in Section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Specifically, the spectral flow of the impurity-localized bound state energies is shown in the upper row, while the second row is dedicated to the invariant itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As is explained in Appendix D, the position of the second impurity is parametrized as R(τ) = (nR − 1)L + ¯R(τ), where nR denotes the number of the unit cell hosting the second imperfection, and ¯R(τ) ∈ [0, L] describes its location within the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The five distinct columns in the above figure correspond to five choices of nR = 1, 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In all of the panels, red dashed lines correspond to the actual solution, while black solid lines relate to the case of two independent impurities (see the approximate formula (36)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As the separation between the impurities becomes of the order of the charge localization length ξg = O(L) (see Appendix D), both adiabatic flows approach the limit of two independent impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 0, while the topological invariant is given by the ap- proximation (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' By the “interacting” impurities, on the other hand, we understand that the exact relations were used to produce the numerical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The numerical technique for evaluation of bulk position space Green’s functions, as well as the topological indices of the form (33), is outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In our calculations, the values of the contour parameters were chosen as η = 1, B = −30 (such a choice of B is motivated by the presence of the bound states below the lowest band ω ∈ (−∞, mink ϵk,1] in our model (37)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The central purpose of our demonstration is to show that upon the increase in the impurity’s separation be- yond the charge localization length ξg = O(L) (see Ap- pendix D), both the topological invariant and the bound state spectrum approach that of a pair of independent impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This effect is a direct consequence of the nearsightedness principle, telling us that a localized cause leads to a localized effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Furthermore, as one may an- ticipate, the discontinuous jumps of the excess charge in- variant occur precisely at the points where bound states enter/leave the occupied part of the energy spectrum, as is explained in Section III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Another interesting obser- vation is the non-zero value of the topological invariant at the beginning of the adiabatic evolution in τ, where the strengths of the electrostatic repulsion are the small- est 0 < Uj ≪ 1 (see Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This feature is a conse- quence of the presence of impurity-localized bound states below the bottom of the lowest energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Such an effect is well-known in the case of attractive scalar impu- rities, whereas here, it is generated by the non-Abelian structure of the model, and, to the best of our knowledge, was not reported previously in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Topological invariants characterizing the boundary charge in unidimensional crystals In this section, we would like to comment on the topo- logical invariants characterizing boundary charges in uni- dimensional crystals, extensively discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In particular, let us consider a d = 1 semi- infinite system described by the Hamiltonian (1), with the boundary placed at x = xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' An appropriate restric- tion of x defines the respective right and left subsystems: x ∈ [xb,∞), right sub-system, (46) x ∈ (−∞,xb], left sub-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (47) In our definition, the primitive unit cell is defined as the one starting at the boundary of the right semi-infinite system UC = [xb, xb+L], with L being the lattice period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In this definition, the left half-system is always obtained from the right one by a local inversion operation, which acts by the inversion of local coordinates within each unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Now we define the boundary charge operators corre- sponding to right and left semi-infinite systems as the envelope-weighted integrals of the expectation values of the appropriate excess charge density operators: Q(R) B =∫ ∞ xb dxf(x)⟨δ̂ρR(x)⟩, (48) Q(L) B =∫ xb −∞ dxf(x)⟨δ̂ρL(x)⟩, (49) where, in analogy with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (17), δ̂ρS(x) = ̂ρS(x) − ¯ρ, and ̂ρS(x) is the density operator referring to the system S = R, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Furthermore, the envelope function f(x) is chosen in accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (22), with xp = xb, and 9 the range of x being restricted according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (46) and (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Let us now consider measuring the total excess charge δQ accumulated around x = xb in a translationally in- variant system x ∈ (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' By the polarization charge neutrality condition QP = 0, demonstrated in Appendix B, the total excess charge also vanishes δQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' On the other hand, we may consider a translationally invari- ant system as a sum of right and left semi-infinite sys- tems with a coupling corresponding to the bulk Hamilto- nian switched in between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This coupling manifests itself as a local perturbation and, by the nearsighted- ness principle of Kohn, is capable of affecting the total charge locally by at most introducing or removing a num- ber of additional bound states, resulting in an integer contribution QI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In this connection, we conclude that δQ = Q(R) B + Q(L) B − QI = 0, where QI is known as the in- terface invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' One of the central results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [10], was to demonstrate that QI =Q(R) B + Q(L) B = −∮C dz 2πi ∂ ∂z log det{G(0)(xb,xb)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (50) That is, the interface invariant, characterizing the bound- ary charge upon local inversions, is a topological quan- tum number given by the winding of the determinant of bulk position space Green’s function evaluated at the location of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Now let us proceed with the transformations of the boundary charge under translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' First, we consider the right boundary charge of the so-called reference sys- tem, starting at xb = 0: Q(R) B (0) = ∫ ∞ 0 dxf(x)(ρ(x) − ¯ρ), (51) and we would like to analyze the changes in this quantity upon the translation of the boundary by xϕ ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In- stead of shifting the boundary, we consider adding the fol- lowing potential ˆV (x) = ˆV0Θ(x)Θ(xϕ − x), ˆV0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' By the Pauli principle, the charge density becomes zero for x ∈ [0,xϕ] as these states sit at infinite energy above the chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' From the definition of the bound- ary charge, we are left with the following contribution: δQ(R) B (xϕ) = ∫ xϕ 0 dxf(x)(0 − ¯ρ) mod 1 = −¯ρxϕ mod 1, (52) where mod 1 contribution again comes from the near- sightedness principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' This analysis allows us to conclude that: Q(R) B (xϕ) − Q(R) B (0) = ¯ρxϕ + I(xϕ), (53) where I(xϕ) is known as the boundary invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' An- other important result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [10] was to show that I(xϕ) = −∮C dz 2πi ∂ ∂z lndetU(xϕ), (54) where U(xϕ) is defined via the path-ordered exponential U(xϕ) =Pexp{∫ x 0 dx′L(x′)}, (55) L(x) =[G(0)(x,x)]−1G(0) 2 (x,x+) − iA(x), (56) and G(0) 2 (x,x′) = ∂x′G(0)(x,x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In other words, the boundary invariant is also a topological quantum number expressed as a winding of the appropriate functional of bulk position space Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In this way, we see that the quantization of the topo- logical invariants characterizing the boundary charge in one-dimensional insulators is a direct consequence of the nearsightedness principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As this intuitive physical prin- ciple holds beyond the single spatial dimension, one ex- pects the excess charges accumulated on inhomogeneities of various spatial co-dimensions in d-dimensional crystals to possess similar topological characterization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Indeed, linear scaling of the boundary charge, along with its discontinuous jumps by a unit of the electron charge at the bound state escape/entrance spectral points, was recently demonstrated in a two-dimensional system [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' CONCLUSIONS AND OUTLOOK In this paper, the quantization of the excess charges on localized scattering centers in d-dimensional insulators was discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Our analysis reveals that an assembly of such imperfections accumulates an integral excess charge, given by a winding number expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' We find that an adiabatic perturbation (no matter how strong) comprised of either relocation of the impurities or a modification of their vertex functions (or both at the same time) results in the change of the total charge by an integer, deter- mined by the saldo of the imperfection-localized bound states that entered or escaped the occupied spectral re- gion, inside the chemical potential-hosting bulk spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The quantization of this topological invariant was shown to be a direct consequence of the nearsightedness principle of the electronic matter, limiting the range of the effect of a localized cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Additionally, this local behavior of the electronic matter in the insulating state was shown to be responsible for the quantization of the topological invariants characterizing the unidimensional boundary charge studied in [7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Furthermore, our study confirms the central paradigm of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [16], namely that localized perturbations in insulators specifi- cally lead to the change in occupancy of the correspond- ing perturbation-localized bound states, modifying the total charge, defined as the macroscopic average on the scales significantly exceeding both the unit cell size L and the charge correlation length ξg, by at most an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As is now obvious, the present paper is of conceptual value only as the evaluation of the suggested topologi- cal invariant (33) for a specific multi-impurity (N ≫ 1) system poses a challenge on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In particular, this concerns questions regarding the regularization schemes 10 for the higher-dimensional equal-argument Green’s func- tions, as well as the basic questions regarding the numer- ical feasibility of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Furthermore, it would be of future interest to study the expansion of the topolog- ical invariant in the interaction between the individual impurities, as generated by the off-diagonal blocks of the M(z) matrix, and analyze its ties with the conventional Born series for the impurity-dressed T-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As it is suggested in the present study, in the insulating state, the impurity density ρI has to be always contrasted with the inverse charge localization length ξg, in such a man- ner that the condition 1 ≫ ρIξd g implies the validity of the Born approximation, treating impurities as independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' ACKNOWLEDGMENTS The author gratefully acknowledges the durable ex- change of ideas with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Pletyukhov and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Schoeller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Further, the author generously thanks S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Miles and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Pletyukhov for their valuable comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Most of the present work was done at the Institut f¨ur Theorie der Statistischen Physik of RWTH Aachen and was financially supported by the Deutsche Forschungsge- meinschaft via RTG 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Appendix A: Contraction of two Green’s functions Quite generically we may represent G = ⨋s ∣s⟩⟨s∣ z − ϵs , (A1) where the meta-index s labels the eigenstates ∣s⟩ and eigenenergies ϵs of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Considering the product of the Green’s function with itself GG = ⨋s ⨋s′ ∣s⟩⟨s′∣ (z − ϵs)(z − ϵs′) ⟨s∣s′⟩ � δ(s,s′) = ⨋s ∣s⟩⟨s∣ (z − ϵs)2 = − ∂ ∂ω ⨋s ∣s⟩⟨s∣ z − ϵs = − ∂ ∂ω G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (A2) Taking the position space matrix elements ⟨x∣GG∣x′′⟩ = − ∂ ∂ω G(x,x′′), and inserting 1 = ∫Rd d(d)x∣x⟩⟨x∣, (A3) we obtain the desired identity ∫Rd d(d)x′G(x,x′)G(x′,x′′) = − ∂ ∂ω G(x,x′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (A4) Appendix B: Polarization charge We consider ∫Rd d(d)xf(x)(ρ(0)(x) − ¯ρ) = ∑ m ∫UC d(d)¯xf(¯x + Rm)(ρ(0)(¯x) − ¯ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (B1) Above we parametrized the position space variable x as x = Rm+¯x, for some vector of integers m, and ¯x is the lo- cal coordinate within the unit cell ¯x ∈ UC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Furthermore, we used the periodicity property of ρ(0)(x), implied by the periodicity of the equal-argument Green’s function G(0)(x, x) = G(0)(x + Rm, x + Rm), ∀m ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (B2) The envelope function varies significantly only in the crossover region ∣Rm∣ = O(Lp), allowing us to approxi- mate f(¯x + Rm) ≈ f(Rm) + ¯x ⋅ ∇f(Rm), (B3) leading to ∫Rd d(d)xf(x)(ρ(0)(x) − ¯ρ) = ∫UC d(d)¯x∑ m (¯x ⋅ ∇f(Rm))(ρ(0)(¯x) − ¯ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (B4) Now approximating ∑ m (¯x ⋅ ∇f(Rm)) ≈ 1 VUC ∫Rd d(d)y(¯x ⋅ ∇yf(y)) = 1 VUC ∫Rd d(d)y∇y ⋅ (¯xf(y)) = 0, (B5) where in the last step we used Gauss’ divergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Appendix C: Contour integral representation First, we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (32) as δQ = − 1 2πi ∫ µ −∞ dω ∂ ∂ω log det{M(z)} + 1 2πi ∫ µ −∞ dω ∂ ∂ω (log det{M(z)})∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (C1) Now we remind ourselves that (log f(z))∗ = log (f(z))∗ ≡ log f ∗(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (C2) Furthermore, one has (det{M(z)})∗ = det{M†(z∗)}, (C3) where, as before, the Hermitian conjugate does not affect the z-variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Now we have det{M†(z∗)} = det{(1 − G(0)(z∗)˜V(0) )†} = det{1 − ˜V(0)G(0)(z∗)} = det{M(z∗)}, (C4) 11 where to get from the pre-last to the last lines we em- ployed the Weinstein–Aronszajn identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' It hence follows that δQ = − 1 2πi ∫ µ −∞ dω ∂ ∂ω log det{M(ω + iη)} − 1 2πi ∫ −∞ µ dω ∂ ∂ω log det{M(ω − iη)} = − ∮C dz 2πi ∂ ∂z log det{M(z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (C5) Above, C is the counterclockwise rectangular contour de- fined as a union of four segments: C =[B + iη,µ + iη) ∪ [µ + iη,µ − iη) ∪ [µ − iη,B − iη) ∪ [B − iη,B + iη), B → −∞, η → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (C6) We note that the integral in (C5) remains unaffected under continuous contour deformations, so long as the analytic structure of the integrand within the patch of the complex plane enclosed by contour C remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In this connection, we may replace C with an arbitrary non-self-intersecting curve crossing the real axis at two points only, at any energy below the lowest eigenvalue of the full Hamiltonian, and at the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Appendix D: Parameters and protocols In the numerical example provided in Section III B, the parameters of the model were chosen according to q = 2π(ex + κey) λ √ 1 + κ2 , A0 = A0(κex − ey) √ 1 + κ2 , m = 1, (D1) κ = 1 + √ 5 2 , λ = 4, e cA0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='17, kR = ⎛ ⎜ ⎝ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='24 ⎞ ⎟ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (D2) Note that as we have set the electron’s mass m = 1 to unity (in addition to the electric charge e = 1 and reduced Plank’s constant ̵h = 1), we work in Hartree’s atomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In this way, the electromagnetic wave is propagating in the x − y plane, with the corresponding magnetic field being B(x) = 2πA0 λ ez sin(q ⋅ x + ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (D3) By definition, the corresponding lattice period is given by L = λ √ 1 + κ2 = 2 √ 2(5 + √ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (D4) To produce the data, we used the following pumping protocol for the impurities’ separation R(τ) = (nR − 1)L + ¯R(τ), ¯R(τ) = L T τ, L T ≪ vF , (D5) where vF is the Fermi velocity and nR is an integer spec- ifying the number of the unit cell hosting the second impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' For the impurities’ vertex functions, we fur- ther make an assumption of the equivalent impurities: U (1)(τ) = U (2)(τ) =∶ U(τ) and ∣B(1) eff (τ)∣ = ∣B(2) eff (τ)∣ =∶ BI(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' The direction of the magnetic moments, on the other hand, is allowed to be different in two scattering centers and is parametrized in spherical polar coordinates B(j) eff (τ) BI(τ) = ⎛ ⎜ ⎝ cosφ(j)(τ)sinθ(j)(τ) sinφ(j)(τ)sinθ(j)(τ) cosθ(j)(τ) ⎞ ⎟ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (D6) In the following, we assume that, as is the case with the location of the second impurity within the unit cell num- ber nR, the impurity strength also grows linearly with τ U(τ) = U0 + δU τ T , δU T ≪ ϵ2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (D7) On the other hand, we assume the effective magnetic field of the impurity to oscillate as BI(τ) = B0 + δB sin(6πτ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (D8) The direction of the spins is prescribed by φ(1)(τ) = φ(2)(τ) = 2π sin(8πτ T ), (D9) θ(j)(τ) = π 2 (1 + (−1)j τ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (D10) The rest of the parameters are chosen as U0 = 0, δU = 10, e cB0 = 3, e cδB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' (D11) Now let us estimate the charge localization length ξg for the second bulk spectral gap, where the chemical po- tential µ is assumed to be placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' According to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' [10], the Fermi velocity may roughly be estimated as vF ≈ kF m ≈ 2π mL ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='825816.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Pletyukhov, “Universal properties of boundary and in- terface charges in multichannel one-dimensional models without symmetry constraints”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' B 104, 125447 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 9 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Miles, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Kennes, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Schoeller, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Pletyukhov, “Universal properties of boundary and interface charges in continuum models of one-dimensional insulators”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' B 104, 155409 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 10 K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Jackiw, “Delta-function potentials in two- and three- dimensional quantum mechanics” MAB B´eg memorial vol- ume (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 24 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Friedman, “Perturbations of the Schr¨odinger equa- tion by potentials with small support”, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 10, 346 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 25 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Kivelson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Schrieffer, “Fractional charge, a sharp quantum observable”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' B 25, 6447 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 26 In the case of one-dimensional (d = 1) single-channel (Nc = 1) systems, it is not possible for the bound state poles to coexist with the continuum (or extended) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' As a result, the effect of the bound state pole merging with the band is to increase the order of the branching pole at the band edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In multichannel (Nc > 1) and/or multidi- mensional (d > 1) systems, on the other hand, the poles may, in principle, coexist with the band continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' In ei- ther case, as a result, the band contribution to the winding number gets modified by unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' 27 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Hou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Weber, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Kennes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Loss, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Schoeller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Klinovaja, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content=' Pletyukhov, “Realization of a three- dimensional quantum Hall effect in a Zeeman-induced sec- ond order topological insulator on a torus”, arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} +page_content='09053, (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfnATd/content/2301.03305v1.pdf'} diff --git a/7dAyT4oBgHgl3EQf2vnp/content/tmp_files/2301.00758v1.pdf.txt b/7dAyT4oBgHgl3EQf2vnp/content/tmp_files/2301.00758v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1a4edb28bfc35e6adddaaa89622b2911b18e540 --- /dev/null +++ b/7dAyT4oBgHgl3EQf2vnp/content/tmp_files/2301.00758v1.pdf.txt @@ -0,0 +1,1713 @@ +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Analysis of a HAPS-Aided GNSS in Urban Areas +using a RAIM Algorithm + +Hongzhao Zheng, Member, IEEE, Mohamed Atia, Senior Member, IEEE, and Halim Yanikomeroglu, Fellow, IEEE +Abstract—The global averaged civilian positioning accuracy is +still at meter level for all existing Global Navigation Satellite +Systems (GNSSs), and the performance is even worse in urban +areas. At lower altitudes than satellites, high altitude platform +stations (HAPS) offer several benefits, such as lower latency, less +pathloss, and likely smaller overall estimation error for the +parameters associated in the pseudorange equation. HAPS can +support GNSSs in many ways, and in this paper we treat the HAPS +as another type of ranging source. In so doing, we examine the +positioning performance of a HAPS-aided GPS system in an urban +area using both a simulation and physical experiment. The HAPS +measurements are unavailable today; therefore, they are modeled +in a rather simple but logical manner in both the simulation and +physical experiment. We show that the HAPS can improve the +horizontal dilution of precision (HDOP), the vertical dilution of +precision (VDOP), and the 3D positioning accuracy of GPS in both +suburban and dense urban areas. We also demonstrate the +applicability of a RAIM algorithm for the HAPS-aided GPS +system, especially in the dense urban area. + +Index Terms—High altitude platform station (HAPS), horizontal +dilution of precision (HDOP), pseudorange, receiver autonomous +integrity monitoring (RAIM), vertical dilution of precision (VDOP). +I. INTRODUCTION +ODAY, many countries and the European union have +their own global navigation satellite systems (GNSSs). +However, 95 percent of the time, the global averaged +horizontal positioning accuracy of existing GNSSs is still at the +meter level, and it is even worse for the vertical positioning +accuracy [1]-[4] due to the nature of the satellite geometry. +Although vertical positioning performance is less important +than horizontal positioning performance today, it might be very +important in the future, for instance, for unmanned aerial +vehicles (UAVs) flying in the 3D aerial highways [5]. Thanks +to ongoing research on localization and navigation fields, there +are a number of techniques developed which can bring the +positioning accuracy of systems involving satellites to the +centimeter level. For example, Li et al. have shown that +centimeter-level positioning accuracy can be achieved using the +multi-constellation GNSS consisting of Beidou, Galileo, +GLONASS and GPS with precise point positioning (PPP) [6]. +Because most civilian applications use single-frequency, low- +cost receivers for localization and navigation, many advanced +positioning algorithms, including PPP that delivers centimeter + +This paper was supported in part by Huawei Canada. +H. Zheng, M. Atia, and H. Yanikomeroglu are with the Department of +Systems and Computer Engineering, Carleton University, Ottawa, ON K1S +5B6, +Canada +(e-mail: +hongzhaozheng@cmail.carleton.ca; +Mohamed.Atia@carleton.ca; halim@sce.carleton.ca). +level positioning accuracy, cannot be implemented. Therefore, +the single point positioning (SPP) is the most commonly used +algorithm in civilian applications. But this is poised to change. +As increasing numbers of low-Earth-orbit (LEO) satellites are +launched into space, researchers are investigating the feasibility +of utilizing LEO satellites to aid the positioning service. For +instance, Li et al. have shown that a centimeter level Signal-In- +Space Ranging Error (SISRE) in the real-time PPP application +can be achieved using a LEO enhanced GNSS [7]. In the event +that GNSS signals are unavailable in urban areas, researchers +are also interested in building navigation systems that +exclusively rely on LEO satellite signals. For example, a +position root mean squared error (RMSE) of 14.8 m for a UAV +has been proven feasible using only two Orbcomm LEO +satellites with the carrier phase differential algorithm [8]. +Compared to medium-Earth-orbit (MEO) satellites, which are +typically used in GNSSs, LEO satellites offer several +advantages, such as lower latency and less pathloss due to +shorter distance to ground users. LEO satellites also offer +greater availability due to the large number of them. +To further enhance high bandwidth networking coverage in +areas with obstacles, such as urban areas, another option is the +use of high altitude platform stations (HAPS1), which refer to +aerial platforms positioned in the stratosphere with a typical +altitude of about 20 km. HAPS can be utilized for many +technologies coming in 5G even 6G and beyond such as +computation offloading [9], edge computing [10], and aerial +base station [11]. As urban areas are where GNSS positioning +performance degrades severely, while also being where most +people live, we could improve the positioning performance of +GNSS by placing several HAPS above metro cities and +equipping them with satellite-grade atomic clocks so that HAPS +can be deployed as another type of ranging source. Even though +atomic clocks on satellites are highly accurate, they are not +perfect due to the time dilation postulates made in both +Einstein’s special theory of relativity and the general theory of +relativity. According to Einstein’s special theory of relativity, +an atomic clock on a fast-moving satellite runs slower than a +clock on Earth by around 7 microseconds per day. On the other +hand, according to the general theory of relativity, an atomic +clock which experiences weaker gravity on a distant satellite +runs faster than a clock which experiences greater gravity on +1 In this paper, the acronym "HAPS" is used to denote “high altitude +platforms station” in both singular and plural forms, in line with the convention +adopted in the ITU (International Telecommunications Union) documents. +T + +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +Earth by about 45 microseconds per day [12]. As HAPS operate +at an altitude of around 20 km and can be quasi-stationary, the +time dilation is negligible from the perspective of special +relativity and greatly reduced from the perspective of general +relativity. Therefore, the atomic clocks on HAPS will likely be +more accurate than that on satellites, which can make the +estimation error of the HAPS clock offset smaller than that of +the satellite clock offset. Since HAPS are positioned much +closer to the Earth than satellites, the pathloss of a HAPS is +expected to be much less, which will likely make the received +signal power of a HAPS stronger than that of a satellite, thereby +reducing the estimation error of the parameters associated in the +pseudorange measurement of the HAPS signal. The movement +of a HAPS can be confined to a cylindrical region with a radius +of 400 m and a height of about 700 m [13], which can reduce +the number of handovers during the course of navigation and +increase the utilization efficiency during its operation life. As +HAPS are positioned in the stratosphere, which is below the +ionosphere, their signals will likely be free of the ionospheric +effect, which is known to be one of the major sources of error +in pseudorange measurements. +Therefore, the overall +estimation error for a HAPS will likely be smaller than that of +a satellite. Similar to the pseudorange measurement for a +satellite, which incorporates the satellite position error, we +should also consider the position error in the pseudorange +measurement for HAPS. Fortunately, researchers have been +investigating the positioning of HAPS and have demonstrated +that HAPS positioning errors are comparable to or lesser than +satellite orbit errors. For example, Dovis et al. prove that 0.5 m +positioning accuracy (circular error probable [CEP] 68 percent) +for a HAPS is achievable using the modified RTK method [14]. +There are a handful of papers in the literature that have +investigated the HAPS-aided GNSS [15]-[18]; however, to the +best of our knowledge, this paper is the first to provide a +comprehensive study of the positioning performance of a +HAPS-aided GNSS in urban areas. +There are plenty of operational GPS satellites that could fail +due to the degraded signal quality for reasons such as +obstruction, multipath, intentional or unintentional attacks, +thereby impacting the positioning performance of the GNSS. In +this case, a signal selection algorithm like the receiver +autonomous integrity monitoring (RAIM) algorithm, which can +detect and exclude poor quality signals, can be helpful in +improving the positioning performance. For example, about 35 +percent decrease in RMS positioning error of the GPS-only case +and 50 percent decrease in RMS positioning error of the +GPS/GLONASS case in a severe urban scenario have been +achieved on smartphone GNSS chips by using a RAIM +algorithm [19]. Moreover, Yang and Xu propose a robust +estimation-based RAIM algorithm that can detect and exclude +multiple faulty satellites effectively with efficiency higher than +the conventional least squares (LS)-based RAIM algorithm +[20]. In this paper, we make three postulations: 1) a HAPS +signal is free of the ionospheric effect; 2) the estimation error +of the HAPS clock offset is smaller than that of the satellite +clock offset; and 3) the received signal power of a HAPS is +higher than that of a satellite, all of which contribute toward the + +Fig. 1. System model of the HAPS-aided GPS. + + +assumption that the overall estimation error of the parameters +associated in the pseudorange equation for the HAPS is smaller +than that for the satellite. Under this assumption, we use the SPP +algorithm developed in our prior work [21] to show that HAPS +can indeed improve the positioning performance of legacy +GNSSs in urban areas through both a simulation and a physical +experiment. We also demonstrate the applicability of the RAIM +algorithm to a HAPS-aided GPS system, especially in dense +urban areas. Since the HAPS measurements are unavailable so +far, they are simulated in a rather simple but logical way in both +the simulation and physical experiment. The contributions of +this paper are listed below. +• +First, using a commercial GNSS simulator, we +simulate the GPS pseudorange signals and generate +the positions of HAPS. By using the default system +parameters as well as a proper manipulation of the +number of visible satellites, we show that the +positioning performance of the GPS-only system in +both the suburban and dense urban areas are close to +the real scenario. Moreover, we show the positioning +performance of different systems where different +numbers of HAPS are used with or without the GPS +system. The issue of the ranging source geometry is +revealed from the simulation results. +• +Next, we apply the SPP algorithm to the real GPS data +collected using two commercial GNSS receivers as +well as the HAPS data generated using the commercial +GNSS software. In so doing, we show the advantage +of the HAPS-aided GPS system in the sense of the +horizontal dilution of precision (HDOP) and the +vertical dilution of precision (VDOP). +• +Finally, we implement a RAIM algorithm and +demonstrate its effectiveness in improving the 3D +positioning performance of the HAPS-aided GPS +system, especially in dense urban areas. +The rest of the paper is organized as follows: in Section Ⅱ, the +system model, the SPP algorithm, and the RAIM algorithm are +described. In Section Ⅲ, the simulation setup of the HAPS- + +Ionosphere +HAPS +HAPS +20km +HAPSfootprint +HAPSfootprint +15° +cell. +cell> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +aided GPS system and the simulation results are presented. In +Section Ⅳ, the physical experiment setup and results, including +both the DOP analysis and the 3D positioning accuracy +analysis, are provided. Finally, Section V offers some +conclusions and a discussion of future research directions. For +simplicity, the GNSS signal only involves the GPS C/A L1 +signal. +II. SYSTEM MODEL +The system model of the HAPS-aided GPS system is +depicted in Fig. 1. There are four satellites shown in Fig. 1, this +is just a reminder that at least four satellites are required to +perform precise 3D localization using GNSS. The typical +choice for the elevation mask is 10 degrees. However, we use +15 degrees as the elevation mask for the satellites and HAPS +due to the following reasons: 1) the atmospheric error owing to +the signal refraction can be neglected if the elevation of a +satellite is greater than 15 degrees [22], which is likely true for +a HAPS as well; 2) As there is a higher chance of ensuring the +required number of ranging source with HAPS, we can improve +the positioning performance further by only using those satellite +signals with better quality. The pseudorange equation for +satellite is given by + + +𝑝𝑆𝐴𝑇 = 𝜌𝑆𝐴𝑇 + 𝑑𝑆𝐴𝑇 + 𝑐(𝑑𝑡 − 𝑑𝑇𝑆𝐴𝑇) + 𝑑𝑖𝑜𝑛,𝑆𝐴𝑇 + 𝑑𝑡𝑟𝑜𝑝,𝑆𝐴𝑇 ++ 𝜖𝑚𝑝,𝑆𝐴𝑇 + 𝜖𝑝 + + +(1) +where 𝑝𝑆𝐴𝑇 denotes the satellite pseudorange measurement, +𝜌𝑆𝐴𝑇 is the geometric range between the satellite and receiver, +𝑑𝑆𝐴𝑇 represents the satellite orbit error, 𝑐 is the speed of light, +𝑑𝑡 is the receiver clock offset from GPS time, 𝑑𝑇𝑆𝐴𝑇 is the +satellite clock offset from GPS time, 𝑑𝑖𝑜𝑛,𝑆𝐴𝑇 denotes the +ionospheric delay for satellite signals, 𝑑𝑡𝑟𝑜𝑝,𝑆𝐴𝑇 denotes the +tropospheric delay for satellite signals, 𝜖𝑚𝑝,𝑆𝐴𝑇 is the delay +caused by the multipath for satellite signals, and 𝜖𝑝 is the delay +caused by the receiver noise. The pseudorange equation for +HAPS can be expressed as follows: + + +𝑝𝐻𝐴𝑃𝑆 = 𝜌𝐻𝐴𝑃𝑆 + 𝑑𝐻𝐴𝑃𝑆 + 𝑐(𝑑𝑡 − 𝑑𝑇𝐻𝐴𝑃𝑆) + 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆 ++ 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 + 𝜖𝑝 + +(2) +where 𝑝𝐻𝐴𝑃𝑆 denotes the HAPS pseudorange measurement, +𝜌𝐻𝐴𝑃𝑆 represents the geometric range between the HAPS and +the receiver, 𝑑𝐻𝐴𝑃𝑆 represents the HAPS position error, 𝑑𝑇𝐻𝐴𝑃𝑆 +is the HAPS clock offset from GPS time, 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆 denotes the +tropospheric delay for HAPS signals, 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 is the delay +caused by the multipath for HAPS signals. The simulated +vehicle trajectory originates at Carleton University, which is in +a suburban area, and ends at Rideau Street, which is in a dense +urban part of Ottawa (see Fig. 2). There are six simulated HAPS +shown as transmitters on Fig. 3. As we can see, one HAPS is +positioned over downtown Ottawa; the other five HAPS are +positioned nearby, over populated areas and conservation areas. +HAPS is quasi-stationary, meaning that it will still be moving +in a variety of manners. In this work, all the HAPS are + +Fig. 2. Vehicle trajectory. + + +Fig. 3. Locations of the simulated HAPS. + + +simulated to be following a circular trajectory with a radius of +300 m. The elevation and azimuth angles of all the HAPS at the +beginning of the simulation are listed in Table I. The positions +of HAPS were chosen to provide a rich diversity in azimuth +angles. With one HAPS at the zenith and the others having +relatively low elevation angles, this constitutes a near Zenith + +Horizon (ZH) geometry, which can deliver a reasonably good +DOP [23]. To make sure the entire urban area is well covered, +HAPS are placed not too far away from the urban area. To better +understand the concept of DOP, the visual illustrations of the +HDOP and VDOP of the simulated HAPS constellation are +provided in Fig. 4 and Fig. 5, respectively. Due to various errors +impacting the pseudorange measurement, the estimated +distance between a HAPS and a receiver can be smaller or +larger than the geometric range. Objects with a higher elevation +angle will likely result in more uncertainty for the vertical + +OSM + relief shading +Tracks: +Ottawa,On to Rideau Centre, +Ottawa,ON +Lyon +Denseurban +Areas +yview +Fost +onburg +Suburban +Areas +Ottawa +Cene:45.40751.75.0087m +Google +Map created at CSVisualizer.com +一净用多款Transmitter5 +deroure +Buckinohan +Transmitter6 +arcde +otineau +Transmitter3 +174 +Transmitter1 +Simulator +417 +Embrun +Russell +Metcalfe +Transmitter2 +Transmitter4> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 4. HDOP of the simulated HAPS constellation (top view). + + +Fig. 5. VDOP of the simulated HAPS constellation (front view). + + +component and less uncertainty for the horizontal component +from the point of view of geometry, and vice versa. The shaded +area is where the receiver is estimated to be. + +A. The Single Point Positioning (SPP) Algorithm +The single point positioning algorithm is implemented on the +basis of the SPP package developed by Napat Tongkasem [24] +with proper modifications [21] so that HAPS can be +incorporated in the SPP algorithm. Fig. 6 shows the flowchart +of the single point positioning algorithm. We should point out +that the implemented single point positioning algorithm is not +the best positioning algorithm, and that the objective of this +work is to show the significance of HAPS in aiding the +positioning performance of a legacy GNSS. The implemented +SPP algorithm can be improved in many ways. For example, if +the knowledge of the measurement error variance is available, +we can apply the weighted least squares (WLS) algorithm to +enhance the positioning performance of the SPP algorithm by +lowering the weights of those observations with higher +variances [25]. If the knowledge of the measurement error +variance is unavailable, the computational complexity of the +SPP algorithm can be reduced by imposing the Cholesky +decomposition for the matrix inversion in (9) [26]. We can also +TABLE I +ELEVATION AND AZIMUTH OF THE HAPS AT THE START OF THE SIMULATION + +HAPS index +Elevation angle +Azimuth angle +HAPS #1 +81.087° +-14.210° +HAPS #2 +24.054° +-128.878° +HAPS #3 +27.952° +68.022° +HAPS #4 +32.450° +171.477° +HAPS #5 +36.554° +2.204° +HAPS #6 +33.805° +-57.884° + + +use the carrier phase measurement to enhance the positioning +performance of the HAPS-aided GPS system, since carrier +performance of the HAPS-aided GPS system, since carrier +phase measurements come with much higher precision, which +usually delivers a more accurate position solution. Since the +HAPS clock offset in this work is not explicitly simulated, we +simply use 𝑑𝑇 to denote the satellite clock offset. From the data +collected by the GNSS receiver, we shall obtain both the +receiver independent exchange (RINEX) format observation +file and the RINEX navigation file, from which we can obtain +satellite information, such as the satellite pseudorange 𝒑𝑺𝑨𝑻, the +ionospheric parameters 𝜶 , the Keplerian parameters, the +pseudo-random noise (𝑷𝑹𝑵) code, which represents the unique +number of each satellite, the day of year ( 𝐷𝑂𝑌 ) which +represents the day of year at the time of measurement. We write +𝑷𝑹𝑵 in bold to represent a vector containing the pseudo- +random noise code of all visible satellites at the current epoch. +We are able to compute the satellite positions, 𝑷𝑺𝑨𝑻 , and +satellite clock offset, 𝒅𝑻 , using the Keplerian parameters +contained in the navigation file. 𝑷𝑯𝑨𝑷𝑺 denotes a vector +containing the positions of all HAPS, which are generated using +the Skydel GNSS simulator [27], and 𝒑𝑯𝑨𝑷𝑺 denotes a vector +containing the HAPS pseudorange, which will be explained in +Section III. To compute the position solution 𝒙 , we first +initialize the receiver position to the center of the Earth; then +we initialize the receiver clock offset to zero and the change in +estimates 𝒅𝒙 to infinity. For each epoch of measurement, we +first check if the number of available ranging sources is more +than three, as at least four ranging sources are required to +perform precise 3D localization. Since the receiver position is +iteratively estimated, we calculate the elevation angles for both +satellites and HAPS with respect to the recently estimated +receiver position. Since both the tropospheric delay and the +ionospheric delay are functions of the receiver position, these +two atmospheric delays are estimated iteratively as well. The +elevation angle, satellite pseudorange, HAPS pseudorange, +satellite position, satellite clock offset, tropospheric delay +𝒅𝒕𝒓𝒐𝒑 , ionospheric delay 𝒅𝒊𝒐𝒏 , and pseudo-random noise +(𝑷𝑹𝑵) code are modified iteratively on the basis of the re- +computed elevation angles for both satellites and HAPS. To +prepare the parameters needed for the least square method, the +pseudorange needs to be corrected as follows: + + +𝒑𝑺𝑨𝑻 +𝒄 += 𝒑𝑺𝑨𝑻 + 𝑐 ∙ 𝒅𝑻 − 𝒅𝒕𝒓𝒐𝒑,𝑺𝑨𝑻 − 𝒅𝒊𝒐𝒏,𝑺𝑨𝑻 + (3) + +where 𝒑𝑺𝑨𝑻 +𝒄 + represents the corrected pseudorange for the + +Transmitter 5 +Transmitter 6 +Transmitter 3 +Transmitter 2 +Transmitter 4Transmitter 5 +Transmitter 6 +Transmitter 1 +Transmitter 3 +Transmitter 2 +Transmitter 4> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 6. Flow chart of the single point positioning algorithm. + + +satellites, and 𝒑𝑺𝑨𝑻 represents the uncorrected pseudorange for +the satellites. In this work, the HAPS pseudorange is modeled +as the sum of the geometric range and the pseudorange +error,which represents the overall estimation error of the +parameters in the HAPS pseudorange equation. Accordingly, +the HAPS pseudorange does not need to be corrected. Due to +the Earth’s rotation, the positions of satellites and HAPS at the +signal emission time are different from their positions at the +signal reception time; this is known as the Sagnac effect [28]. +The coordinates of satellites/HAPS can be transformed from the +signal emission time to the signal reception time by [28] + + +∆𝑡𝑅𝑂𝑇 = 𝑡𝑟𝑥 − 𝑡𝑡𝑥 +(4) + + +𝑃𝑖,𝑟𝑥 = 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)𝑃𝑖,𝑡𝑥 +(5) + +where ∆𝑡𝑅𝑂𝑇 denotes the signal propagation time, 𝑡𝑟𝑥 +represents the signal reception time, 𝑡𝑡𝑥 represents the signal +emission time, 𝑃𝑖,𝑟𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates at the +signal reception time, 𝑃𝑖,𝑡𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates +at the signal emission time, 𝜔𝐸 denotes the Earth’s rotation rate, +and 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) is known as the rotation matrix, which +is described as follows: + + +𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) += [ +cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) +sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) +0 +− sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) +0 +cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) +0 +0 +1 +] . + + +(6) +The line-of-sight vector 𝒗, and the true range between ranging +sources and receiver 𝝆, are then calculated to compute the a +priori range residual vector 𝒃 and the design matrix 𝑯, where + + +𝒃 = 𝒑𝒄 − 𝝆 +(7) + + +𝑯 = [𝒗, 𝟏𝑙𝑒𝑛𝑔𝑡ℎ(𝑷𝐜)×1] +(8) + +where 𝒑𝒄 is the corrected satellite pseudorange combined with +the corrected HAPS pseudorange, 𝟏𝑙𝑒𝑛𝑔𝑡ℎ(𝑷𝐜)×1 denotes a +column vector of length being the total number of visible +ranging sources, and 𝑷𝐜 is a vector containing the corrected +positions of the visible ranging sources (i.e., satellite + HAPS). +Finally, the least square solution is computed as follows: + + +𝑸 = (𝑯′𝑯)−1 +(9) + + +𝒅𝒙 = 𝑸𝑯′𝒃 +(10) + + +𝑑𝑡 = 𝒅𝒙(4)/𝑐 +(11) + +where Q is known as the covariance matrix, and 𝒅𝒙(4) denotes +the fourth element in the vector 𝒅𝒙. To prevent the algorithm +from getting numerical issues, we should ensure the term being +inversed in (9) is non-singular; in other words, the design matrix +𝑯 should be non-singular. With the extra observations by +utilizing HAPS as additional ranging sources, the chance of 𝑯 +being singular is likely reduced; the non-singular design matrix +can be ensured by avoiding the use of collinear observations, +which means that two or more observations have about the +same azimuth and elevation angle. We observe that the term +being inversed in (9), 𝑯′𝑯, is a Hermitian, positive definite +matrix, therefore the Cholesky decomposition can be imposed +to reduce the computational complexity [26]. The covariance +matrix, 𝑸, is described by + + +𝑸 = +[ + + + + 𝜎𝑥 +2 +𝜎𝑥𝑦 +𝜎𝑥𝑧 +𝜎𝑥𝑡 +𝜎𝑥𝑦 +𝜎𝑦 +2 +𝜎𝑦𝑧 +𝜎𝑦𝑡 +𝜎𝑥𝑧 +𝜎𝑦𝑧 +𝜎𝑧 +2 +𝜎𝑧𝑡 +𝜎𝑥𝑡 +𝜎𝑦𝑡 +𝜎𝑧𝑡 +𝜎𝑡 +2 ] + + + + + +(12) + +Initialization +PsAT-PHAPS,PRN,DOY +x = 04x1 +Input +dt = x(4)/c +PHAPS, PsAT, dT,α +dx = x + Inf +stop = 0 +Exit +No +NsAr + NHAPs ≥ 4 +IYes +No +dx(1:3)|> 0.01 +Yes +Finding parameters +For sotellites +For HAPS +OsAT,dtrop.dion +HAPS +Applying elevation mask +For sotellites +For HAPS +sAT,dtrep,dion,PRN,dT,PsAr +HAPS,PHAPS +Pseudorange correction +PSAT-PHAPS +Combining the +corrected pseudoranges +p' = [psar.phaes] +国 +Correcting for the Sagnaceffect +(i.e., Earth rotation) +PSAT,PHAPS +Combining the corrected +ranging source positions +P° = [PSAT.PHAPs] +Finding parameters +V,p,b,H,Q +andno +Computing the position solution +using the Least Square method +3p'x +x,dt> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +where receiver coordinates x, y, z in the Earth-centered Earth- +fixed (ECEF) coordinate frame and the receiver clock offset, +respectively. The least square solution will be found when the +norm of the change in receiver position 𝒅𝒙(1: 3) is sufficiently +small. In this work, this threshold is set as 0.01 m. We use the +HDOP, the VDOP and the 3D positioning accuracy as the +metrics to show the advantage of the proposed HAPS-aided +GPS system; the 3D positioning accuracy is used to show the +applicability of the RAIM to the HAPS-aided GPS system. To +compute the HDOP, we need to convert the covariance matrix +into the local north-east-down (NED) coordinate frame, which +can be done with the following equations [29]: + + +𝑸𝑵𝑬𝑫 = 𝑹′𝑸̃𝑹 = [ +𝜎𝑛 +2 +𝜎𝑛𝑒 +𝜎𝑛𝑑 +𝜎𝑛𝑒 +𝜎𝑒 +2 +𝜎𝑒𝑑 +𝜎𝑛𝑑 +𝜎𝑒𝑑 +𝜎𝑑 +2 +] +(13) + + +𝑸̃ = [ +𝜎𝑥 +2 +𝜎𝑥𝑦 +𝜎𝑥𝑧 +𝜎𝑥𝑦 +𝜎𝑦 +2 +𝜎𝑦𝑧 +𝜎𝑥𝑧 +𝜎𝑦𝑧 +𝜎𝑧 +2 +] +(14) + + +𝑹 = [ +−sin 𝜆 +cos 𝜆 +0 +− cos 𝜆 sin 𝜑 +− sin 𝜆 sin 𝜑 +cos 𝜑 +cos 𝜆 cos 𝜑 +sin 𝜆 cos 𝜑 +sin 𝜑 +] +(15) + +where 𝜎𝑛, 𝜎𝑒, and 𝜎𝑑 represent the receiver position errors in +the local north, east, and down directions, respectively. 𝜆 and 𝜑 +represent the longitude and latitude of the receiver, +respectively. Then, the HDOP is described by + + +𝐻𝐷𝑂𝑃 = √𝜎𝑛2 + 𝜎𝑒2 +(16) + +and the VDOP is described by + + +𝑉𝐷𝑂𝑃 = √𝜎𝑑 +2. +(17) + +B. The Receiver Autonomous Integrity Monitoring (RAIM) +Algorithm +The RAIM algorithm is a signal selection algorithm that can +detect and even exclude abnormal observations using redundant +measurements. It can detect an abnormal observation when the +number of observations is at least five; it can exclude this +abnormal observation when the number of observations is at +least six. The RAIM algorithm is typically applied to multi- +constellation GNSSs where the number of ranging sources is +more than enough to perform precise 3D localization, and it is +typically applied to cases where there likely exists at least one +observation that differs from the expected value significantly. +Such cases include urban areas, where the pseudorange + + + + + + + + +measurement is highly subject to the multipath effect. With the +assistance from HAPS, the chance of enabling the RAIM +function will likely increase. Typical RAIM algorithms tend to +use the standard deviation of the target observable, which is the +pseudorange measurement in our work. As knowledge of the +standard deviation of the satellite pseudorange is unavailable on +the receivers we use, in this work the RAIM algorithm is +implemented on the basis of [30], which considers a 𝐶/𝑁0- +based variance model and a computationally efficient method, +namely the modified Danish estimation method. 2 The +implemented 𝐶/𝑁0-based RAIM algorithm is given in Alg. 1, +where 𝑁 denotes the number of visible ranging sources. The +input to this algorithm consists of the position fix computed +using the SPP algorithm 𝒙, and the 𝑪/𝑵𝟎 of the ranging source +signal. Since HAPS are located at much lower altitudes than +2 To the best of our knowledge, RAIM is the most common algorithm used +for integrity monitoring. Since we only have the 𝐶/𝑁0 data which can be +utilized for the integrity monitoring, we could not identify in the literature any +other appropriate RAIM-like algorithm for comparison. However, we believe +that the other RAIM algorithms would also be applicable if the knowledge of +the standard deviation of the satellite pseudorange happens to be available. +Algorithm 1 The 𝐶/𝑁0-based RAIM Algorithm +Input: The SPP estimated position solution 𝒙 and 𝑪/𝑵𝟎; +Output: The SPP and RAIM jointly estimated position +solution 𝒙̂. +1: +Initialize the parameters 𝑠𝑡𝑜𝑝 and 𝒅𝒙; +2: +while |𝒅𝒙(1: 3)| > 0.01 do +3: + +Same procedures as the SPP algorithm until +“Finding parameters” after correcting for the +Sagnac effect; +4: + +for 𝑖 = 1 to 𝑁 do +5: + + +if 𝑠𝑡𝑜𝑝 == 1 do +6: + + + +Find the variance of the observation 𝑖, 𝑠𝑖, +according to (19); +7: + + +end if +8: + +end for +9: + +Find the weight matrix 𝑾 and the design matrix +𝑯 , and calculate the covariance matrix 𝑸 +according to (20); +10: + +Calculate the change in estimates 𝒅𝒙 according +to (21), and update the position solution 𝒙; +11: + +Calculate the pseudorange residual 𝒗̂ according +to (22), and the covariance matrix of the residuals +𝑪𝒗̂ according to (24); +12: + +for 𝑖 = 1 to 𝑁 do +13: + + +Find the normalized residual of observation 𝑖 +at the current iteration 𝑘, 𝑤̅𝑖,𝑘 according to +(26); +14: + + +if |𝑤̅𝑖,𝑘| > 𝑛1−(𝛼0/2) do +15: + + + +Update the variance of the observation 𝑖 +for the next iteration 𝑘 + 1 , 𝜎𝑖,𝑘+1 +2 +, +according to (25); +16: + + +end if +17: + +end for +18: +end while + +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +satellites, in practice the 𝐶/𝑁0 value of the HAPS might be +higher than that for any satellite. As it is possible that a handful +of HAPS signals might suffer from severe multipath effects, we +can exclude those HAPS signals whose 𝐶/𝑁0 values are much +lower than the higher ones. In this work, the multipath effect is +not explicitly simulated for the HAPS signal; therefore, we +assume that the 𝐶/𝑁0 of each HAPS is equal to the maximum +𝐶/𝑁0 value of the available satellites at each epoch, meaning +that the signal quality for a HAPS will always be better than +that for any satellites. The variance covariance matrix (VCM) +𝜮 of the observations (pseudoranges) 𝒑 is defined as follows: + + +𝜮 = 𝑑𝑖𝑎𝑔(𝑠1, 𝑠2, … , 𝑠𝑛) +(18) + + +𝑠𝑖 = 10 + 1502 ∗ 10(−𝐶/𝑁0,𝑖)/10 +(19) + +where 𝑠𝑖 denotes the variance of the observation 𝑖. We assume +that the observations are uncorrelated, and that the errors follow +the normal distribution with 𝑁(𝟎, 𝜮). The weight matrix, 𝑾, is +defined as the inverse of the VCM, 𝜮−1 . The least square +equations become + + +𝑸 = (𝑯′𝑾𝑯)−1 +(20) + + +𝒅𝒙 = 𝑸𝑯′𝑾𝑷. +(21) + +The least square residuals of the pseudorange 𝒗̂ can be obtained +as follows: + + +𝒗̂ = 𝑯 ∙ 𝒅𝒙 − 𝑷 +(22) + + +𝑷 = 𝒑𝒄 − 𝝆 +(23) + +where 𝑯 represents the design matrix, 𝒅𝒙 represents the +change in estimates, 𝒑𝒄 denotes the corrected pseudoranges, +and 𝝆 denotes the geometric range between ranging sources +and the receiver. The covariance matrix of the residuals, 𝑪𝒗̂, is +computed as + + +𝑪𝒗̂ = 𝜮 − 𝑯(𝑯𝑇𝜮−1𝑯)−1𝑯𝑇. +(24) + +To detect and exclude the abnormal observations, we follow the +modified Danish estimation method proposed in [30]. + + +𝜎𝑖,𝑘+1 +2 += 𝜎𝑖,0 +2 ∙ {exp ( +|𝑤̅𝑖,𝑘| +𝑇 ) , |𝑤̅𝑖,𝑘| > 𝑛1−(𝛼0/2) +1, |𝑤̅𝑖,𝑘| ≤ 𝑛1−(𝛼0/2) + +(25) + +with + + +𝑤̅𝑖,𝑘 = +𝒗̂𝑖,𝑘 +√(𝑪𝒗̂𝒊,𝟏)𝑖𝑖 + +(26) + +where 𝜎𝑖,0 +2 denotes the a priori variance of the observation 𝑖 +(i.e., s𝑖), 𝑤̅𝑖,𝑘 denotes the normalized residual of observation 𝑖 +at iteration 𝑘,√(𝑪𝒗̂𝒊,𝟏)𝑖𝑖 represents the standard deviation of +observation 𝑖 from the first iteration, 𝑛1−(𝛼0/2) denotes the 𝛼0- +quantile of the standard normal distribution, which is also called +the critical value, 𝛼0 is the predetermined false alarm rate +which is 0.5 % in this work. The modified Danish method is an +iteratively reweighted LS algorithm that implements a robust +estimator. This method detects and excludes abnormal +observations by comparing the absolute value of each +normalized pseudorange residual, |𝑤̅𝑖,𝑘|, with the critical value, +𝑛1−(𝛼0/2), in each iteration. The variances for the observations +whose normalized residuals are greater than the critical value +are multiplied with exponential terms, making the variances of +those observations larger, hence lowering the weight of those +observations. By iteratively multiplying the variances of the +abnormal observations by exponential terms, the weight of the +abnormal observations will likely become much smaller than +that of the normal observations; therefore, the abnormal +observations can be considered as being excluded. +III. SIMULATION OF THE HAPS-AIDED GPS SYSTEM +In this section, we will describe the simulation setup used in +the Skydel GNSS software [27] and present the simulation +results in terms of 3D positioning accuracy for several hybrid +systems and two standalone systems in both a suburban +scenario and a dense urban scenario. + +A. Simulation Setup +The system model is established using the default Earth +orientation parameters of the Skydel GNSS simulation software +[27], which considers all GPS satellites orbiting around the +Earth and transmitting the L1 C/A code. The Saastamoinen +model is chosen to emulate the tropospheric effect, and the +Klobuchar model is chosen to emulate the ionospheric effect +using the default Klobuchar parameters that come with the +software. The output from Skydel contains the ECEF +coordinates of satellites at the signal emission time, the +ionospheric corrections, the tropospheric corrections, the +satellite clock offsets, the ECEF coordinates of the receiver, the +signal emission time, and so forth, at each time stamp from the +start of the simulation. The receiver clock offset in the +simulation is zero by default. The correction terms in the +pseudorange equation of satellite including the satellite orbit +error, the multipath error, and the receiver noise are not +separately considered in the simulation; instead, a pseudorange +error is introduced to reflect the presence of those effects. The +pseudorange error of satellite is featured using the built-in first +order Gauss-Markov process with a default time constant of 10 +s, and the standard deviation of 6 m. The continuous model for +the first order Gauss-Markov process is described by [31]: + + +������̇ = − +1 +𝑇𝑐 ������̇ + 𝑤 +(27) + +where ������̇ represents a random process with zero mean, +correlation time 𝑇𝑐, and noise 𝑤. The autocorrelation of the first + +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +TABLE Ⅱ +DETAILS OF THE SIMULATION SETUP + +Item +Processing strategy +Earth orientation parameter +Software default Earth orientation parameters +Satellite signal +GPS L1 C/A +Tropospheric model +Saastamoinen model +Ionospheric model +Klobuchar model +Sampling rate +12.5 MS/s +Satellite pseudorange error +1st order Gauss-Markov process (time constant = 10 s; standard deviation = 6 m) +HAPS pseudorange error +Gaussian noise (mean = 0 m; std = 2 m for the suburban scenario; std = 5 m for the dense urban +scenario) +Number of GPS satellites (dense urban +scenario) +8-10 +Number of GPS satellites (suburban +scenario) +4 +Total number of HAPS +6 +order Gauss-Markov process is described by [32]: + +𝑅(𝛥𝑡) = 𝜎2𝑒−|𝛥𝑡| +𝜏 (28) + +where 𝛥𝑡 represents the sampling interval, 𝜎 and 𝜏 denote the +standard deviation and the time constant of the first order +Gauss-Markov process, respectively. The characteristics of the +pseudorange errors for satellites are set to be the same in both +the suburban scenario and the dense urban scenario. However, +we randomly select four satellites in the dense urban scenario +in order to emulate the dense urban area in a rather simple way. +We verify that by doing so, the standard deviation of the 3D +positioning accuracy for the GPS-only system in the simulation +is close to that in the physical experiment. The pseudorange +error for the HAPS is modeled using the Gaussian noise with +standard deviations of 2 m and 5 m, representing the suburban +and the dense urban scenario, respectively. Under the +assumption that the overall estimation error of the HAPS is less +than that of the satellite, the standard deviation for the HAPS +pseudorange error is deliberately set to be smaller than that of +the satellite pseudorange error in the suburban and dense urban +scenarios. To investigate the impact of the number of HAPS on +the positioning performance of the HAPS-aided GPS system, +we consider four hybrid systems with different numbers of +randomly selected HAPS at each epoch. We also consider the +HAPS-only system for the completeness of a research problem. +Under this setting, we examine the 3D positioning performance +of different systems in the suburban and dense urban scenarios. +In the suburban scenario, the number of visible satellites varies +between eight and ten, while in the dense urban scenario the +number of visible satellites is set to four. The details of the +simulation setup are given in Table Ⅱ. + +B. Simulation Results +Fig. 7 shows the cumulative distribution function (CDF) of +the 3D positioning accuracy for different positioning systems in +the suburban scenario. With the assumption that the +pseudorange error for a HAPS is smaller than that of a satellite, +we can see from Fig. 7 that all the hybrid systems (HAPS + + +Fig. 7. CDF of the 3D positioning accuracy for different systems (suburban +scenario). + + +Fig. 8. CDF of the 3D positioning accuracy for different systems (dense urban +scenario). + + +GPS) outperform the GPS-only system; the more HAPS, the +better the positioning performance of the HAPS-aided GPS + +Suburbanscenario(stdofHAPSpseudorangeerror=2m) +0.9 +0.8 +0.7 +0.6 +D +0.5 +0.4 +0.3 +0.2 +GPS-only system +1-HAPS with GPS system +2-HAPS with GPS system +3-HAPS with GPS system +0.1 +4-HAPS with GPS system +4-HAPS-only system +0 +5 +10 +15 +20 +25 +30 +3D positional accuracy (m) in local NED frameDenseurbanscenario(stdofHAP +pseudorangeerror=5m) +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +4-GPS-only system +1-HAPS with 4-GPS system +2-HAPS with 4-GPS system +0.1 +3-HAPS with 4-GPS system +4-HAPS with 4-GPS system +4-HAPS-only system +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +3D positionalaccuracy (m)inlocal NEDframe> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 9. Positioning performance of M8T and M8U for horizontal and vertical +planes. + + +system. Nevertheless, we observe that the positioning +performance of the 4-HAPS-only system, where all ranging +sources have a much smaller pseudorange error than the +satellite, is not the best and can occasionally be very poor. This +may be due to the following reasons: 1) the 4-HAPS-only +system has much fewer ranging sources for computing receiver +positions; 2) the HAPS geometry can be poor occasionally since +we are randomly selecting four HAPS at each epoch. There are +several cases where the HAPS geometry is considered poor. For +example, when the four randomly selected HAPS are on the +same side of the receiver. The CDF of the 3D positioning +accuracy for different positioning systems in the dense urban +scenario is shown in Fig. 8, from which we can see a similar +trend that the more HAPS the better the positioning +performance of the HAPS-aided GPS system. In the dense +urban scenario, where only four GPS satellites are selected for +positioning, the 4-HAPS-only system achieves better +positioning performance than the 4-GPS-only system due to the +better signal quality for the HAPS. However, we should +consider using the HAPS-aided GPS system for the best +positioning performance. +IV. FIELD EXPERIMENTS +To verify and support the simulation results, we process the +raw GNSS data collected using two commercial GNSS +receivers. In this section, we present the experiment setup, the +modeling of HAPS signals, and the HAPS pseudorange error. +We also provide an analysis of the DOP and the 3D positioning +accuracy for both the GPS-only system and the HAPS-aided +GPS system. + +A. Experiment Setup +The raw GNSS data is collected along a vehicle trajectory +similar to the one shown in Fig. 2, except for a slight difference +due to a partial road closure on the day of data collection. The +raw GNSS data is collected using both the Ublox EVK-M8T +TABLE Ⅲ +EVK-M8T GNSS UNIT SPECIFICATIONS [35] + +Parameter +Specification +Serial Interfaces +1 USB V2.0 +1 RS232, max.baud rate 921,6 kBd +DB9 +/- 12 V level +14 pin – 3.3 V logic +1 DDC (I2C compatible) max. 400 kHz +1 SPI-clock signal max. 5,5 MHz – SPI DATA +max. 1 Mbit/s +Timing Interfaces +2 Time-pulse outputs +1 Time-mark input +Dimensions +105 × 64 × 26 mm +Power Supply +5 V via USB or external powered via extra power +supply pin 14 (V5_IN) 13 (GND) +Normal Operating +Temperature +−40℃ to +65℃ + + +and the Ublox EVK-M8U GNSS units. The Ublox EVK-M8T +unit is a timing product that can provide users with precise +timing information for post-processing; the Ublox EVK-M8U +unit is a dead reckoning product equipped with inertial +measurement units (IMUs) such that the positioning +performance of this product will not be degraded much even in +the dense urban area. Fig. 9 shows the positioning performance +along both horizontal and vertical planes for both M8T and +M8U during the entire observation period. From Fig. 9, we can +see that M8U outperforms M8T for both horizontal positioning +accuracy and vertical positioning accuracy. As only M8T +provides the timing information required for post-processing, +the receiver positions computed by EVK-M8U are used as the +ground truth data for the analysis of the positioning +performance. The raw GNSS data is processed using the single +point positioning algorithm described in Section Ⅱ. Table Ⅲ +gives the specifications of the EVK-M8T GNSS unit. To +emulate realistic LOS conditions for HAPS in the urban area, +the LOS probability as a function of the HAPS elevation angle +in the urban area is implemented on the basis of [33] and [34]. +It is worth mentioning that the LOS probability model for the +HAPS provided by [33] is proposed based on the city of +Chicago; imposing this LOS probability model for the dense +urban area of Ottawa might be too harsh considering their +incompatible city scales. Since there is no LOS probability +corresponding to the suburban area in [34], the one for rural area +in [34] is used as the LOS probability for the HAPS in the +suburban area. The LOS probability for the HAPS in the rural +area in [34] is verified as being consistent with the LOS +probability for the HAPS in the suburban area in [33]. The +pseudorange of the HAPS in the experiment is modeled as the +addition of the geometric range between the satellite and +receiver, the receiver clock offset multiplied by the speed of +light, and the pseudorange error representing the sum of all +kinds of estimation errors. The pseudorange errors for the +HAPS in the suburban and dense urban areas are simulated as +Gaussian noise with zero mean and standard deviations of 2 m +and 5 m, respectively. Since the vehicle trajectory involves both + +The positioning performance of M8T and M8U +12 +Horizontal positioning accuracy (M8T) +Horizontalpositioningaccuracy(M8U) +Vertical positioningaccuracy(M8T) +10 +Verticalpositioningaccuracy(M8U) +g accuracy (m) +8 +6 +Positioning +2 +0 +100 +200 +300 +400 +500 +600 +700 +epoch (s)> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + + +Fig. 11. HDOP (top) and VDOP (bottom). + + +Fig. 10. HAPS availability during the entire observation period. + + +suburban and dense urban areas, the entire route is divided into +two segments, where the first segment is considered as the +suburban scenario while the second segment is considered as +the dense urban scenario (see Fig. 2). By observing the +positioning performance of the GPS-only system using the real +GPS data, the LOS probability for the suburban area is applied +to the HAPS for epochs less than 380 s, and the LOS probability +for the dense urban area is applied to the HAPS for epochs +greater than or equal to 380 s. Since the GNSS receivers we use +do not provide an accurate receiver clock offset with respect to +the GPS time, the receiver clock offset in each epoch is +estimated by using the ground truth receiver positions provided +from Ublox EVK-M8U and the precise timing information, +such as the receiver clock drift and the receiver clock bias, +provided from Ublox EVK-M8T. We should note that the +pseudorange of the HAPS in the experiment is modeled as a +function of the receiver clock offset, which is estimated with +best effort. Nevertheless, additional errors should be expected +in the pseudorange of the HAPS with the magnitude depending +on the quality of all visible satellite signals and the ground truth +receiver position. As we would expect the quality of the satellite +signals in the suburban area to be better than that in the dense +urban area, we should also expect the receiver clock offset to be +estimated with higher accuracy in the suburban area than in the +dense urban area. + +B. Experiment Results +With more ranging sources, we should expect the availability +of the HAPS-aided GPS system to be higher than the GPS-only +system. The availability of HAPS and GPS satellite during the +entire course of observation is shown in Fig. 10. As we can see, +the availability of the HAPS-aided GPS system during the +entire observation period is 100 %, while the availability of the +GPS-only system is 99.71 % as there are two epochs (circled in +a black ellipse) where the number of GPS satellites is three. +While the difference between the availability of the HAPS- +aided GPS system and the GPS-only system is not significant, +this is probably because Ottawa is a relatively small metro city +compared to the metro cities like Chicago. In the following, we +first present a comparison of the HDOP and VDOP between the +HAPS-aided GPS system and the GPS-only system. Next, we +analyze the 3D positioning performance for both the GPS-only +system and the HAPS-aided GPS system. To show the +improvements brought by the RAIM algorithm, we compare the +RAIM-enabled positioning systems, where both the SPP and +the RAIM algorithms are applied, with the RAIM-disabled +positioning systems, where only the SPP algorithm is applied. + +1) Dilution of Precision Analysis +Fig. 11 shows the HDOP and VDOP of the GPS-only system +and the HAPS-aided GPS system. As we can see, both the +HDOP and VDOP of the HAPS-aided GPS system are better +than that of the GPS-only system. In particular, we notice that +there are fewer spikes on the HDOP and VDOP performance of +the HAPS-aided GPS system, which demonstrates that the + +15 +GPS-only system +HAPS-aided GPS system +HDOP +5 +0 +0 +100 +200 +300 +400 +500 +600 +700 +epoch (s)20 +GPS-only system +HAPS-aided GPS system +(w) +15 +VDOI +10 +5 +DAM +0 +0 +100 +200 +300 +400 +500 +600 +700 +epoch(s)11 +10 +HAPSs/satellites +9 +8 +visible +6 +5 +of +Number +4 +3 +2 +HAPS +GPSsatellite +1 +0 +100 +200 +300 +400 +500 +600 +700 +epoch (s)> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + + +Fig. 12. CDF of the 3D positioning accuracy in the suburban area. + + +Fig. 13. CDF of the 3D positioning accuracy in the dense urban area. + + +HDOP and VDOP performance of the HAPS-aided GPS system +is more stable than the GPS-only system. + +2) 3D Positioning Accuracy Analysis +The CDFs of the 3D positioning accuracy for both the +suburban area and the dense urban area are shown in Fig. 12 +and Fig. 13, respectively. As we can see, without enabling the +RAIM, the 90-percentile 3D positioning accuracy of the GPS- +only system can be improved by 36 % in the suburban area, and +33.64 % in the dense urban area with the assistance from the +HAPS. With the RAIM turned on, we can observe that the +positioning performance of both the GPS-only system and the +HAPS-aided GPS system can be further improved. Yet we +notice that the improvement brought by the RAIM in the +suburban area for the GPS-only system is almost negligible, +while it is more tangible for the HAPS-aided GPS system. This +is because the quality of signals in the suburban area tends to be +great, and the HAPS-aided GPS system has a relatively higher +chance to enable the RAIM as there are more ranging sources +in the system. This observation is also applicable to the dense +urban scenario where the 90-percentile 3D positioning accuracy +of the HAPS-aided GPS system is improved by 45.2 %, which +is much more significant than that for the GPS-only system. For +the dense urban scenario where the multipath is severe, the +RAIM algorithm plays a more significant role, especially in the +HAPS-aided GPS system. The reason behind this is that the +number of visible satellites in the dense urban area is low, +making the RAIM algorithm for the GPS-only system less +effective. Since the implemented RAIM algorithm detects and +excludes an abnormal observation by multiplying its variance +with an exponential term if the absolute value of its normalized +pseudorange residual surpasses the critical value, we count the +number of times where the absolute value of the normalized +pseudorange residuals surpass the critical value for both +systems considered and for both the suburban scenario and the +dense urban scenario. For convenience, we rephrase the number +of times where the absolute value of the normalized +pseudorange residuals surpass the critical value as the number +of times the RAIM is enabled. With the system model +considered in this work, we find that the number of times the +RAIM is enabled for the GPS-only system is roughly 23.84 % +as many as the number of times the RAIM is enabled for the +HAPS-aided GPS system in the suburban area; and the number +of times the RAIM is enabled for the GPS-only system is about +50.72 % as many as the number of times the RAIM is enabled +for the HAPS-aided GPS system in the dense urban area. This +demonstrates the applicability of the RAIM algorithm on the +HAPS-aided GPS system, especially in the dense urban area. +V. CONCLUSION +HAPS have a number of advantages over satellites, including +(but not limited to) lower latency, lower pathloss, smaller +pseudorange errors, and HAPS can provide continuous +coverage to reduce the number of handovers for users. This +makes HAPS an excellent candidate to serve as another type of +ranging source. Since urban areas are where GNSS positioning +performance degrades severely, while also being where most +people live, deploying several HAPS as additional ranging +sources above metropolitan cities would improve GNSS +positioning performance and maximize the profit of the extra +payloads on HAPS. From both the simulation and physical +experiment results, we observed that HAPS can indeed improve +the HDOP, the VDOP, and the 3D positioning accuracy of a +legacy GNSS. With the system model considered in this work, +we showed that the 90-percentile 3D positioning accuracy of +the GPS-only system can be improved by around 35 % in both +suburban and dense urban areas. We demonstrated the +applicability of the RAIM algorithm for the HAPS-aided GPS +system, especially in the dense urban areas. To enhance the +simulation of the HAPS-aided GPS, the receiver clock offset +should be estimated with higher accuracy. We think the +effectiveness of the RAIM algorithm can be improved if the +standard deviation of the target observable is available. To +further improve the positioning performance for urban areas, +we can make use of terrestrial signals such as cellular network +signals and multipath signals. This would constitute a vertical + +Suburbanarea +0.9 +0.8 +0.7 +0.6 +DF +0.5 +0.4 +0.3 +0.2 +GPS-only system (RAIM off) +GPS-only system (RAIM on) +0.1 +HAPS-aided GPS system (RAIM off) +HAPS-aidedGPSsystem (RAIMon) +0 +0 +5 +10 +15 +20 +25 +3Dpositioningerror(m)Dense urban area +0.9 +0.8 +0.7 +0.6 +DF +0.5 +0.4 +0.3 +0.2 +GPS-only system (RAIM off) +GPS-only system (RAIM on) +0.1 +HAPS-aided GPS system (RAIM off) +HAPS-aided GPS system (RAIM on) +0 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +3D positioning error (m)> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +heterogeneous network (V-Het-Net) positioning system, which +we believe will yield a lower VDOP based on the DOP +illustration presented in this paper. +ACKNOWLEDGMENT +The Skydel software is a formal donation from Orolia to +Carleton University. +REFERENCES +[1] “Global Positioning System standard positioning service performance +standard,” GPS.GOV, Washington, DC, USA, Apr. 2020. 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Ferrer, “A +comparison between different error modeling of MEMS applied to +GPS/INS integrated systems,” Sens., vol. 13, no. 8, pp. 9549-9588, Jul. +2013. +[32] O. G. Crespillo, M. Joerger, and S. Langel, “Overbounding GNSS/INS +integration with uncertain GNSS Gauss-Markov error parameters,” +IEEE/ION Position, Locat. Navig. Symp. (PLANS), Portland, Oregon, +2020, pp. 481-489. +[33] F. Hsieh and M. Rybakowski, “Propagation model for high altitude +platform systems based on ray tracing simulation,” in Proc. 13th Eur. +Conf. Antennas Propag. (EuCAP), Krakow, Poland, 2019, pp. 1-5. +[34] S. Alfattani, W. Jaafar, Y. Hmamouche, H. Yanikomeroglu, and A. +Yongacoglu, “Link budget analysis for reconfigurable smart surfaces in +aerial platforms,” IEEE Open J. Commun. Soc., vol. 2, pp. 1980-1995, +2021. +[35] “EVK-M8T user guide,” Ublox, May 2018. Accessed on: Dec. 31, 2022. +[Online]. +Available: +https://content.u- +blox.com/sites/default/files/products/documents/EVK- +M8T_UserGuide_%28UBX-14041540%29.pdf + + + +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +Hongzhao +Zheng +(Member, +IEEE) +received the B. Eng. (Hons.) degree in +engineering physics from the Carleton +University, Ottawa, ON, Canada, in 2019. +He is currently a PhD student at Carleton +University. His research interest is the +urban positioning using sensor-enabled +heterogeneous wireless infrastructure. + + +Mohamed Atia (Senior Member, IEEE) +received the B.S. and M.Sc. degrees in +computer systems from Ain Shams +University, Cairo, Egypt, in 2000 and +2006, respectively, and the Ph.D. degree +in electrical and computer engineering +from Queen’s University, Kingston, ON, +Canada, in 2013. He is currently an +Associate Professor with the Department of Systems and +Computer Engineering, Carleton University. He is also the +Founder and the Director of the Embedded and Multi-Sensory +Systems Laboratory (EMSLab), Carleton University. His +research interests include sensor fusion, navigation systems, +artificial intelligence, and robotics. + + +Halim Yanikomeroglu (Fellow, IEEE) +received the BSc degree in electrical and +electronics engineering from the Middle +East Technical University, Ankara, Turkey, +in 1990, and the MASc degree in electrical +engineering (now ECE) and the PhD degree +in electrical and computer engineering from +the University of Toronto, Canada, in 1992 +and 1998, respectively. Since 1998 he has +been with the Department of Systems and Computer +Engineering at Carleton University, Ottawa, Canada, where he +is now a Full Professor. His research interests cover many +aspects of wireless communications and networks. He has given +110+ invited seminars, keynotes, panel talks, and tutorials in the +last five years. Dr. Yanikomeroglu’s collaborative research +with industry resulted in 39 granted patents. Dr. Yanikomeroglu +is a Fellow of the IEEE, the Engineering Institute of Canada +(EIC), and the Canadian Academy of Engineering (CAE). He is +a Distinguished Speaker for the IEEE Communications Society +and the IEEE Vehicular Technology Society, and an Expert +Panelist of the Council of Canadian Academies (CCA|CAC). +Dr. Yanikomeroglu is currently serving as the Chair of the +Steering Committee of IEEE’s flagship wireless event, +Wireless Communications and Networking Conference +(WCNC). He is also a member of the IEEE ComSoc GIMS, +IEEE ComSoc Conference Council, and IEEE PIMRC Steering +Committee. He served as the General Chair and Technical +Program Chair of several IEEE conferences. He has also served +in the editorial boards of various IEEE periodicals. +Dr. Yanikomeroglu received several awards for his research, +teaching, and service, including the IEEE ComSoc Fred W. +Ellersick Prize (2021), IEEE VTS Stuart Meyer Memorial +Award (2020), and IEEE ComSoc Wireless Communications +TC Recognition Award (2018). He received best paper awards +at IEEE Competition on Non-Terrestrial Networks for B5G and +6G in 2022 (grand prize), IEEE ICC 2021, IEEE WISEE 2021 +and 2022. + + + + diff --git a/7dAyT4oBgHgl3EQf2vnp/content/tmp_files/load_file.txt b/7dAyT4oBgHgl3EQf2vnp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..54e627f34c59d07756c68e1a0786d4aa7a5745a2 --- /dev/null +++ b/7dAyT4oBgHgl3EQf2vnp/content/tmp_files/load_file.txt @@ -0,0 +1,815 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf,len=814 +page_content='> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Analysis of a HAPS-Aided GNSS in Urban Areas using a RAIM Algorithm Hongzhao Zheng, Member, IEEE, Mohamed Atia, Senior Member, IEEE, and Halim Yanikomeroglu, Fellow, IEEE Abstract—The global averaged civilian positioning accuracy is still at meter level for all existing Global Navigation Satellite Systems (GNSSs), and the performance is even worse in urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' At lower altitudes than satellites, high altitude platform stations (HAPS) offer several benefits, such as lower latency, less pathloss, and likely smaller overall estimation error for the parameters associated in the pseudorange equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' HAPS can support GNSSs in many ways, and in this paper we treat the HAPS as another type of ranging source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In so doing, we examine the positioning performance of a HAPS-aided GPS system in an urban area using both a simulation and physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The HAPS measurements are unavailable today;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' therefore, they are modeled in a rather simple but logical manner in both the simulation and physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We show that the HAPS can improve the horizontal dilution of precision (HDOP), the vertical dilution of precision (VDOP), and the 3D positioning accuracy of GPS in both suburban and dense urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We also demonstrate the applicability of a RAIM algorithm for the HAPS-aided GPS system, especially in the dense urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Index Terms—High altitude platform station (HAPS), horizontal dilution of precision (HDOP), pseudorange, receiver autonomous integrity monitoring (RAIM), vertical dilution of precision (VDOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' INTRODUCTION ODAY, many countries and the European union have their own global navigation satellite systems (GNSSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' However, 95 percent of the time, the global averaged horizontal positioning accuracy of existing GNSSs is still at the meter level, and it is even worse for the vertical positioning accuracy [1]-[4] due to the nature of the satellite geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Although vertical positioning performance is less important than horizontal positioning performance today, it might be very important in the future, for instance, for unmanned aerial vehicles (UAVs) flying in the 3D aerial highways [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Thanks to ongoing research on localization and navigation fields, there are a number of techniques developed which can bring the positioning accuracy of systems involving satellites to the centimeter level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For example, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' have shown that centimeter-level positioning accuracy can be achieved using the multi-constellation GNSS consisting of Beidou, Galileo, GLONASS and GPS with precise point positioning (PPP) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Because most civilian applications use single-frequency, low- cost receivers for localization and navigation, many advanced positioning algorithms, including PPP that delivers centimeter This paper was supported in part by Huawei Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Zheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Atia, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Yanikomeroglu are with the Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada (e-mail: hongzhaozheng@cmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='carleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='ca;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Mohamed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='Atia@carleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='ca;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' halim@sce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='carleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' level positioning accuracy, cannot be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Therefore, the single point positioning (SPP) is the most commonly used algorithm in civilian applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' But this is poised to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As increasing numbers of low-Earth-orbit (LEO) satellites are launched into space, researchers are investigating the feasibility of utilizing LEO satellites to aid the positioning service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For instance, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' have shown that a centimeter level Signal-In- Space Ranging Error (SISRE) in the real-time PPP application can be achieved using a LEO enhanced GNSS [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In the event that GNSS signals are unavailable in urban areas, researchers are also interested in building navigation systems that exclusively rely on LEO satellite signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For example, a position root mean squared error (RMSE) of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='8 m for a UAV has been proven feasible using only two Orbcomm LEO satellites with the carrier phase differential algorithm [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Compared to medium-Earth-orbit (MEO) satellites, which are typically used in GNSSs, LEO satellites offer several advantages, such as lower latency and less pathloss due to shorter distance to ground users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' LEO satellites also offer greater availability due to the large number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To further enhance high bandwidth networking coverage in areas with obstacles, such as urban areas, another option is the use of high altitude platform stations (HAPS1), which refer to aerial platforms positioned in the stratosphere with a typical altitude of about 20 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' HAPS can be utilized for many technologies coming in 5G even 6G and beyond such as computation offloading [9], edge computing [10], and aerial base station [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As urban areas are where GNSS positioning performance degrades severely, while also being where most people live, we could improve the positioning performance of GNSS by placing several HAPS above metro cities and equipping them with satellite-grade atomic clocks so that HAPS can be deployed as another type of ranging source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Even though atomic clocks on satellites are highly accurate, they are not perfect due to the time dilation postulates made in both Einstein’s special theory of relativity and the general theory of relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' According to Einstein’s special theory of relativity, an atomic clock on a fast-moving satellite runs slower than a clock on Earth by around 7 microseconds per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' On the other hand, according to the general theory of relativity, an atomic clock which experiences weaker gravity on a distant satellite runs faster than a clock which experiences greater gravity on 1 In this paper, the acronym "HAPS" is used to denote “high altitude platforms station” in both singular and plural forms, in line with the convention adopted in the ITU (International Telecommunications Union) documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' T > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Earth by about 45 microseconds per day [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As HAPS operate at an altitude of around 20 km and can be quasi-stationary, the time dilation is negligible from the perspective of special relativity and greatly reduced from the perspective of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Therefore, the atomic clocks on HAPS will likely be more accurate than that on satellites, which can make the estimation error of the HAPS clock offset smaller than that of the satellite clock offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since HAPS are positioned much closer to the Earth than satellites, the pathloss of a HAPS is expected to be much less, which will likely make the received signal power of a HAPS stronger than that of a satellite, thereby reducing the estimation error of the parameters associated in the pseudorange measurement of the HAPS signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The movement of a HAPS can be confined to a cylindrical region with a radius of 400 m and a height of about 700 m [13], which can reduce the number of handovers during the course of navigation and increase the utilization efficiency during its operation life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As HAPS are positioned in the stratosphere, which is below the ionosphere, their signals will likely be free of the ionospheric effect, which is known to be one of the major sources of error in pseudorange measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Therefore, the overall estimation error for a HAPS will likely be smaller than that of a satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Similar to the pseudorange measurement for a satellite, which incorporates the satellite position error, we should also consider the position error in the pseudorange measurement for HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Fortunately, researchers have been investigating the positioning of HAPS and have demonstrated that HAPS positioning errors are comparable to or lesser than satellite orbit errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For example, Dovis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' prove that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='5 m positioning accuracy (circular error probable [CEP] 68 percent) for a HAPS is achievable using the modified RTK method [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' There are a handful of papers in the literature that have investigated the HAPS-aided GNSS [15]-[18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' however, to the best of our knowledge, this paper is the first to provide a comprehensive study of the positioning performance of a HAPS-aided GNSS in urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' There are plenty of operational GPS satellites that could fail due to the degraded signal quality for reasons such as obstruction, multipath, intentional or unintentional attacks, thereby impacting the positioning performance of the GNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In this case, a signal selection algorithm like the receiver autonomous integrity monitoring (RAIM) algorithm, which can detect and exclude poor quality signals, can be helpful in improving the positioning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For example, about 35 percent decrease in RMS positioning error of the GPS-only case and 50 percent decrease in RMS positioning error of the GPS/GLONASS case in a severe urban scenario have been achieved on smartphone GNSS chips by using a RAIM algorithm [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Moreover, Yang and Xu propose a robust estimation-based RAIM algorithm that can detect and exclude multiple faulty satellites effectively with efficiency higher than the conventional least squares (LS)-based RAIM algorithm [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In this paper, we make three postulations: 1) a HAPS signal is free of the ionospheric effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2) the estimation error of the HAPS clock offset is smaller than that of the satellite clock offset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' and 3) the received signal power of a HAPS is higher than that of a satellite, all of which contribute toward the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' System model of the HAPS-aided GPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' assumption that the overall estimation error of the parameters associated in the pseudorange equation for the HAPS is smaller than that for the satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Under this assumption, we use the SPP algorithm developed in our prior work [21] to show that HAPS can indeed improve the positioning performance of legacy GNSSs in urban areas through both a simulation and a physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We also demonstrate the applicability of the RAIM algorithm to a HAPS-aided GPS system, especially in dense urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since the HAPS measurements are unavailable so far, they are simulated in a rather simple but logical way in both the simulation and physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The contributions of this paper are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' • First, using a commercial GNSS simulator, we simulate the GPS pseudorange signals and generate the positions of HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' By using the default system parameters as well as a proper manipulation of the number of visible satellites, we show that the positioning performance of the GPS-only system in both the suburban and dense urban areas are close to the real scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Moreover, we show the positioning performance of different systems where different numbers of HAPS are used with or without the GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The issue of the ranging source geometry is revealed from the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' • Next, we apply the SPP algorithm to the real GPS data collected using two commercial GNSS receivers as well as the HAPS data generated using the commercial GNSS software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In so doing, we show the advantage of the HAPS-aided GPS system in the sense of the horizontal dilution of precision (HDOP) and the vertical dilution of precision (VDOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' • Finally, we implement a RAIM algorithm and demonstrate its effectiveness in improving the 3D positioning performance of the HAPS-aided GPS system, especially in dense urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The rest of the paper is organized as follows: in Section Ⅱ, the system model, the SPP algorithm, and the RAIM algorithm are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In Section Ⅲ, the simulation setup of the HAPS- Ionosphere HAPS HAPS 20km HAPSfootprint HAPSfootprint 15° cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' cell> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < aided GPS system and the simulation results are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In Section Ⅳ, the physical experiment setup and results, including both the DOP analysis and the 3D positioning accuracy analysis, are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Finally, Section V offers some conclusions and a discussion of future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For simplicity, the GNSS signal only involves the GPS C/A L1 signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' SYSTEM MODEL The system model of the HAPS-aided GPS system is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' There are four satellites shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 1, this is just a reminder that at least four satellites are required to perform precise 3D localization using GNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The typical choice for the elevation mask is 10 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' However, we use 15 degrees as the elevation mask for the satellites and HAPS due to the following reasons: 1) the atmospheric error owing to the signal refraction can be neglected if the elevation of a satellite is greater than 15 degrees [22], which is likely true for a HAPS as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2) As there is a higher chance of ensuring the required number of ranging source with HAPS, we can improve the positioning performance further by only using those satellite signals with better quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The pseudorange equation for satellite is given by 𝑝𝑆𝐴𝑇 = 𝜌𝑆𝐴𝑇 + 𝑑𝑆𝐴𝑇 + 𝑐(𝑑𝑡 − 𝑑𝑇𝑆𝐴𝑇) + 𝑑𝑖𝑜𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑆𝐴𝑇 + 𝑑𝑡𝑟𝑜𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑆𝐴𝑇 + 𝜖𝑚𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑆𝐴𝑇 + 𝜖𝑝 (1) where 𝑝𝑆𝐴𝑇 denotes the satellite pseudorange measurement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝜌𝑆𝐴𝑇 is the geometric range between the satellite and receiver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑑𝑆𝐴𝑇 represents the satellite orbit error,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑐 is the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑑𝑡 is the receiver clock offset from GPS time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑑𝑇𝑆𝐴𝑇 is the satellite clock offset from GPS time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑑𝑖𝑜𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑆𝐴𝑇 denotes the ionospheric delay for satellite signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑑𝑡𝑟𝑜𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑆𝐴𝑇 denotes the tropospheric delay for satellite signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝜖𝑚𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑆𝐴𝑇 is the delay caused by the multipath for satellite signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' and 𝜖𝑝 is the delay caused by the receiver noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The pseudorange equation for HAPS can be expressed as follows: 𝑝𝐻𝐴𝑃𝑆 = 𝜌𝐻𝐴𝑃𝑆 + 𝑑𝐻𝐴𝑃𝑆 + 𝑐(𝑑𝑡 − 𝑑𝑇𝐻𝐴𝑃𝑆) + 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆 + 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 + 𝜖𝑝 (2) where 𝑝𝐻𝐴𝑃𝑆 denotes the HAPS pseudorange measurement, 𝜌𝐻𝐴𝑃𝑆 represents the geometric range between the HAPS and the receiver, 𝑑𝐻𝐴𝑃𝑆 represents the HAPS position error, 𝑑𝑇𝐻𝐴𝑃𝑆 is the HAPS clock offset from GPS time, 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆 denotes the tropospheric delay for HAPS signals, 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 is the delay caused by the multipath for HAPS signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The simulated vehicle trajectory originates at Carleton University, which is in a suburban area, and ends at Rideau Street, which is in a dense urban part of Ottawa (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' There are six simulated HAPS shown as transmitters on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As we can see, one HAPS is positioned over downtown Ottawa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' the other five HAPS are positioned nearby, over populated areas and conservation areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' HAPS is quasi-stationary, meaning that it will still be moving in a variety of manners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In this work, all the HAPS are Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Vehicle trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Locations of the simulated HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' simulated to be following a circular trajectory with a radius of 300 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The elevation and azimuth angles of all the HAPS at the beginning of the simulation are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The positions of HAPS were chosen to provide a rich diversity in azimuth angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' With one HAPS at the zenith and the others having relatively low elevation angles, this constitutes a near Zenith + Horizon (ZH) geometry, which can deliver a reasonably good DOP [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To make sure the entire urban area is well covered, HAPS are placed not too far away from the urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To better understand the concept of DOP, the visual illustrations of the HDOP and VDOP of the simulated HAPS constellation are provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Due to various errors impacting the pseudorange measurement, the estimated distance between a HAPS and a receiver can be smaller or larger than the geometric range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Objects with a higher elevation angle will likely result in more uncertainty for the vertical OSM + relief shading Tracks: Ottawa,On to Rideau Centre, Ottawa,ON Lyon Denseurban Areas yview Fost onburg Suburban Areas Ottawa Cene:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='40751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='0087m Google Map created at CSVisualizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='com 一净用多款Transmitter5 deroure Buckinohan Transmitter6 arcde otineau Transmitter3 174 Transmitter1 Simulator 417 Embrun Russell Metcalfe Transmitter2 Transmitter4> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' HDOP of the simulated HAPS constellation (top view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' VDOP of the simulated HAPS constellation (front view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' component and less uncertainty for the horizontal component from the point of view of geometry, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The shaded area is where the receiver is estimated to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The Single Point Positioning (SPP) Algorithm The single point positioning algorithm is implemented on the basis of the SPP package developed by Napat Tongkasem [24] with proper modifications [21] so that HAPS can be incorporated in the SPP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 6 shows the flowchart of the single point positioning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We should point out that the implemented single point positioning algorithm is not the best positioning algorithm, and that the objective of this work is to show the significance of HAPS in aiding the positioning performance of a legacy GNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The implemented SPP algorithm can be improved in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For example, if the knowledge of the measurement error variance is available, we can apply the weighted least squares (WLS) algorithm to enhance the positioning performance of the SPP algorithm by lowering the weights of those observations with higher variances [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' If the knowledge of the measurement error variance is unavailable, the computational complexity of the SPP algorithm can be reduced by imposing the Cholesky decomposition for the matrix inversion in (9) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We can also TABLE I ELEVATION AND AZIMUTH OF THE HAPS AT THE START OF THE SIMULATION HAPS index Elevation angle Azimuth angle HAPS #1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='087° 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='210° HAPS #2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='054° 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='878° HAPS #3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='952° 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='022° HAPS #4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='450° 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='477° HAPS #5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='554° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='204° HAPS #6 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='805° 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='884° use the carrier phase measurement to enhance the positioning performance of the HAPS-aided GPS system, since carrier performance of the HAPS-aided GPS system, since carrier phase measurements come with much higher precision, which usually delivers a more accurate position solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since the HAPS clock offset in this work is not explicitly simulated, we simply use 𝑑𝑇 to denote the satellite clock offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' From the data collected by the GNSS receiver, we shall obtain both the receiver independent exchange (RINEX) format observation file and the RINEX navigation file, from which we can obtain satellite information, such as the satellite pseudorange 𝒑𝑺𝑨𝑻, the ionospheric parameters 𝜶 , the Keplerian parameters, the pseudo-random noise (𝑷𝑹𝑵) code, which represents the unique number of each satellite, the day of year ( 𝐷𝑂𝑌 ) which represents the day of year at the time of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We write 𝑷𝑹𝑵 in bold to represent a vector containing the pseudo- random noise code of all visible satellites at the current epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We are able to compute the satellite positions, 𝑷𝑺𝑨𝑻 , and satellite clock offset, 𝒅𝑻 , using the Keplerian parameters contained in the navigation file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑷𝑯𝑨𝑷𝑺 denotes a vector containing the positions of all HAPS, which are generated using the Skydel GNSS simulator [27], and 𝒑𝑯𝑨𝑷𝑺 denotes a vector containing the HAPS pseudorange, which will be explained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To compute the position solution 𝒙 , we first initialize the receiver position to the center of the Earth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' then we initialize the receiver clock offset to zero and the change in estimates 𝒅𝒙 to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For each epoch of measurement, we first check if the number of available ranging sources is more than three, as at least four ranging sources are required to perform precise 3D localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since the receiver position is iteratively estimated, we calculate the elevation angles for both satellites and HAPS with respect to the recently estimated receiver position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since both the tropospheric delay and the ionospheric delay are functions of the receiver position, these two atmospheric delays are estimated iteratively as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The elevation angle, satellite pseudorange, HAPS pseudorange, satellite position, satellite clock offset, tropospheric delay 𝒅𝒕𝒓𝒐𝒑 , ionospheric delay 𝒅𝒊𝒐𝒏 , and pseudo-random noise (𝑷𝑹𝑵) code are modified iteratively on the basis of the re- computed elevation angles for both satellites and HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To prepare the parameters needed for the least square method, the pseudorange needs to be corrected as follows: 𝒑𝑺𝑨𝑻 𝒄 = 𝒑𝑺𝑨𝑻 + 𝑐 ∙ 𝒅𝑻 − 𝒅𝒕𝒓𝒐𝒑,𝑺𝑨𝑻 − 𝒅𝒊𝒐𝒏,𝑺𝑨𝑻 (3) where 𝒑𝑺𝑨𝑻 𝒄 represents the corrected pseudorange for the Transmitter 5 Transmitter 6 Transmitter 3 Transmitter 2 Transmitter 4Transmitter 5 Transmitter 6 Transmitter 1 Transmitter 3 Transmitter 2 Transmitter 4> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Flow chart of the single point positioning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' satellites, and 𝒑𝑺𝑨𝑻 represents the uncorrected pseudorange for the satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In this work, the HAPS pseudorange is modeled as the sum of the geometric range and the pseudorange error,which represents the overall estimation error of the parameters in the HAPS pseudorange equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Accordingly, the HAPS pseudorange does not need to be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Due to the Earth’s rotation, the positions of satellites and HAPS at the signal emission time are different from their positions at the signal reception time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' this is known as the Sagnac effect [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The coordinates of satellites/HAPS can be transformed from the signal emission time to the signal reception time by [28] ∆𝑡𝑅𝑂𝑇 = 𝑡𝑟𝑥 − 𝑡𝑡𝑥 (4) 𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑟𝑥 = 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑡𝑥 (5) where ∆𝑡𝑅𝑂𝑇 denotes the signal propagation time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑡𝑟𝑥 represents the signal reception time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑡𝑡𝑥 represents the signal emission time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑟𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates at the signal reception time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝑃𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='𝑡𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates at the signal emission time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝜔𝐸 denotes the Earth’s rotation rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' and 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) is known as the rotation matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' which is described as follows: 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) = [ cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) 0 − sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) 0 cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) 0 0 1 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' (6) The line-of-sight vector 𝒗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' and the true range between ranging sources and receiver 𝝆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' are then calculated to compute the a priori range residual vector 𝒃 and the design matrix 𝑯,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' where 𝒃 = 𝒑𝒄 − 𝝆 (7) 𝑯 = [𝒗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝟏𝑙𝑒𝑛𝑔𝑡ℎ(𝑷𝐜)×1] (8) where 𝒑𝒄 is the corrected satellite pseudorange combined with the corrected HAPS pseudorange,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝟏𝑙𝑒𝑛𝑔𝑡ℎ(𝑷𝐜)×1 denotes a column vector of length being the total number of visible ranging sources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' and 𝑷𝐜 is a vector containing the corrected positions of the visible ranging sources (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=', satellite + HAPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Finally, the least square solution is computed as follows: 𝑸 = (𝑯′𝑯)−1 (9) 𝒅𝒙 = 𝑸𝑯′𝒃 (10) 𝑑𝑡 = 𝒅𝒙(4)/𝑐 (11) where Q is known as the covariance matrix, and 𝒅𝒙(4) denotes the fourth element in the vector 𝒅𝒙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To prevent the algorithm from getting numerical issues, we should ensure the term being inversed in (9) is non-singular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' in other words, the design matrix 𝑯 should be non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' With the extra observations by utilizing HAPS as additional ranging sources, the chance of 𝑯 being singular is likely reduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' the non-singular design matrix can be ensured by avoiding the use of collinear observations, which means that two or more observations have about the same azimuth and elevation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We observe that the term being inversed in (9), 𝑯′𝑯, is a Hermitian, positive definite matrix, therefore the Cholesky decomposition can be imposed to reduce the computational complexity [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The covariance matrix, 𝑸, is described by 𝑸 = [ 𝜎𝑥 2 𝜎𝑥𝑦 𝜎𝑥𝑧 𝜎𝑥𝑡 𝜎𝑥𝑦 𝜎𝑦 2 𝜎𝑦𝑧 𝜎𝑦𝑡 𝜎𝑥𝑧 𝜎𝑦𝑧 𝜎𝑧 2 𝜎𝑧𝑡 𝜎𝑥𝑡 𝜎𝑦𝑡 𝜎𝑧𝑡 𝜎𝑡 2 ] (12) Initialization PsAT-PHAPS,PRN,DOY x = 04x1 Input dt = x(4)/c PHAPS, PsAT, dT,α dx = x + Inf stop = 0 Exit No NsAr + NHAPs ≥ 4 IYes No dx(1:3)|> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='01 Yes Finding parameters For sotellites For HAPS OsAT,dtrop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content="dion HAPS Applying elevation mask For sotellites For HAPS sAT,dtrep,dion,PRN,dT,PsAr HAPS,PHAPS Pseudorange correction PSAT-PHAPS Combining the corrected pseudoranges p' = [psar." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='phaes] 国 Correcting for the Sagnaceffect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=', Earth rotation) PSAT,PHAPS Combining the corrected ranging source positions P° = [PSAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content="PHAPs] Finding parameters V,p,b,H,Q andno Computing the position solution using the Least Square method 3p'x x,dt> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < where receiver coordinates x, y, z in the Earth-centered Earth- fixed (ECEF) coordinate frame and the receiver clock offset, respectively." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The least square solution will be found when the norm of the change in receiver position 𝒅𝒙(1: 3) is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In this work, this threshold is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='01 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We use the HDOP, the VDOP and the 3D positioning accuracy as the metrics to show the advantage of the proposed HAPS-aided GPS system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' the 3D positioning accuracy is used to show the applicability of the RAIM to the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To compute the HDOP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' we need to convert the covariance matrix into the local north-east-down (NED) coordinate frame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' which can be done with the following equations [29]: 𝑸𝑵𝑬𝑫 = 𝑹′𝑸̃𝑹 = [ 𝜎𝑛 2 𝜎𝑛𝑒 𝜎𝑛𝑑 𝜎𝑛𝑒 𝜎𝑒 2 𝜎𝑒𝑑 𝜎𝑛𝑑 𝜎𝑒𝑑 𝜎𝑑 2 ] (13) 𝑸̃ = [ 𝜎𝑥 2 𝜎𝑥𝑦 𝜎𝑥𝑧 𝜎𝑥𝑦 𝜎𝑦 2 𝜎𝑦𝑧 𝜎𝑥𝑧 𝜎𝑦𝑧 𝜎𝑧 2 ] (14) 𝑹 = [ −sin 𝜆 cos 𝜆 0 − cos 𝜆 sin 𝜑 − sin 𝜆 sin 𝜑 cos 𝜑 cos 𝜆 cos 𝜑 sin 𝜆 cos 𝜑 sin 𝜑 ] (15) where 𝜎𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝜎𝑒,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' and 𝜎𝑑 represent the receiver position errors in the local north,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' east,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' and down directions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝜆 and 𝜑 represent the longitude and latitude of the receiver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Then, the HDOP is described by 𝐻𝐷𝑂𝑃 = √𝜎𝑛2 + 𝜎𝑒2 (16) and the VDOP is described by 𝑉𝐷𝑂𝑃 = √𝜎𝑑 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' (17) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The Receiver Autonomous Integrity Monitoring (RAIM) Algorithm The RAIM algorithm is a signal selection algorithm that can detect and even exclude abnormal observations using redundant measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' It can detect an abnormal observation when the number of observations is at least five;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' it can exclude this abnormal observation when the number of observations is at least six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The RAIM algorithm is typically applied to multi- constellation GNSSs where the number of ranging sources is more than enough to perform precise 3D localization, and it is typically applied to cases where there likely exists at least one observation that differs from the expected value significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Such cases include urban areas, where the pseudorange measurement is highly subject to the multipath effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' With the assistance from HAPS, the chance of enabling the RAIM function will likely increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Typical RAIM algorithms tend to use the standard deviation of the target observable, which is the pseudorange measurement in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As knowledge of the standard deviation of the satellite pseudorange is unavailable on the receivers we use, in this work the RAIM algorithm is implemented on the basis of [30], which considers a 𝐶/𝑁0- based variance model and a computationally efficient method, namely the modified Danish estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2 The implemented 𝐶/𝑁0-based RAIM algorithm is given in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 1, where 𝑁 denotes the number of visible ranging sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The input to this algorithm consists of the position fix computed using the SPP algorithm 𝒙, and the 𝑪/𝑵𝟎 of the ranging source signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since HAPS are located at much lower altitudes than 2 To the best of our knowledge, RAIM is the most common algorithm used for integrity monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since we only have the 𝐶/𝑁0 data which can be utilized for the integrity monitoring, we could not identify in the literature any other appropriate RAIM-like algorithm for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' However, we believe that the other RAIM algorithms would also be applicable if the knowledge of the standard deviation of the satellite pseudorange happens to be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Algorithm 1 The 𝐶/𝑁0-based RAIM Algorithm Input: The SPP estimated position solution 𝒙 and 𝑪/𝑵𝟎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Output: The SPP and RAIM jointly estimated position solution 𝒙̂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 1: Initialize the parameters 𝑠𝑡𝑜𝑝 and 𝒅𝒙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2: while |𝒅𝒙(1: 3)| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='01 do 3: Same procedures as the SPP algorithm until “Finding parameters” after correcting for the Sagnac effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 4: for 𝑖 = 1 to 𝑁 do 5: if 𝑠𝑡𝑜𝑝 == 1 do 6: Find the variance of the observation 𝑖, 𝑠𝑖, according to (19);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 7: end if 8: end for 9: Find the weight matrix 𝑾 and the design matrix 𝑯 , and calculate the covariance matrix 𝑸 according to (20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 10: Calculate the change in estimates 𝒅𝒙 according to (21), and update the position solution 𝒙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 11: Calculate the pseudorange residual 𝒗̂ according to (22), and the covariance matrix of the residuals 𝑪𝒗̂ according to (24);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 12: for 𝑖 = 1 to 𝑁 do 13: Find the normalized residual of observation 𝑖 at the current iteration 𝑘, 𝑤̅𝑖,𝑘 according to (26);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 14: if |𝑤̅𝑖,𝑘| > 𝑛1−(𝛼0/2) do 15: Update the variance of the observation 𝑖 for the next iteration 𝑘 + 1 , 𝜎𝑖,𝑘+1 2 , according to (25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 16: end if 17: end for 18: end while > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < satellites, in practice the 𝐶/𝑁0 value of the HAPS might be higher than that for any satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As it is possible that a handful of HAPS signals might suffer from severe multipath effects, we can exclude those HAPS signals whose 𝐶/𝑁0 values are much lower than the higher ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In this work, the multipath effect is not explicitly simulated for the HAPS signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' therefore, we assume that the 𝐶/𝑁0 of each HAPS is equal to the maximum 𝐶/𝑁0 value of the available satellites at each epoch, meaning that the signal quality for a HAPS will always be better than that for any satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The variance covariance matrix (VCM) 𝜮 of the observations (pseudoranges) 𝒑 is defined as follows: 𝜮 = 𝑑𝑖𝑎𝑔(𝑠1, 𝑠2, … , 𝑠𝑛) (18) 𝑠𝑖 = 10 + 1502 ∗ 10(−𝐶/𝑁0,𝑖)/10 (19) where 𝑠𝑖 denotes the variance of the observation 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We assume that the observations are uncorrelated, and that the errors follow the normal distribution with 𝑁(𝟎, 𝜮).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The weight matrix, 𝑾, is defined as the inverse of the VCM, 𝜮−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The least square equations become 𝑸 = (𝑯′𝑾𝑯)−1 (20) 𝒅𝒙 = 𝑸𝑯′𝑾𝑷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' (21) The least square residuals of the pseudorange 𝒗̂ can be obtained as follows: 𝒗̂ = 𝑯 𝒅𝒙 − 𝑷 (22) 𝑷 = 𝒑𝒄 − 𝝆 (23) where 𝑯 represents the design matrix, 𝒅𝒙 represents the change in estimates, 𝒑𝒄 denotes the corrected pseudoranges, and 𝝆 denotes the geometric range between ranging sources and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The covariance matrix of the residuals, 𝑪𝒗̂, is computed as 𝑪𝒗̂ = 𝜮 − 𝑯(𝑯𝑇𝜮−1𝑯)−1𝑯𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' (24) To detect and exclude the abnormal observations, we follow the modified Danish estimation method proposed in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 𝜎𝑖,𝑘+1 2 = 𝜎𝑖,0 2 ∙ {exp ( |𝑤̅𝑖,𝑘| 𝑇 ) , |𝑤̅𝑖,𝑘| > 𝑛1−(𝛼0/2) 1, |𝑤̅𝑖,𝑘| ≤ 𝑛1−(𝛼0/2) (25) with 𝑤̅𝑖,𝑘 = 𝒗̂𝑖,𝑘 √(𝑪𝒗̂𝒊,𝟏)𝑖𝑖 (26) where 𝜎𝑖,0 2 denotes the a priori variance of the observation 𝑖 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=', s𝑖), 𝑤̅𝑖,𝑘 denotes the normalized residual of observation 𝑖 at iteration 𝑘,√(𝑪𝒗̂𝒊,𝟏)𝑖𝑖 represents the standard deviation of observation 𝑖 from the first iteration, 𝑛1−(𝛼0/2) denotes the 𝛼0- quantile of the standard normal distribution, which is also called the critical value, 𝛼0 is the predetermined false alarm rate which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='5 % in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The modified Danish method is an iteratively reweighted LS algorithm that implements a robust estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' This method detects and excludes abnormal observations by comparing the absolute value of each normalized pseudorange residual, |𝑤̅𝑖,𝑘|, with the critical value, 𝑛1−(𝛼0/2), in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The variances for the observations whose normalized residuals are greater than the critical value are multiplied with exponential terms, making the variances of those observations larger, hence lowering the weight of those observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' By iteratively multiplying the variances of the abnormal observations by exponential terms, the weight of the abnormal observations will likely become much smaller than that of the normal observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' therefore, the abnormal observations can be considered as being excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' SIMULATION OF THE HAPS-AIDED GPS SYSTEM In this section, we will describe the simulation setup used in the Skydel GNSS software [27] and present the simulation results in terms of 3D positioning accuracy for several hybrid systems and two standalone systems in both a suburban scenario and a dense urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Simulation Setup The system model is established using the default Earth orientation parameters of the Skydel GNSS simulation software [27], which considers all GPS satellites orbiting around the Earth and transmitting the L1 C/A code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The Saastamoinen model is chosen to emulate the tropospheric effect, and the Klobuchar model is chosen to emulate the ionospheric effect using the default Klobuchar parameters that come with the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The output from Skydel contains the ECEF coordinates of satellites at the signal emission time, the ionospheric corrections, the tropospheric corrections, the satellite clock offsets, the ECEF coordinates of the receiver, the signal emission time, and so forth, at each time stamp from the start of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The receiver clock offset in the simulation is zero by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The correction terms in the pseudorange equation of satellite including the satellite orbit error, the multipath error, and the receiver noise are not separately considered in the simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' instead, a pseudorange error is introduced to reflect the presence of those effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The pseudorange error of satellite is featured using the built-in first order Gauss-Markov process with a default time constant of 10 s, and the standard deviation of 6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The continuous model for the first order Gauss-Markov process is described by [31]: ������̇ = − 1 𝑇𝑐 ������̇ + 𝑤 (27) where ������̇ represents a random process with zero mean, correlation time 𝑇𝑐, and noise 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The autocorrelation of the first > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < TABLE Ⅱ DETAILS OF THE SIMULATION SETUP Item Processing strategy Earth orientation parameter Software default Earth orientation parameters Satellite signal GPS L1 C/A Tropospheric model Saastamoinen model Ionospheric model Klobuchar model Sampling rate 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='5 MS/s Satellite pseudorange error 1st order Gauss-Markov process (time constant = 10 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' standard deviation = 6 m) HAPS pseudorange error Gaussian noise (mean = 0 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' std = 2 m for the suburban scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' std = 5 m for the dense urban scenario) Number of GPS satellites (dense urban scenario) 8-10 Number of GPS satellites (suburban scenario) 4 Total number of HAPS 6 order Gauss-Markov process is described by [32]: 𝑅(𝛥𝑡) = 𝜎2𝑒−|𝛥𝑡| 𝜏 (28) where 𝛥𝑡 represents the sampling interval, 𝜎 and 𝜏 denote the standard deviation and the time constant of the first order Gauss-Markov process, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The characteristics of the pseudorange errors for satellites are set to be the same in both the suburban scenario and the dense urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' However, we randomly select four satellites in the dense urban scenario in order to emulate the dense urban area in a rather simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We verify that by doing so, the standard deviation of the 3D positioning accuracy for the GPS-only system in the simulation is close to that in the physical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The pseudorange error for the HAPS is modeled using the Gaussian noise with standard deviations of 2 m and 5 m, representing the suburban and the dense urban scenario, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Under the assumption that the overall estimation error of the HAPS is less than that of the satellite, the standard deviation for the HAPS pseudorange error is deliberately set to be smaller than that of the satellite pseudorange error in the suburban and dense urban scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To investigate the impact of the number of HAPS on the positioning performance of the HAPS-aided GPS system, we consider four hybrid systems with different numbers of randomly selected HAPS at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We also consider the HAPS-only system for the completeness of a research problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Under this setting, we examine the 3D positioning performance of different systems in the suburban and dense urban scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In the suburban scenario, the number of visible satellites varies between eight and ten, while in the dense urban scenario the number of visible satellites is set to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The details of the simulation setup are given in Table Ⅱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Simulation Results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 7 shows the cumulative distribution function (CDF) of the 3D positioning accuracy for different positioning systems in the suburban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' With the assumption that the pseudorange error for a HAPS is smaller than that of a satellite, we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 7 that all the hybrid systems (HAPS + Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' CDF of the 3D positioning accuracy for different systems (suburban scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' CDF of the 3D positioning accuracy for different systems (dense urban scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' GPS) outperform the GPS-only system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' the more HAPS, the better the positioning performance of the HAPS-aided GPS Suburbanscenario(stdofHAPSpseudorangeerror=2m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='6 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='2 GPS-only system 1-HAPS with GPS system 2-HAPS with GPS system 3-HAPS with GPS system 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='1 4-HAPS with GPS system 4-HAPS-only system 0 5 10 15 20 25 30 3D positional accuracy (m) in local NED frameDenseurbanscenario(stdofHAP pseudorangeerror=5m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='2 4-GPS-only system 1-HAPS with 4-GPS system 2-HAPS with 4-GPS system 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='1 3-HAPS with 4-GPS system 4-HAPS with 4-GPS system 4-HAPS-only system 0 10 20 30 40 50 60 70 80 90 100 3D positionalaccuracy (m)inlocal NEDframe> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Positioning performance of M8T and M8U for horizontal and vertical planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Nevertheless, we observe that the positioning performance of the 4-HAPS-only system, where all ranging sources have a much smaller pseudorange error than the satellite, is not the best and can occasionally be very poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' This may be due to the following reasons: 1) the 4-HAPS-only system has much fewer ranging sources for computing receiver positions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2) the HAPS geometry can be poor occasionally since we are randomly selecting four HAPS at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' There are several cases where the HAPS geometry is considered poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For example, when the four randomly selected HAPS are on the same side of the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The CDF of the 3D positioning accuracy for different positioning systems in the dense urban scenario is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 8, from which we can see a similar trend that the more HAPS the better the positioning performance of the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In the dense urban scenario, where only four GPS satellites are selected for positioning, the 4-HAPS-only system achieves better positioning performance than the 4-GPS-only system due to the better signal quality for the HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' However, we should consider using the HAPS-aided GPS system for the best positioning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' FIELD EXPERIMENTS To verify and support the simulation results, we process the raw GNSS data collected using two commercial GNSS receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In this section, we present the experiment setup, the modeling of HAPS signals, and the HAPS pseudorange error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We also provide an analysis of the DOP and the 3D positioning accuracy for both the GPS-only system and the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Experiment Setup The raw GNSS data is collected along a vehicle trajectory similar to the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2, except for a slight difference due to a partial road closure on the day of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The raw GNSS data is collected using both the Ublox EVK-M8T TABLE Ⅲ EVK-M8T GNSS UNIT SPECIFICATIONS [35] Parameter Specification Serial Interfaces 1 USB V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='0 1 RS232, max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='baud rate 921,6 kBd DB9 +/- 12 V level 14 pin – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='3 V logic 1 DDC (I2C compatible) max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 400 kHz 1 SPI-clock signal max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 5,5 MHz – SPI DATA max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 1 Mbit/s Timing Interfaces 2 Time-pulse outputs 1 Time-mark input Dimensions 105 × 64 × 26 mm Power Supply 5 V via USB or external powered via extra power supply pin 14 (V5_IN) 13 (GND) Normal Operating Temperature −40℃ to +65℃ and the Ublox EVK-M8U GNSS units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The Ublox EVK-M8T unit is a timing product that can provide users with precise timing information for post-processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' the Ublox EVK-M8U unit is a dead reckoning product equipped with inertial measurement units (IMUs) such that the positioning performance of this product will not be degraded much even in the dense urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 9 shows the positioning performance along both horizontal and vertical planes for both M8T and M8U during the entire observation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 9, we can see that M8U outperforms M8T for both horizontal positioning accuracy and vertical positioning accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As only M8T provides the timing information required for post-processing, the receiver positions computed by EVK-M8U are used as the ground truth data for the analysis of the positioning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The raw GNSS data is processed using the single point positioning algorithm described in Section Ⅱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Table Ⅲ gives the specifications of the EVK-M8T GNSS unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To emulate realistic LOS conditions for HAPS in the urban area, the LOS probability as a function of the HAPS elevation angle in the urban area is implemented on the basis of [33] and [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' It is worth mentioning that the LOS probability model for the HAPS provided by [33] is proposed based on the city of Chicago;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' imposing this LOS probability model for the dense urban area of Ottawa might be too harsh considering their incompatible city scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since there is no LOS probability corresponding to the suburban area in [34], the one for rural area in [34] is used as the LOS probability for the HAPS in the suburban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The LOS probability for the HAPS in the rural area in [34] is verified as being consistent with the LOS probability for the HAPS in the suburban area in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The pseudorange of the HAPS in the experiment is modeled as the addition of the geometric range between the satellite and receiver, the receiver clock offset multiplied by the speed of light, and the pseudorange error representing the sum of all kinds of estimation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The pseudorange errors for the HAPS in the suburban and dense urban areas are simulated as Gaussian noise with zero mean and standard deviations of 2 m and 5 m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since the vehicle trajectory involves both The positioning performance of M8T and M8U 12 Horizontal positioning accuracy (M8T) Horizontalpositioningaccuracy(M8U) Vertical positioningaccuracy(M8T) 10 Verticalpositioningaccuracy(M8U) g accuracy (m) 8 6 Positioning 2 0 100 200 300 400 500 600 700 epoch (s)> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' HDOP (top) and VDOP (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' HAPS availability during the entire observation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' suburban and dense urban areas, the entire route is divided into two segments, where the first segment is considered as the suburban scenario while the second segment is considered as the dense urban scenario (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' By observing the positioning performance of the GPS-only system using the real GPS data, the LOS probability for the suburban area is applied to the HAPS for epochs less than 380 s, and the LOS probability for the dense urban area is applied to the HAPS for epochs greater than or equal to 380 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since the GNSS receivers we use do not provide an accurate receiver clock offset with respect to the GPS time, the receiver clock offset in each epoch is estimated by using the ground truth receiver positions provided from Ublox EVK-M8U and the precise timing information, such as the receiver clock drift and the receiver clock bias, provided from Ublox EVK-M8T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We should note that the pseudorange of the HAPS in the experiment is modeled as a function of the receiver clock offset, which is estimated with best effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Nevertheless, additional errors should be expected in the pseudorange of the HAPS with the magnitude depending on the quality of all visible satellite signals and the ground truth receiver position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As we would expect the quality of the satellite signals in the suburban area to be better than that in the dense urban area, we should also expect the receiver clock offset to be estimated with higher accuracy in the suburban area than in the dense urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Experiment Results With more ranging sources, we should expect the availability of the HAPS-aided GPS system to be higher than the GPS-only system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The availability of HAPS and GPS satellite during the entire course of observation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As we can see, the availability of the HAPS-aided GPS system during the entire observation period is 100 %, while the availability of the GPS-only system is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='71 % as there are two epochs (circled in a black ellipse) where the number of GPS satellites is three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' While the difference between the availability of the HAPS- aided GPS system and the GPS-only system is not significant, this is probably because Ottawa is a relatively small metro city compared to the metro cities like Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In the following, we first present a comparison of the HDOP and VDOP between the HAPS-aided GPS system and the GPS-only system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Next, we analyze the 3D positioning performance for both the GPS-only system and the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To show the improvements brought by the RAIM algorithm, we compare the RAIM-enabled positioning systems, where both the SPP and the RAIM algorithms are applied, with the RAIM-disabled positioning systems, where only the SPP algorithm is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 1) Dilution of Precision Analysis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 11 shows the HDOP and VDOP of the GPS-only system and the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As we can see, both the HDOP and VDOP of the HAPS-aided GPS system are better than that of the GPS-only system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' we notice that there are fewer spikes on the HDOP and VDOP performance of the HAPS-aided GPS system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' which demonstrates that the 15 GPS-only system HAPS-aided GPS system HDOP 5 0 0 100 200 300 400 500 600 700 epoch (s)20 GPS-only system HAPS-aided GPS system (w) 15 VDOI 10 5 DAM 0 0 100 200 300 400 500 600 700 epoch(s)11 10 HAPSs/satellites 9 8 visible 6 5 of Number 4 3 2 HAPS GPSsatellite 1 0 100 200 300 400 500 600 700 epoch (s)> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' CDF of the 3D positioning accuracy in the suburban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' CDF of the 3D positioning accuracy in the dense urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' HDOP and VDOP performance of the HAPS-aided GPS system is more stable than the GPS-only system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2) 3D Positioning Accuracy Analysis The CDFs of the 3D positioning accuracy for both the suburban area and the dense urban area are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 12 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 13, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' As we can see, without enabling the RAIM, the 90-percentile 3D positioning accuracy of the GPS- only system can be improved by 36 % in the suburban area, and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='64 % in the dense urban area with the assistance from the HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' With the RAIM turned on, we can observe that the positioning performance of both the GPS-only system and the HAPS-aided GPS system can be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Yet we notice that the improvement brought by the RAIM in the suburban area for the GPS-only system is almost negligible, while it is more tangible for the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' This is because the quality of signals in the suburban area tends to be great, and the HAPS-aided GPS system has a relatively higher chance to enable the RAIM as there are more ranging sources in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' This observation is also applicable to the dense urban scenario where the 90-percentile 3D positioning accuracy of the HAPS-aided GPS system is improved by 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='2 %, which is much more significant than that for the GPS-only system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For the dense urban scenario where the multipath is severe, the RAIM algorithm plays a more significant role, especially in the HAPS-aided GPS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' The reason behind this is that the number of visible satellites in the dense urban area is low, making the RAIM algorithm for the GPS-only system less effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since the implemented RAIM algorithm detects and excludes an abnormal observation by multiplying its variance with an exponential term if the absolute value of its normalized pseudorange residual surpasses the critical value, we count the number of times where the absolute value of the normalized pseudorange residuals surpass the critical value for both systems considered and for both the suburban scenario and the dense urban scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' For convenience, we rephrase the number of times where the absolute value of the normalized pseudorange residuals surpass the critical value as the number of times the RAIM is enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' With the system model considered in this work, we find that the number of times the RAIM is enabled for the GPS-only system is roughly 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='84 % as many as the number of times the RAIM is enabled for the HAPS-aided GPS system in the suburban area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' and the number of times the RAIM is enabled for the GPS-only system is about 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='72 % as many as the number of times the RAIM is enabled for the HAPS-aided GPS system in the dense urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' This demonstrates the applicability of the RAIM algorithm on the HAPS-aided GPS system, especially in the dense urban area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' CONCLUSION HAPS have a number of advantages over satellites, including (but not limited to) lower latency, lower pathloss, smaller pseudorange errors, and HAPS can provide continuous coverage to reduce the number of handovers for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' This makes HAPS an excellent candidate to serve as another type of ranging source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since urban areas are where GNSS positioning performance degrades severely, while also being where most people live, deploying several HAPS as additional ranging sources above metropolitan cities would improve GNSS positioning performance and maximize the profit of the extra payloads on HAPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' From both the simulation and physical experiment results, we observed that HAPS can indeed improve the HDOP, the VDOP, and the 3D positioning accuracy of a legacy GNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' With the system model considered in this work, we showed that the 90-percentile 3D positioning accuracy of the GPS-only system can be improved by around 35 % in both suburban and dense urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We demonstrated the applicability of the RAIM algorithm for the HAPS-aided GPS system, especially in the dense urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To enhance the simulation of the HAPS-aided GPS, the receiver clock offset should be estimated with higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' We think the effectiveness of the RAIM algorithm can be improved if the standard deviation of the target observable is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' To further improve the positioning performance for urban areas, we can make use of terrestrial signals such as cellular network signals and multipath signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' This would constitute a vertical Suburbanarea 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='6 DF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='2 GPS-only system (RAIM off) GPS-only system (RAIM on) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='1 HAPS-aided GPS system (RAIM off) HAPS-aidedGPSsystem (RAIMon) 0 0 5 10 15 20 25 3Dpositioningerror(m)Dense urban area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='6 DF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='2 GPS-only system (RAIM off) GPS-only system (RAIM on) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='1 HAPS-aided GPS system (RAIM off) HAPS-aided GPS system (RAIM on) 0 0 20 40 60 80 100 120 140 160 180 3D positioning error (m)> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < heterogeneous network (V-Het-Net) positioning system, which we believe will yield a lower VDOP based on the DOP illustration presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' ACKNOWLEDGMENT The Skydel software is a formal donation from Orolia to Carleton University.' metadata={'source': 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Alfattani, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Jaafar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Hmamouche, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Yanikomeroglu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Yongacoglu, “Link budget analysis for reconfigurable smart surfaces in aerial platforms,” IEEE Open J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 1980-1995, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' [35] “EVK-M8T user guide,” Ublox, May 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Accessed on: Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' 31, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Available: https://content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='u- blox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='com/sites/default/files/products/documents/EVK- M8T_UserGuide_%28UBX-14041540%29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='pdf > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Hongzhao Zheng (Member, IEEE) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' (Hons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=') degree in engineering physics from the Carleton University, Ottawa, ON, Canada, in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' He is currently a PhD student at Carleton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' His research interest is the urban positioning using sensor-enabled heterogeneous wireless infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Mohamed Atia (Senior Member, IEEE) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' degrees in computer systems from Ain Shams University, Cairo, Egypt, in 2000 and 2006, respectively, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' degree in electrical and computer engineering from Queen’s University, Kingston, ON, Canada, in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' He is currently an Associate Professor with the Department of Systems and Computer Engineering, Carleton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' He is also the Founder and the Director of the Embedded and Multi-Sensory Systems Laboratory (EMSLab), Carleton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' His research interests include sensor fusion, navigation systems, artificial intelligence, and robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Halim Yanikomeroglu (Fellow, IEEE) received the BSc degree in electrical and electronics engineering from the Middle East Technical University, Ankara, Turkey, in 1990, and the MASc degree in electrical engineering (now ECE) and the PhD degree in electrical and computer engineering from the University of Toronto, Canada, in 1992 and 1998, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Since 1998 he has been with the Department of Systems and Computer Engineering at Carleton University, Ottawa, Canada, where he is now a Full Professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' His research interests cover many aspects of wireless communications and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' He has given 110+ invited seminars, keynotes, panel talks, and tutorials in the last five years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Yanikomeroglu’s collaborative research with industry resulted in 39 granted patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Yanikomeroglu is a Fellow of the IEEE, the Engineering Institute of Canada (EIC), and the Canadian Academy of Engineering (CAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' He is a Distinguished Speaker for the IEEE Communications Society and the IEEE Vehicular Technology Society, and an Expert Panelist of the Council of Canadian Academies (CCA|CAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Yanikomeroglu is currently serving as the Chair of the Steering Committee of IEEE’s flagship wireless event, Wireless Communications and Networking Conference (WCNC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' He is also a member of the IEEE ComSoc GIMS, IEEE ComSoc Conference Council, and IEEE PIMRC Steering Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' He served as the General Chair and Technical Program Chair of several IEEE conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' He has also served in the editorial boards of various IEEE periodicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Yanikomeroglu received several awards for his research, teaching, and service, including the IEEE ComSoc Fred W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' Ellersick Prize (2021), IEEE VTS Stuart Meyer Memorial Award (2020), and IEEE ComSoc Wireless Communications TC Recognition Award (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} +page_content=' He received best paper awards at IEEE Competition on Non-Terrestrial Networks for B5G and 6G in 2022 (grand prize), IEEE ICC 2021, IEEE WISEE 2021 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQf2vnp/content/2301.00758v1.pdf'} diff --git a/8dE2T4oBgHgl3EQf8Ait/content/tmp_files/2301.04215v1.pdf.txt b/8dE2T4oBgHgl3EQf8Ait/content/tmp_files/2301.04215v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3b86cd081d18655bcd4fdcde961116286408f61 --- /dev/null +++ b/8dE2T4oBgHgl3EQf8Ait/content/tmp_files/2301.04215v1.pdf.txt @@ -0,0 +1,2767 @@ +arXiv:2301.04215v1 [astro-ph.SR] 10 Jan 2023 +Astronomy & Astrophysics manuscript no. harps_high-mass_binaries +©ESO 2023 +January 12, 2023 +High-mass eclipsing binaries: a testbed for models of interior +structure and evolution⋆ ⋆⋆ +Accurate fundamental properties and surface chemical composition for +V1034 Sco, GL Car, V573 Car and V346 Cen +K. Pavlovski1, J. Southworth2, A. Tkachenko3, T. Van Reeth3, and E. Tamajo4 +1 Department of Physics, Faculty of Science, University of Zagreb, 10 000 Zagreb, Croatia +e-mail: pavlovski@phy.hr +2 Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK +3 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium +4 University of Applied Sciences, 10 410 Velika Gorica, Croatia +January 12, 2023 +ABSTRACT +Aims. The surface chemical compositions of stars are affected by physical processes which bring the products of thermonuclear burn- +ing to the surface. Despite their potential in understanding the structure and evolution of stars, elemental abundances are available for +only a few high-mass binary stars. We aim to enlarge this sample by determining the physical properties and photospheric abundances +for four eclipsing binary systems containing high-mass stars: V1034 Sco, GL Car, V573 Car and V346 Cen. The components have +masses 8–17 M⊙ and effective temperatures from 22 500 to 32 200 K, and are all on the main sequence. +Methods. We present new high-resolution and high signal-to-noise spectroscopy from HARPS, and analyse them using spectral +disentangling and NLTE spectral synthesis. We model existing light curves and new photometry from the TESS satellite. +Results. We measure the stellar masses to 0.6–2.0% precision, radii to 0.8–1.7% precision, effective temperatures to 1.1–1.6% preci- +sion, and abundances of C, N, O, Mg and Si. The abundances are similar to those found in our previous studies of high-mass eclipsing +binaries; our sample now comprises 25 high-mass stars in 13 binary systems. We also find tidally-excited pulsations in V346 Cen. +Conclusions. We reinforce our previous conclusions: interior chemical element transport is not as efficient in binary star components +as in their single-star counterparts in the same mass regime and evolutionary stage, possibly due to the effects of tidal forces. Our ulti- +mate goal is to provide a larger sample of OB-type stars in binaries which would enable a thorough comparison to stellar evolutionary +models, as well as to single high-mass stars. +Key words. stars: fundamental parameters – stars: evolution – binaries: spectroscopic – binaries: eclipsing – stars: abundances +1. Introduction +The interior structure and evolution of a star are largely deter- +mined by its mass and chemical composition at formation. Pre- +cise and accurate observational constraints on these fundamen- +tal physical quantities are required for the validation, calibration +and improvement of theoretical models of the interior structure +and evolution of stars. Despite being much more complex than +single stars, binary star systems are a treasure trove for testing +stellar structure and evolution models and understanding how +these might be improved. In the case of eclipsing binaries (EBs) +where both components are detected spectroscopically, it is pos- +sible to measure their masses and radii with high precision and +accuracy using only orbital mechanics and geometry. Detached +systems are particularly valuable as they are expected to evolve +as single stars without alteration of their evolution by mass trans- +fer episodes. +⋆ Based on observations made with the ESO 3.6 m Telescope and +the HARPS spectrograph, operated on La Silla, Chile by the European +Southern Observatory. +⋆⋆ Table A.1 is only available in electronic form at the CDS +via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via +https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/ +The role of precise empirical mass measurements is difficult +to overstate for validating and calibrating modern and sophisti- +cated stellar models. Herrero et al. (1992) presented a study of +25 luminous galactic OB-type stars and reported a discrepancy +between the masses inferred from their spectra (via wind theory) +and those predicted by evolutionary models. The authors termed +the effect the “mass discrepancy” and emphasised the difficulty +in attributing it to either of the two theories (wind and stellar +evolution) involved. Since then, many attempts have been made +to diagnose the cause of the mass discrepancy in intermediate- +to high-mass stars. +Given the high precision and accuracy that SB2 detached +eclipsing binaries (dEBs) allow us to achieve in measurements of +mass and surface gravity (e.g., Torres et al. 2010), these objects +are important in studying the mass discrepancy. Burkholder et al. +(1997) studied seven early-type spectroscopic binaries with +masses below 15 M⊙ and reported a good agreement between +masses inferred from binary dynamics and those estimated +with evolutionary models in all cases where the stars are non- +interacting. Guinan et al. (2000) and Pavlovski et al. (2009) pre- +sented independent studies of the high-mass SB2 dEB V380 Cyg +and reported a substantial mass discrepancy for the evolved pri- +Article number, page 1 of 25 + +A&A proofs: manuscript no. harps_high-mass_binaries +Table 1: Basic characteristics of binary systems studied in this work. +Binary +Other +Orbital +Vmax +Spectral +Age +Cluster +Apsidal +system +designation +period (d) +(mag) +types +(Myr) +membership +period (yr) +V1034Sco +CPD−41o7742 +2.44 +8.80 +O9.5 V + B1-1.5 V +3–8 +NGC 6231 +23.4 ± 0.8 +GL Car +HD 306168 +2.42 +9.74 +B0.5 V + B1 V +2.0 +NGC 3572 +25.20 ± 0.02 +V573 Car +CPD−59o2628 +1.47 +9.47 +O9.5 V + B0.3 V +2 +Trumpler16 +- +V346 Cen +HD 101837 +6.32 +9.57 +B1/3II/III +10 +Stock 14 +306 ± 4 +Notes. References to the quantities are given in Section 2. +mary component, in the sense that its dynamical mass is too low +compared to the predictions of standard stellar models. Whereas +Guinan et al. (2000) showed that the discrepancy could be re- +solved by introducing extra near-core mixing in the form of con- +vective core overshooting, Pavlovski et al. (2009) found that ro- +tationally induced mixing in models was insufficient to explain +the mass discrepancy. Indeed, Tkachenko et al. (2014b) demon- +strated that only the combined effects of rotation and convec- +tive core overshooting can account for the mass discrepancy ob- +served in V380 Cyg. +Recently, +Massey et al. +(2012), +Morrell et al. +(2014), +Mahy et al. (2015), and Pavlovski et al. (2018) reported sys- +tematic discrepancies between the Keplerian and evolutionary +masses of stars less massive than 30 M⊙. Mahy et al. (2020b) +found a good agreement between the spectroscopic and dynam- +ical masses for 26 early-type binary components whereas their +evolutionary masses appear to be systematically overestimated. +These results hint towards models of interior structure and +evolution being the primary cause of the mass discrepancy. +Tkachenko et al. (2020) and Johnston (2021) demonstrated that +the problem cannot be attributed to differences in observation +and analysis methods between research groups, and instead +showed that the mass discrepancy progressively increases with +the evolutionary stage of the star. In particular, the authors +found that higher convective core masses were required in +models of stellar structure and evolution for stars that are born +with a convective core. The effects of excess core mass can be +efficiently mimicked with an enhanced mixing in the near-core +regions, irrespective of the true cause(s) of the mixing. +Connecting the treatment of interior mixing in stellar evo- +lution models and the mass discrepancy requires extra obser- +vational constraints. Surface chemical composition measure- +ments are ideal because chemical abundance patterns are ex- +pected to be substantially altered by various mechanisms of in- +terior mixing and chemical element transport in stars. For ex- +ample, Heap et al. (2006) found surface nitrogen enrichment in +80% of their sample stars and speculated on the role of rotation +in causing this phenomenon. Overall, these observational find- +ings are in good agreement with predictions from rotating stellar +evolution models for high-mass stars (Meynet & Maeder 2000; +Maeder & Meynet 2000; Heger et al. 2000; Heger & Langer +2000; Langer 2012) with the caveat that the observed nitrogen +enrichments (Heap et al. 2006) are larger than those predicted +by the models. +Hunter et al. (2008, 2009) studied a large sample of +intermediate- to high-mass stars in the Magellanic Clouds. Some +of their findings corroborate the theory of rotationally-induced +mixing while others contradict it (e.g., rapidly and slowly ro- +tating stars without and with substantial surface nitrogen en- +richment, respectively). Slowly-rotating, nitrogen-enriched stars +were also found by Markova et al. (2018) and the authors sug- +gested that inadequacies of the models in these particular cases +might be related to the efficiency of rotational mixing. At the +same time, Pavlovski et al. (2018) presented a detailed study +of the surface chemical compositions in several high-mass SB2 +dEBs and found no dependence of the abundances of carbon (C), +nitrogen (N) and oxygen (O) on either the projected rotational +velocity (v sin i) or surface gravity (log g) of the star. +Whilst changes in the photospheric CNO abundances of +high-mass B-type single stars have been found (Przybilla et al. +2010; Nieva & Przybilla 2012; Maeder et al. 2014; Cazorla et al. +2017a,b; Markova et al. 2018), the role of rotationally-induced +mixing in the formation of these chemical abundance patterns +remains poorly quantified. At the same time, Aerts et al. (2014) +demonstrated that neither v sin i nor the rotational frequency of a +star has significant predictive power for the surface N abundance. +Instead, the latter correlates strongly with the effective tempera- +ture (Teff) of the star and the frequency of its dominant acoustic +oscillation mode. Furthermore, Rogers et al. (2013) showed that +internal gravity waves (IGWs) excited at the convective-radiative +boundary near the core in high-mass stars are efficient in trans- +porting angular momentum and chemicals on short timescales +and over large distances. Pedersen et al. (2018) demonstrated +that the IGW-driven functional form of the interior mixing pro- +file is a good candidate to simultaneously explain the observed +properties of gravity-mode oscillations and surface abundances +in B-type stars. +SB2 dEBs are at the forefront of efforts to resolve deficien- +cies in theoretical stellar models. The masses and radii of the +component stars can be measured precisely and independently +of models, and the requirement for the stars to have the same age +and initial chemical composition at formation provides an addi- +tional stringent constraint on theoretical models. Moreover, mea- +sured masses and radii give a precise surface gravity which can +be used to break the degeneracy between Teff and log g in spec- +tral analysis, boosting the accuracy of measurements of the sur- +face chemical compositions of the stars. The DEBCat1 catalogue +of dEBs (Southworth 2015) currently lists approximately 300 +examples with precisely-measured masses and radii, but only a +small fraction of high-mass systems have useful constraints on +their photospheric chemical abundances (Serenelli et al. 2021). +In this study, we aim to enlarge the sample of high-mass SB2 +dEBs with accurately determined surface chemical abundance +patterns. In Sections 2 and 3, we present the sample and high- +quality spectroscopic data used in this study. Section 4 covers the +determination of the spectroscopic orbits of the stars, Sections 5 +and 6 the inference of their atmospheric parameters and chemical +abundances, and Section 7 the light curve analysis. The chemical +compositions, ages and distances to the binary stars analysed are +compared in Section 8 to the properties of their parent clusters. +We finish with a discussion (Section 9) and conclusions (Sec- +tion 10). It is important to state that the various analyses pre- +1 http://www.astro.keele.ac.uk/jkt/debcat/ +Article number, page 2 of 25 + +Pavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +sented in this work were performed iteratively to ensure internal +consistency in the derived results. +2. Sample +We selected four main-sequence (MS) dEBs for study, based on +the masses of the components, their membership of open clusters +or associations, and on their visibility during the telescope time +we were allocated. Basic information on these targets is given in +Table 1. All eight stars have a spectral type of late-O or early-B +and a surface gravity between 3.7 or 4.2 dex, so are in a rela- +tively early evolutionary phase. All systems except V573 Car re- +side in eccentric orbits. All are confirmed members of open clus- +ters, although we did not impose any additional constraints on +their ages and/or chemical compositions from the cluster mem- +bership. All of our targets except GL Car were included in the +homogeneous sample of Tkachenko et al. (2020). +V1034Sco is located in the core of the open cluster +NGC 6231, which in turn is near the centre of the Sco OB1 asso- +ciation. A detailed spectroscopic and X-ray study was presented +by Sana et al. (2003). Light curves have been presented and anal- +ysed by Sana et al. (2005) and Bouzid et al. (2005). The most re- +cent analysis is that published by Rosu et al. (2022b), who deter- +mined physical properties of the components and measured ap- +sidal motion from our spectra (retrieved from the ESO archive) +and the light curves from Bouzid et al. (2005). +GL Car is a dEB studied by the Copenhagen group +(Giménez & Clausen 1986) using uvby photometry. The system +shows a significant orbital eccentricity (e = 0.157) and fast apsi- +dal motion with U = 25.20±0.02 yr (Giménez & Garcia-Pelayo +1983; Giménez & Clausen 1986; Wolf et al. 2008). We know of +no previous time-series spectroscopy of the system, so the cur- +rent work provides the first measurements of its physical proper- +ties. +V573 Car is one of the brightest stars in very young +open cluster Trumpler16, although its membership is disputed +(Kaltcheva & Georgiev 1993). Its spectroscopic binary nature +was found by Walborn (1982), and the discovery of eclipses was +made by Freyhammer et al. (2001) during a study of the nearby +massive binary system η Carinae. Freyhammer et al. (2001) ob- +tained extensive uvby photometry and, combined with radial ve- +locities (RVs) from Levato et al. (1991), determined the physical +properties of the system. +V346 Cen +contains +early +B-type +components +(Houk & Cowley 1975) in an orbit with a significant eccentric- +ity. Apsidal motion is present with a period of U = 306 ± 4 yr +(Giménez et al. 1986a; Drobek et al. 2013). High-quality light +curves in the Strömgren uvby system were obtained and +analysed by the Copenhagen group (Giménez et al. 1986a,b). +The only full spectroscopic dataset for this system is our own +HARPS data, available through the ESO archive, and which +were already analysed by Mayer et al. (2016). +3. Observations +3.1. Spectroscopy +The spectra presented in this work were all taken in one +observing run2 over the nights 2–7 April 2009 using the +High-Accuracy Radial-velocity Planet Searcher (HARPS) cross- +dispersed échelle spectrograph (Mayor et al. 2003) at the 3.6-m +2 ESO proposal 083.D-0040(A), PI J. Southworth +telescope at ESO La Silla. HARPS achieves extreme RV preci- +sion due to a high mechanical stability, being fed by two opti- +cal fibres, sited in a vacuum chamber, and calibrated by a Th- +Ar emission lamp. We operated HARPS in the high-efficiency +EGGS mode, which has a resolving power of R = 80 000, and +used the second fibre to obtain the sky background during each +observation. Each spectrum consists of 72 orders incident on two +CCDs, covering 3780–6900Å with a gap at 5304–5337Å be- +tween the CCDs. +We reduced the spectra using semi-automatic IRAF3 scripts. +Reduction of the spectra included the standard steps: bias sub- +traction, flat-field correction, spectral order localisation, extrac- +tion, and wavelength calibration. Normalisation of extracted +spectral orders was performed by fitting ninth-order polynomial +functions to selected continuum points in the blaze function. +Since the Balmer lines cover up to three consecutive spectral or- +ders, these were normalised by interpolating the blaze functions +from adjacent orders as described by Kolbas et al. (2015). The +HARPS blaze functions are very stable so the normalisation and +merging of even these difficult orders produced very satisfactory +results. +3.2. Photometry +Our analysis below originally relied on published ground-based +light curves, as will be discussed in Section 7. In the course +of this work, additional data became available from the NASA +Transiting Exoplanet Survey Satellite (TESS), a space-based +mission that has observed most of the celestial sphere in sec- +tors of 27.4 d duration (Ricker et al. 2015). The TESS datasets +used in the current study are shown in Fig. 1. Additional data +are shown in Fig. A.1, and may be useful in future for period or +apsidal motion studies. +TESS observed V1034 Sco in sectors 12 (1800 s cadence) +and 39 (600 s cadence). We extracted the light curves using cus- +tom aperture masks. V1034 Sco is in a crowded field and the +TESS pixels subtend a large angle (21′′) so the light curves con- +tain a significant amount of third light. Our analysis was based +on sector 39 due to the better temporal sampling. +V346 Cen was observed using TESS in sectors 10 and 11 +(1800 s cadence), and 37 and 38 (600 s cadence). The light +curves available for download from MAST4 are very affected +by the field crowding but are nevertheless much better than the +ground-based data for this object. We based our analysis on the +data from sectors 37 and 38. +TESS observed V573 Car in sectors 10, 36 and 37. Because +this object is very close to the extremely bright η Car binary sys- +tem, the standard data products from TESS (Jenkins et al. 2016) +are unreliable. We therefore extracted photometry from the halo +of V573 Car by making a careful customised pixel selection for +the aperture mask, with the aim to maximise the collected flux +of V573 Car compared to the flux of η Car. The resulting light +curves are of relatively low quality and suffer from a large and +varying amount of third light, so we did not use these in our anal- +ysis. We note that V573 Car was just outside the field of view of +TESS during sector 11 but we were still able to extract a light +curve using halo photometry. +GL Car was observed by TESS in sectors 10 and 11 (1800 s +cadence), and 37 (600 s cadence). The light curves available on +3 IRAF is distributed by the National Optical Astronomy Observatory, +which are operated by the Association of the Universities for Research +in Astronomy, Inc., under cooperative agreement with the NSF. +4 https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html +Article number, page 3 of 25 + +A&A proofs: manuscript no. harps_high-mass_binaries +Fig. 1: Light curves used in the current study from our reduction of data from the TESS satellite. They have been normalised to +zero magnitude for display purposes. +MAST have eclipses that are too deep so we again extracted our +own photometry from the TESS full-frame images using custom +aperture masks. +4. Spectroscopic orbits +The spectra of binary systems containing high-mass stars are dif- +ficult to analyse for several reasons. First, the v sin i values are +typically large, smearing out the spectral lines and causing the +lines from the two components to blend together even around the +phases of maximum RV difference. Second, there are relatively +few spectral lines that are strong enough to provide useful RV in- +formation. We therefore determined the spectroscopic orbits of +the stars using the method of spectral disentangling (SPD). This +method was introduced by Simon & Sturm (1994) in wavelength +space and by Hadrava (1995) in Fourier space. It represents the +observed composite spectra of a binary system as a sum of the in- +dividual spectra of the two stars shifted in RV according to their +orbital motion. SPD makes it possible to quantitatively anal- +yse time-series spectra of SB2 systems even when line blending +is strong (Hensberge et al. 2000; Pavlovski & Hensberge 2005). +Note that no template spectra are needed for SPD, thus avoiding +any biases due to template mismatch (Hensberge & Pavlovski +2007). +We used the FDBinary5 code (Ilijic et al. 2004) to perform +SPD in Fourier space using Fast Fourier Transform (FFT). For +each object we analysed all spectra simultaneously to determine +the disentangled spectra of the two stars and their spectroscopic +orbital parameters. We fitted directly for the orbital parame- +ters, without the intermediate step of calculating RVs. The or- +bital parameters were the orbital period, P, time of periastron +pasage, Tperi, eccentricity, e, argument of periastron, ω, and ve- +locity semiamplitudes, KA and KB. The orbital periods were held +fixed as they are well determined from previous analyses. The +orbital solutions are given in Table 2 in which the mass ratio +(q = KA/KB) is also given. +We also disentangled individual short segments of spectra in +order to concentrate on spectral lines of interest, avoid interstel- +lar lines, and achieve reasonable computation times. The Balmer +lines were not used in the determination of the spectroscopic +orbits because they are much wider than the changes in RV of +the stars over an orbital cycle. The best fits were obtained us- +ing the downhill simplex algorithm (Press et al. 1992). We found +100 runs with 1000 iterations each to be sufficient to ensure the +5 http://sail.zpf.fer.hr/fd3 +Article number, page 4 of 25 + +Pavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +Fig. 2: Visualisation of the spectroscopic orbits of our targets. The best-fitting orbits are shown with black lines and the RVs of the +stars at the times of observation with red symbols for the primary component, and blue symbols for the secondary component. Note +that these are not measured RVs, hence the uncertainties in RVs are not assigned to individual symbols, because we calculate orbital +parameters directly from all observed spectra for each system (see Section 4). +Table 2: Parameters of the spectroscopic orbits for the four targets determined by SPD. +Binary +P +Tperi +e +ω +KA +KB +q +system +(d) +(BJD) +(deg) +(kms1) +(kms1) +V1034Sco +2.440656 +51934.356±0.032 +0.029 ± 0.003 +191 ± 12 +168.3 ± 0.3 +299.0 ± 1.1 +0.563 ± 0.002 +GL Car +2.422238 +54901.182±0.015 +0.146 (fixed) +32.2 ± 2.7 +244.6 ± 1.8 +259.3 ± 1.6 +0.943 ± 0.009 +V573 Car +1.469332 +- +0.0 +90 +250.61 ± 0.71 +306.3 ± 1.1 +0.818 ± 0.011 +V346 Cen +6.321835 +50452.543±0.016 +0.289 ± 0.006 +22.2 ± 1.3 +135.3 ± 0.6 +190.1 ± 0.7 +0.712 ± 0.038 +global minimum was found whilst keeping the required com- +putation time manageable. Convergence was achieved quickly +because of the high quality of the HARPS spectra and the avail- +ability of preliminary orbital parameters from the literature. Un- +certainties in the results were obtained using 10 000 bootstrap- +ping simulations (Pavlovski et al. 2018). Fig. 2 is a visualisation +of the spectroscopic orbits of the four targets and the phase dis- +tribution of our spectra. +4.1. V1034 Sco +V1034 Sco has been found to have a small eccentricity +(Hill et al. 1974; Levato & Morrell 1983). We have been able +to measure precise velocity amplitudes for the components (Ta- +ble 2) which highlight the low mass ratio of the system. +Our results are in good agreement with those from Sana et al. +(2003, 2005) and agree within the errorbars with those from +Rosu et al. (2022b). We conclude that the RV semi-amplitudes of +the components of V1034Sco are now well-determined since the +the accuracy achieved is about 0.2% for the primary and 0.4% for +the secondary. The argument of periastron is quite uncertain due +to the small eccentricity, and is much better determined from the +photometric analysis in Section 7. +Article number, page 5 of 25 + +A&A proofs: manuscript no. harps_high-mass_binaries +4.2. GL Car +To the best of our knowledge, our spectroscopic orbit for GL Car +is the first one published. The mass ratio is in fairly good agree- +ment with the photometric value of q = 0.943 ± 0.009 found by +Giménez & Clausen (1986). We fixed the eccentricity to a value +of 0.146, which is precisely known from analyses of its apsidal +motion (Wolf et al. 2008). We fitted for the argument of perias- +tron, which is well-determined when the eccentricity is fixed. +4.3. V537 Car +This is the only binary system with a circular orbit in our sample. +The HARPS spectra densely cover both quadratures. We also +obtained spectra during the primary and secondary minimum, +but did not use these in our analysis because the eclipses are not +total. +A spectroscopic orbit for V573 Car has previously been +published by Freyhammer et al. (2001) but based on only +two newly obtained spectra and eight spectra taken from +Levato & Malaroda (1982). The velocity amplitudes measured +by these authors are quite uncertain but agree with ours to within +the errorbars. +4.4. V346 Cen +V346 Cen has a significant eccentricity of e = 0.289 ± 0.006. +Our HARPS spectra have good phase coverage and SPD quickly +converged to a stable solution (Table 2). Our results are in rea- +sonable agreement with the only previous spectroscopic analysis +of this system (Mayer et al. 2016), as expected because they used +the same spectra. +However, the velocity amplitudes we measured are both +1.5 km s−1 lower than those of Mayer et al. (2016). We attribute +this to differences in the methods employed in the two analyses. +In particular, Mayer et al. (2016) employed cross-correlation +(Zucker & Mazeh 1994) to determine RVs, using as templates +the disentangled spectra themselves. This approach is mathemat- +ically incorrect and extensive numerical experiments have shown +that it is not reliable (Ilijic et al. 2001); SPD has instead been +shown to be the best approach to determining spectroscopic or- +bits (Southworth & Clausen 2007). Our approach yields masses +that are smaller by 0.26 M⊙ and 0.11 M⊙ than those found by +Mayer et al. (2016) for the primary and secondary component of +the system, respectively, which is larger than the quoted uncer- +tainties. +5. Atmospheric parameters +For determination of the atmospheric parameters and individ- +ual abundances of C, N, O, Mg and Si, we employed a hy- +brid NLTE approach as described in detail in Nieva & Przybilla +(2007, 2012). A hybrid NLTE approach means that the mod- +elling combines hydrostatic, plane-parallel, and line-blanketed +model atmospheres in local thermodynamic equilibrium (LTE) +with line formation calculated in NLTE. We used the Atlas9 +code (Kurucz 1979; Castelli & Kurucz 2003) for the calculations +of model atmospheres. Then emergent fluxes and line profiles +were calculated with the codes Detail and Surface (Giddings +1980; Butler & Giddings 1985). In Detail the coupled radia- +tive transfer and statistical equilibrium equations are solved, +while Surface was used for the calculations of NLTE syn- +thetic spectra. The following model atoms were used in these +calculations: H i (Przybilla & Butler 2004), He i/ii (Przybilla +Table 3: The atmospheric parameters derived from optimal fit- +ting of disentangled spectra of the components to a grid of NLTE +spectra. +Star +Teff +v sin i +ξt +(K) +(km s−1) +(km s−1) +V1034 Sco A +32 200 ± 500 +169.8 ± 2.6 +5 ± 1 +V1034 Sco B +25 800 ± 300 +94.5 ± 3.3 +5 ± 1 +GL Car A +30 950 ± 500 +180.1 ± 2.2 +4 ± 1 +GL Car B +30 400 ± 500 +134.6 ± 3.5 +2 ± 1 +V573 Car A +31 900 ± 400 +184.6 ± 2.7 +5 ± 1 +V573 Car B +28 700 ± 350 +155.4 ± 3.1 +3 ± 1 +V346 Cen A +26 100 ± 300 +165.2 ± 2.8 +5 ± 1 +V346 Cen B +22 500 ± 300 +89.1 ± 2.3 +5 ± 1 +2005), C ii/iii (Nieva & Przybilla 2006), N ii (Przybilla & Butler +2001), O i/ii (Becker & Butler 1988; Przybilla et al. 2000), Mg ii +(Przybilla et al. 2001), and Si ii/iii/iv (Becker & Butler 1990). +We used the disentangled spectra generated in the previous +section to determine the Teff, v sin i, and microturbulent veloc- +ity (ξt) for each of the eight stars in our sample. This process +was greatly helped by the availability of log g values from the +measured masses and radii (see Section 7) so our analysis was +performed iteratively. The disentangled spectra were still in the +common continuum of the binary system so needed to be renor- +malised to the continuum of the individual component stars. This +was done iteratively alongside the light curve analysis, to arrive +at light ratios that were consistent between the two types of the +analysis (Ilijic et al. 2004; Pavlovski & Hensberge 2005). +The exception to the process above was GL Car, for which +the light curve solutions suffered from a degeneracy which +caused the light ratio to be highly uncertain. We therefore fitted +the disentangled spectra to obtain the Teff, v sin i and log g values +and the light ratio directly, using the approach of Tamajo et al. +(2011) and Kolbas et al. (2015). After iteration with the light +curve solution, log g was fixed for the final measurements of the +remaining parameters. We have found that such spectroscopi- +cally determined light ratios can be competitive with those from +light curve analysis (Pavlovski et al. 2009, 2018, 2022). +Since we are dealing with late-O, and early-B type stars, the +helium ionisation balance (He i/He ii) is a sensitive indicator of +Teff. Our spectra cover a broad spectral range and thus allowed +us to use a large number of lines: 4009, 4026, 4388, 4437, 4471, +4713, 4921, 5015, 5047, 5875 and 6678 Å for He i and 4200, +4541, 4686, and 5411 Å for He ii. Once a first set of parameters +was obtained, we made the light ratio a free parameter to check +its reliability. As a further check we also fitted the Hδ, Hγ and Hβ +lines, during which we excluded wavelengths affected by inter- +stellar absorption (specifically the red wing of Hβ). We did not +base our Teff measurements on the Balmer lines because their +large width makes them susceptible to errors due to continuum +normalisation. +The He line strengths also depend on ξt, which for hot stars +can be obtained by minimising the scatter in the O abundances. +We started with the assumptions of solar He abundance and +ξt = 2 km s−1, and subsequently relaxed each of them before +refitting. Convergence was fast, taking either one or two itera- +tions for all eight stars. Once this was achieved, we repeated our +optimal fitting of the disentangled spectra described above. The +results of this process are given in Table 3. Below we compare +our results to published determinations for each system, except +Article number, page 6 of 25 + +Pavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +Fig. 3: Fits to the He i 4388Å and He ii 4541 Å lines. The ionisation balance of He i and He ii was used in the determination of Teff +for the stars. The blue data are the disentangled spectra and the red lines the best fits. The upper row is for the primary stars and the +lower row is for the secondary stars. The lower S/N for the secondary components arises because they are fainter than the primary +components. The absence of He ii 4541 Å absorption in V1034 Sco B and V346 Cen B is obvious and indicates that Teff < 23 000K. +for GL Car for which there is no other analysis based on modern +spectroscopy. +5.1. V1034 Sco +In the most recent study, Rosu et al. (2022b) analysed disen- +tangled spectra of the components obtained from the HARPS +spectra obtained in our observing run, and available at the ESO +archive. The Teffs they derived are within 1σ uncertainty of our +results. This is encouraging, especially as Rosu et al. (2022b) +used a different NLTE spectrum synthesis code to us. +Rosu et al. (2022b) also fitted for surface gravity, using the +wings of the Balmer and some He i lines, whereas we prefer +the surface gravities determined with a high precision from the +masses and radii. The two analyses agree to within 2σ, but the +uncertainties of the values from Rosu et al. (2022b) are much +larger (±0.10 dex) than our own (±0.01 dex). +5.2. V573 Car +Freyhammer et al. (2001) studied V573 Car using two high- +resolution spectra from the FEROS spectrograph, taken near op- +posite quadratures. They fitted the spectra with NLTE synthetic +spectra for the He i, He ii, Hδ and Hγ lines. The helium lines of +the components are not completely resolved at quadrature due to +the high v sin i, and the Balmer lines are not resolved. +The agreement between their and our Teff measurements is +well within the 1σ errorbars for component A, but only within +2σ for component B. We attribute this to the very small num- +ber of spectra available to Freyhammer et al. (2001) compared +to our own extensive dataset. We also find that fitting disentan- +gled spectra is superior to fitting individual observed spectra be- +cause it avoids problems with blending of lines from the two +stars. Moreover, disentangled spectra have a higher S/N than in- +dividual observed spectra. +5.3. V346 Cen +The atmospheric parameters for both components of V346 Cen +were determined by Mayer et al. (2016) from the same set of the +HARPS spectra as we used. In their analysis, Mayer et al. (2016) +used optimal fitting of disentangled spectra in similar manner as +we did, for fixed surface gravities and microturbulent velocities +(fixed to ξt = 2 km s−1) and using a grid of synthetic spectra from +Lanz & Hubeny (2007). They found a large discrepancy for the +secondary: their spectroscopic analysis gave 20 991 ± 190 K and +Article number, page 7 of 25 + +1.1 +He I 4541 +He I 4388 +lux +ormalised +0.9 +0.8 +V346 Cen +0.P +6 +-4 +-2 +0 +2 +4 +Relative +wavelength +A1.1 +- He I 4541 +He I 4388 +ormalis +0.9 +V573 + Car B +0.8 +6 +-4 +-2 +0 +2 +4 +Relative +wavelength1.1 +- He I 4541 +ux +He I 4388 +p +ormalis +0.9 +GL Car B +0.8 +6 +-4 +-2 +0 +2 +4 +Relative +wavelength1.1 +He I 4541 +He I 4388 +1lx +ormalised +0.9 +0.8 +V1034 Sc0 B +6 +-4 +0 +2 +4 +Relative +wavelength1.1 +- He I 4541 +He I 4388 +lux +ormalised +0.9 +0.8 +V346 Cen A +0.P +6 +-4 +-2 +0 +2 +4 +Relative +wavelength +A1.1 +- He I 4541 +ux +He I 4388 +pi +ormalis +0.9 +V573 +Car A +0.8 +6 +-4 +-2 +0 +2 +4 +Relative +wavelength1.1 +- He I 4541 +He I 4388 +pa +ormalis +0.9 +GL Car A +0.8 +6 +-4 +-2 +0 +2 +4 +Relative +wavelength1.1 +-He I 4541 +He I 4388 +xni +ormalised +0.9 +0.8 +V1034 Sc0 A +0.P +6 +-4 +-2 +0 +2 +4 +Relative +wavelength +AA&A proofs: manuscript no. harps_high-mass_binaries +Table 4: Abundances determined for the stars in our sample of binary systems. +Star +log ǫ(C) +log ǫ(N) +log ǫ(O) +[N/C] +[N/O] +log ǫ(Mg) +log ǫ(Si) +V1034 Sco A +8.39 ± 0.12 +7.71 ± 0.12 +8.76 ± 0.07 +−0.68 ± 0.17 +−1.05 ± 0.14 +7.67 ± 0.14 +7.56 ± 0.01 +V1034 Sco B +8.27 ± 0.05 +7.67 ± 0.08 +8.69 ± 0.12 +−0.57 ± 0.09 +−1.02 ± 0.14 +7.45 ± 0.07 +7.46 ± 0.14 +GL Car A +8.18 ± 0.08 +7.69 ± 0.14 +8.74 ± 0.12 +−0.67 ± 0.16 +−1.04 ± 0.18 +7.52 ± 0.12 +7.50 ± 0.12 +GL Car B +8.21 ± 0.12 +7.72 ± 0.11 +8.76 ± 0.12 +−0.52 ± 0.16 +−0.85 ± 0.16 +7.50 ± 0.13 +7.44 ± 0.14 +V573 Car A +8.30 ± 0.08 +7.63 ± 0.10 +8.67 ± 0.05 +−0.57 ± 0.12 +−0.52 ± 0.11 +7.58 ± 0.08 +7.57 ± 0.12 +V573 Car B +8.28 ± 0.05 +7.76 ± 0.07 +8.61 ± 0.04 +−0.55 ± 0.09 +−0.94 ± 0.08 +7.45 ± 0.05 +7.54 ± 0.13 +V346 Cen A +8.13 ± 0.05 +7.68 ± 0.05 +8.70 ± 0.04 +−0.45 ± 0.07 +−1.02 ± 0.06 +7.70 ± 0.13 +7.45 ± 0.16 +V346 Cen B +8.33 ± 0.06 +7.72 ± 0.09 +8.80 ± 0.07 +−0.61 ± 0.11 +−1.08 ± 0.11 +7.40 ± 0.14 +7.35 ± 0.17 +This work +8.27 ± 0.08 +7.69 ± 0.05 +8.70 ± 0.06 +−0.58 ± 0.07 +−1.01 ± 0.07 +7.59 ± 0.14 +7.49 ± 0.08 +OB binariesa +8.25 ± 0.07 +7.69 ± 0.06 +8.71 ± 0.05 +−0.56 ± 0.08 +−1.02 ± 0.07 +7.56 ± 0.12 +7.45 ± 0.09 +B starsb +8.33 ± 0.04 +7.79 ± 0.04 +8.76 ± 0.05 +−0.54 ± 0.06 +−0.97 ± 0.06 +7.56 ± 0.05 +7.50 ± 0.05 +Notes. The Teff and log g values used for the construction of the model atmospheres are given in Tables 3 and 7, respectively. +(a) The abundances found for OB binaries in our previous work (Pavlovski et al. 2018) (b) The ‘present-day cosmic abundances’ for B stars +(Nieva & Przybilla 2012) +Fig. 4: Example of fits to N lines for our target stars. In this +case the N ii 3995 Å line is shown for V1034 Sco (primary star +on the left, secondary star on the right). The blue lines show +the renormalised disentangled spectra of the stars. The red lines +show synthetic spectra from our precalculated grid for three dif- +ferent abundances (labelled on the bottom right corner in each +panel). +their light curve analysis gave 25 376 ± 18 K. The former value +is in much better agreement with our result (Table 3), and the +complete absence of the He ii lines demands Teff < 23 000 K +(based on a detailed examination of theoretical spectra for the +He ii 4686 Å line). Mayer et al. (2016) did not discuss the He ii +lines in the secondary’s spectrum at all. They also gave unre- +alistically small uncertainties for Teff: ±25 K for the primary, +±190 K for the secondary from spectroscopy, and ±18 K for the +secondary from the light curve analysis. Such low uncertainties +are typical for formal errors of the fitting algorithms, but are un- +realistic. +6. Abundance analysis +With the Teff, ξt and v sin i from Section 5, and log g from the +masses and radii of the stars (Section 7), we have all quanti- +ties needed for determining surface abundances. We calculated +model atmospheres for the Teff and log g values of the compo- +nents with the atlas9 code. Then a grid of synthetic spectra +was calculated in NLTE with detail, and surface. The follow- +Fig. 5: Same as Fig. 4 but for O lines in the components of +GL Car. The complex blend of O ii lines at 4941 and 4943 and +the O ii line at 4955 Å are shown. +ing species were considered: C, N, O, Mg and Si. Spectra for a +broad range of elemental abundances were calculated, spanning +±0.05 dex in steps of 0.05 dex, around the ‘present-day cosmic +abundances’ determined in Nieva & Przybilla (2012) (log ǫ(C) = +8.25, log ǫ(N) = 7.69, log ǫ(O) = 8.71, log ǫ(Mg) = 7.56, and +log ǫ(Si) = 8.45). These were broadened by the instrumental +broadening, and a rotational kernel. The microturbulent velocity +was taken into account in the line profile calculations with sur- +face as determined from minimising the scatter in the O abun- +dances and given in Table 3. Abundances were determined by +minimising the residuals (χ2 criterion) between the renormalised +disentangled spectrum and the synthetic spectrum. In the renor- +malisation of the disentangled spectra to their individual contin- +uum, the light ratio obtained in the light curve analysis (Table 6) +Article number, page 8 of 25 + +ux +ormalised +66'0 +0.98 +8.60 +0.97 +8.70 +GL Car B +8.80 +0.96 +4930 +4940 +4950 +4960 +Wavelength +[Axn +ormalised +66'0 +0.98 +8.60 +0.97 +8.70 +G +tar +8.80 +0.96 +4930 +4940 +4950 +4960X +0.98 +pasi +0.96 +ormal +0.94 +0.92 +7.60 +7.70 +V1034 Sc0 +7.80 +0.9 +3992 +3994 +3996 +3998 +Wavelength +Aux +.995 +p含 +ormal +66'0 +0.985 +7.60 +7.70 +V1034 +Sco +7.80 +0.98 +3992 +3994 +3996 +3998 +Wavelength. +FAPavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +Fig. 6: +Same as Fig. 4 but for lines in the components of +V573 Car. The upper two panels show O ii lines at 4661, 4673 +and 4676 Å (shown with green lines), which are blended with +C ii lines at 4659, 4663, 4665 and 4673 Å (shown with red lines). +The calculated synthetic spectrum is shown using a black line. +The lower two panels show the N ii 3995 Å lines. +was used, except for GL Car where the spectroscopically deter- +mined light ratio was employed. +The number of lines available for the abundance determina- +tion of a particular element varies due primarily to Teff. For the +Teff range covered by our target stars, the spectral lines of CNO +are quite varied. The most numerous spectral lines are for O, +which is why we used them to determine ξt. The spectral lines +of C are the least numerous, and the broad wavelength coverage +of HARPS spectra is of vital importance. We show examples of +the disentangled and synthetic spectra in Figs. 4 to 7 for selected +CNO lines. +The results for all five elements are given in Table 4, as well +as the indices [N/C] and [N/O]. Uncertainties were calculated +including the standard deviation of the mean for available spec- +tral lines, and the uncertainty due to uncertainties in Teff and ξt. +Fig. 7: Same as Fig. 4 but for V346 Cen. The upper two panels +show the C ii lines at 5133–5151Å, and the bottom two panels +show the N ii 4630.5Å line for the two components. +Uncertainty in the abundances due to uncertainties in the surface +gravity are negligible since log g is determined to high precision +from the masses and radii. +A fairly good agreement between the abundances in both +components of the same binary system is seen from examina- +tion of Table 4. The most notable difference is for the abundance +of magnesium (Mg) which in three cases (V1034 Sco, V573 Car +and V346 Cen) is modestly larger than the uncertainties. The +Mg abundances are inferred from a single line, Mg ii 4481 Å, +so their uncertainties are larger than for other species. For other +elements, the observed abundance differences are mostly below +0.1 dex, well within the 1σ uncertainty interval. Apart from +Mg, the largest deviations are for the C abundance in V346 Cen +(log ǫ(C)A − log ǫ(C)B = −0.20 ± 0.08 dex), the N abundance +in V573 Car (−0.13 ± 0.12), and the O abundance in V346 Cen +(−0.10 ± 0.08). V346 Cen A has the lowest C abundance among +the eight stars, with log ǫ(C) = 8.13 ± 0.05, almost 0.20 dex +less than the mean C abundance. Contrary to this, the N abun- +dance for the same star is normal. It is also worth noting that +V346 Cen A is the most evolved in our sample of eight OB stars. +Article number, page 9 of 25 + +Iux +ised +0.98 +ormali +0.96 +7.70 +7.80 +7.90 +V346 +Cen B +0.94 +4626 +4628 +4630 +4632 +4634 +4636 +Wavelengthux +0.99 +ormalised +0.98 +7.70 +0.97 +7.80 +V346 +Cen A +7.90 +4626 +4628 +4630 +4632 +4634 +4636 +Wavelength +Apast +0.98 +ormal +8.00 +0.96 +8.20 +V346 Cen B +8.40 +5130 +5135 +5140 +5145 +5150 +5155 +Wavelength +[AX +0.995 +ma. +0.99 +0.985 +8.10 +8.20 +V346 +0.98 +Cen +8.30 +5130 +5135 +5140 +5145 +5150 +5156xnl +ormalised +0.98 +0.96 +7.70 +7.80 +7.90 +0668 +3992 +3994 +3996 +3998 +4000 +Wavelengthxn +ormalised +0.99 +0.98 +7.70 +7.80 +V573 +Car +7.90 +0668 +3992 +3994 +3996 +3998 +4000 +Wavelength +Apa +ormalis +0.98 +0.96 +C +V573 +Car B +0.94 +0 +4660 +4665 +4670 +4675 +4680 +Wavelengthxn +1 +pa +STT +0.98 +ormal +0.96 +C +0.94 +:ar +0 +4660 +4665 +4670 +4675 +4680A&A proofs: manuscript no. harps_high-mass_binaries +Fig. 8: Individual abundances of carbon (left), nitrogen (middle), and oxygen (right) as a function of surface gravity. The stars in the +present sample are indicated with filled blue circles while stars taken from our previous studies are shown with open blue circles. +The surface gravity (obtained from the binary solution) is used as a proxy for stellar evolution. Solid red lines show theoretical +evolutionary tracks for a 15 M⊙ star and three values of the initial rotational velocity Ω/Ωcrit = 0.1, 0.3, and 0.5 (Georgy et al. +2013). The cosmic standard abundance values of Nieva & Przybilla (2012) are indicated with horizontal dashed lines. +Fig. 9: Same as in Fig. 8 but for the [N/C] and [N/O] abundance +indices. +Fig. 8 compares individual CNO abundances determined +for the eight stars in this work to our previous abundance +measurements in high-mass binaries. The new determinations +are shown in solid blue circles, whilst our previous results +are represented with open blue circles. Our previous deter- +minations are for 17 high-mass stars in nine binary systems: +V578 Mon (Pavlovski & Hensberge 2005; Garcia et al. 2014; +Pavlovski et al. +2018), +V453 Cyg +(Pavlovski & Southworth +2009; Pavlovski et al. 2018), V380 Cyg (Pavlovski et al. 2009; +Table 5: Comparison of abundances determined for targets in +present work to their parent clusters. +NGC 6231 +Element +Kilian et al. +Mathys et al. +This work +(1994) +(2002) +V1034 Sco +log ǫ(C) +8.37±0.05 +8.29±0.17 +8.33±0.08 +log ǫ(N) +7.85±0.05 +7.85±0.10 +7.69±0.08 +log ǫ(O) +8.61±0.05 +8.30±0.42 +8.73±0.12 +log ǫ(Mg) +7.39±0.04 +– +7.56±0.11 +log ǫ(Si) +7.15±0.06 +– +7.51±0.10 +[N/C] +−0.52±0.07 +−0.44±0.20 +−0.64±0.11 +[N/O] +−0.76±0.07 +−0.45±0.40 +−1.04±0.14 +NGC 3293 +Element +Hunter et al. +Morel et al. +V573 Car +(2009) +(2022) +GL Car +log ǫ(C) +7.97±0.19 +8.13±0.16 +8.24±0.17 +log ǫ(N) +7.60±0.15 +7.72±0.14 +7.70±0.27 +log ǫ(O) +8.65±0.17 +– +8.70±0.18 +log ǫ(Mg) +7.22±0.16 +7.45±0.18 +7.50±0.20 +log ǫ(Si) +7.42±0.09 +7.56±0.25 +7.51±0.26 +[N/C] +−0.37±0.21 +−0.40±0.21 +−0.54±0.27 +[N/O] +−1.05±0.26 +– +−1.00±0.11 +Notes. V1034 Sco is member of the open cluster NGC 6231 for +which abundance analyses were published by Kilian et al. (1994) and +Mathys et al. (2002). For the open clusters NGC 3572 and Trumpler 16, +parent clusters of GL Car and V573 Car, respectively, no abundance +studies are available. We used abundance studies of the open clus- +ter NGC 3293, since it is part of the large Car OB1 complex, as are +NGC 3572 and Trumpler 16. +Tkachenko et al. +2014b), +σ Sco +(Tkachenko et al. +2014a), +α Vir (Tkachenko et al. 2016), CW Cep (Johnston et al. 2019), +AH Cep (Pavlovski et al. 2018), V478 Cyg (Pavlovski et al. +2018) and the primary component in V621 Per (Southworth et +al. in prep.). There are no discernable systematics between the +new and previous sample. This is expected because we are con- +sistently using the same reduction and analysis tools, so the 25 +stars (in 13 binary systems) represent an homogeneous sample. +Article number, page 10 of 25 + +[N/O] +0.5 +[dex +-0.5 +0.3 +0 +4 +3.5 +3 +log g0.5 +[N/C] +0 +xap +0.3 +-0.5 +4 +3.5 +3 +log g +[dexPavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +In the last three rows of Table 4 the mean values for abundances +in the present sample, in OB binaries studied previously by us, +and the ‘present cosmic abundance standard’ – an abundance +pattern evaluated for B-type stars by Nieva & Przybilla (2012) +– are given for comparison. As already mentioned, both our +samples are in perfect agreement and there are no outliers. +However, it can be seen that the CNO abundances for OB stars +in binary systems are below the cosmic abundance standard, +with very few exceptions. Nieva & Przybilla (2012) determined +elemental abundances for sample of sharp-lined early B-type +stars, enabling a very high accuracy. This was not an option +for our work because our sample stars are all in short-period +binary systems so are either moderate or fast rotators. Thus the +spectral lines are usually broad and overlapping, making the +choice of suitable spectral lines for abundance determination +more limited, and thus affecting the accuracy of the results. +Fig. 8 presents a comparison of the inferred CNO abun- +dances with theoretical evolutionary tracks of a 15 M⊙ star com- +puted for three values of the initial rotational velocity Ω/Ωcrit = +0.1, 0.3 and 0.5 (Georgy et al. 2013). In their model calcula- +tions Georgy et al. (2013) used the following initial abundances: +log ǫ(C) = 8.28, log ǫ(N) = 8.67, log ǫ(O) = 8.55. The values for +the abundances of C and N are in fair agreement with our present +and preious findings (c.f. Table 5, but differ by 0.15 dex for the +O abundance. Therefore, we empirically ‘corrected’ the initial O +abundance in the theoretical models, and shift the O abundance +upwards in Fig. 8, and accordingly for [N/O] in Fig. 9. +One can see that the models predict significant depletion and +enhancement of C and N, respectively, as the rotation rate of the +star increases, while only a marginal depletion is predicted for +O. Moreover, these abundance trends are substantial at the start +of the main-sequence evolution already. However, the individual +abundances of CNO elements measured by us do not follow the +relations predicted by the models: instead we observe a scatter of +values around or slightly below the cosmic standard abundance +values of Nieva & Przybilla (2012). +Furthermore, the N to C abundance ratio index [N/C] is a +sensitive probe of the stellar evolution model predictions, as +can be seen in Fig. 9. The models suggest a noticeable in- +crease of the surface N abundance with respect to the abun- +dance of C (top panel) and O (bottom panel) as the rotation +rate of the star Ω/Ωcrit increases. Similar to the individual el- +emental abundances discussed above, we do not observe the in- +crease in the [N/C] and [N/O] indices as the surface gravity of the +star decreases. Moreover, the bulk of our abundance measure- +ments cluster around the mean [N/C] and [N/O] values found by +Nieva & Przybilla (2012) in the solar neighbourhood, with the +spread being significantly smaller than one would expect if the +abundance ratios were altered substantially by the effect of stel- +lar rotation. +No previous abundance determinations are available for any +of the binary systems analysed in this work. However, we can +check our results against published abundances for the open +clusters our sample are members of. Photospheric abundances +for B-type stars in the open cluster NGC 6231 were determined +by Kilian et al. (1994) and Mathys et al. (2002). Results from +these studies are compared to our results for V1034Sco in Ta- +ble 5. The parent clusters for the other three systems have not +(yet) been subject to a chemical composition study, but the open +cluster NGC 3293 (which is part of young association Car OB1 +Turner et al. 1980) is well studied. Hunter et al. (2009) deter- +mined abundances from 50 B-type stars, while in a recent publi- +cation Morel et al. (2022) examined a large sample of about 150 +B-type stars in the framework of the Gaia-ESO Survey. Since the +dEBs V573 Car and GL Car belongs to the Car OB1 association, +we use NGC 3293 as a proxy for the abundance pattern in Car +OB1. It is interesting that the massive sample of B-type stars +analysed in Morel et al. (2022), with a spread in v sin i values, +show a pattern of under-abundances compared to the standard +solar abundances (Asplund et al. 2009), in accordance with our +general abundance pattern. +7. Light curve analysis +We assembled the available light curves of the four targets in this +work and modelled them using a consistent approach in order to +determine their physical properties. The light curves were fit- +ted using the Wilson-Devinney (WD) code (Wilson & Devinney +1971; Wilson 1979), which implements Roche geometry to +determine the shapes of the stars and thus the brightness of +binary systems as a function of orbital phase. We used the +2004 version of the WD code, driven with the jktwd wrapper +(Southworth et al. 2011). +For each system we performed a series of tests to determine +the best approach to modelling it with jktwd. Once we had ar- +rived at the preferred solution, we performed further tests to +determine the range of plausible solutions and thus the uncer- +tainties in the fitted parameters. This step was taken because we +have consistently found that the formal errorbars calculated by +the WD code underestimate the true uncertainty of the fitted pa- +rameters (Pavlovski & Southworth 2009; Pavlovski et al. 2009, +2018; Southworth et al. 2020), as indicated in the user guide to +the code (Wilson & Van Hamme 2004). +Unless otherwise specified we used Mode 0 in the WD code, +which is for detached binary systems where the light contribu- +tions for each star are fitted individually, simple reflection, and +the logarithmic limb darkening (LD) law. We fitted for the po- +tentials and light contributions of the two stars, the orbital in- +clination and a phase shift with respect to the adopted orbital +ephemeris. The mass ratio was fixed at the spectroscopic value, +bolometric albedos were set to 1.0, synchronous rotation was as- +sumed, and the gravity brightening exponents were set to 1.0. A +circular orbit was assumed for V573 Car but the possibility of +an eccentric orbit was checked. The input LD coefficients were +obtained by bilinear interpolation in the tables of van Hamme +(1993). +For the purposes of determining the uncertainties in the fitted +parameters, we ran a series of alternative solutions for differing +choice of WD code mode of operation, choice of numerical reso- +lution, treatment of reflection, choice of LD law, whether the LD +coefficients were fixed or fitted, treatment of third light, variation +of the mass ratio within the uncertainties, and the possibility of +orbital eccentricity (for V573 Car). We also considered the ef- +fects of albedo, rotational velocity and gravity brightening, by +fixing them at different values and also attempting to fit for them +directly. +The net result of this process was a default solution for each +system, accompanied by a measurement of how much each fit- +ted parameter changed between this default solution and each +of the alternative solutions. These changes were then added in +quadrature to arrive at a final robust uncertainty value for each +fitted parameter. The results for all four systems are summarised +in Table 6. The fractional radii are volume-equivalent values ob- +tained from the lc flavour of the WD code. +Article number, page 11 of 25 + +A&A proofs: manuscript no. harps_high-mass_binaries +Table 6: Summary of the parameters for the wd2004 solutions of the light curves of the systems. +Parameter +wd2004 name +V1034 Sco +GL Car +V573 Car +V346 Cen +Control and fixed parameters: +wd2004 operation mode +mode +0 +0 +0 +0 +Treatment of reflection +mref +1 +1 +1 +1 +Number of reflections +nref +1 +1 +1 +1 +Limb darkening law +ld +2 (logarithmic) +2 (logarithmic) +1 (linear) +2 (logarithmic) +Numerical grid size (normal) +n1, n2 +60 +50 +50 +50 +Numerical grid size (coarse) +n1l, n2l +50 +40 +40 +40 +Fixed parameters: +Orbital period (d) +period +2.440646 +2.4222681 +1.4693316 +6.3220088 +Primary eclipse time (HJD) +hjd0 +2451931.2652 +2459321.2994 +2450456.8164 +2459335.5607 +Mass ratio +rm +0.563 +0.943 +0.818 +0.712 +Teff star A (K) +tavh +32 200 +30 960 +31 900 +26 100 +Teff star B (K) +tavh +25 800 +30 390 +28 700 +22 500 +Rotation rates +f1, f2 +1.0, 1.0 +1.67, 1.29 +1.0, 1.0 +2.49, 2.70 +Gravity darkening +gr1, gr2 +1.0, 1.0 +1.0, 1.0 +1.0, 1.0 +1.0, 1.0 +Bolometric albedos +alb1, alb2 +1.0, 1.0 +1.0, 1.0 +1.0, 1.0 +1.0, 1.0 +Fitted parameters: +Phase shift +pshift +−0.0035 +0.0413 +0.0008 +0.0702 +Star A potential +phsv +3.670 ± 0.044 +5.736 ± 0.033 +3.913 ± 0.024 +5.933 ± 0.015 +Star B potential +phsv +4.228 ± 0.031 +5.766 ± 0.064 +4.106 ± 0.027 +8.012 ± 0.032 +Orbital inclination (◦) +xincl +81.80 ± 0.32 +86.57 ± 0.17 +80.52 ± 0.14 +84.97 ± 0.12 +Orbital eccentricity +e +0.027 ± 0.013 +0.1465 ± 0.0004 +0.0 (fixed) +0.2750 ± 0.0006 +Argument of periastron (◦) +perr0 +57 ± 18 +22.47 ± 0.36 +27.53 ± 0.28 +Light from star A (u band) +hlum +8.006 +Light from star B (u band) +clum +4.269 +Light from star A (v band) +hlum +7.861 +Light from star B (v band) +clum +4.449 +Light from star A (b band) +hlum +7.893 +Light from star B(b band) +clum +4.531 +Light from star A (y band) +hlum +7.888 +Light from star B (y band) +clum +4.551 +Light from star A (TESS band) +hlum +8.37 ± 0.10 +6.096 ± 0.081 +8.374 ± 0.086 +Light from star B (TESS band) +clum +1.73 ± 0.08 +5.204 ± 0.102 +1.967 ± 0.006 +Third light (TESS band) +el3 +0.239 ± 0.012 +0.166 ± 0.006 +0.214 ± 0.007 +Fractional radius of star A +0.3300 ± 0.0032 +0.2203 ± 0.0015 +0.3308 ± 0.0025 +0.2088 ± 0.0014 +Fractional radius of star B +0.1901 ± 0.0022 +0.2088 ± 0.0017 +0.2759 ± 0.0029 +0.1106 ± 0.0008 +Notes. Detailed descriptions of the control parameters can be found in the WD code user guide (Wilson & Van Hamme 2004). A and B refer to +the primary and secondary stars, respectively. Uncertainties are only quoted when they have been robustly assessed by comparison with a full set +of alternative solutions. +7.1. V1034 Sco +Two photometric studies of V1034 Sco have been published. +Bouzid et al. (2005) presented light curves taken in the Ström- +gren uvby filters, with 409, 645, 1058 and 1036 datapoints, +respectively. Sana et al. (2005) obtained light curves in two +narrow-band filters, designated λ4685 and λ6051, containing +112 and 138 datapoints, respectively. For our exploratory so- +lutions we used the Sana et al. (2005) data as the Bouzid et al. +(2005) data are not available. +In the course of this work a new light curve became avail- +able from sector 39 of the TESS satellite (see Section 3). As the +TESS data are of much higher quality than the other photome- +try, we have based our final results for V1034 Sco on these data. +Before doing so, we performed a preliminary fit with jktebop +(Southworth 2013) to obtain an orbital ephemeris then phase- +binned these data down to 500 bins to decrease the computa- +tion time. Our final solution is for an eccentric orbit, including +third light, and the logarithmic LD law and the linear LD co- +efficient fitted for each star and passband. The main contribu- +tors to the uncertainty in the fractional radii are the treatment +of albedo and gravity darkening. Uncertainties arising from the +choice of numerical resolution, WD program mode, rotation rate +and LD were all significantly smaller and therefore contributed +negligibly when all uncertainties for each parameter were added +in quadrature. The parameters and their uncertainties are given +in Table 6, and the best fits are shown in Fig. 10. +7.2. GL Car +Light curves of GL Car in the Strömgren uvby system were ob- +tained by Giménez et al. (1985) using the 0.5 m Strömgren Au- +tomated Telescope at ESO La Silla. They comprise 526 ob- +servations through each filter, 234 in the 1982 observing sea- +son and 308 in the 1983 season. These data were analysed by +Giménez & Clausen (1986) using the wink code (Wood 1971). +The observations were obtained in electronic form from the +archive of J. V. Clausen and used in the current work to ob- +Article number, page 12 of 25 + +Pavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +Fig. 10: The light curves and best WD models for V1034 Sco. +The differential magnitudes are plotted versus orbital phase and +are colour-coded according to the central wavelengths of the +passbands. The source and passband of each light curve is la- +belled. The residuals of the fit are shown at the base of the figure. +Offsets have been applied between the light curves and residuals +for clarity. +tain a preliminary solution. We found values and uncertainties +for the fitted parameters in good agreement with those from +Giménez & Clausen (1986). +Subsequent to our analysis of the uvby data a new light curve +of GL Car became available from TESS. We phase-binned this +and modelled it using wd, fitting for an eccentric orbit and third +light. The rotation rates (F1 and F2) were set to the ratios of +the measured rotational velocities (Table 3) and the synchronous +values, determined iteratively. Unlike the uvby data, the TESS +light curve shows a strong correlation between the light ratio +and the amount of third light. We therefore applied the light ra- +tio from our spectroscopic analysis. Because there is no mech- +anism to explicitly apply a spectroscopic light ratio in wd2004 +we propagated the light ratio from our spectral interval (which +corresponds closely to the Johnson B band) to the Strömgren +uvby bands (see Southworth 2010) using atlas9 theoretical spec- +tra (Castelli et al. 1997) and passband response functions from +Maíz Apellániz (2006). We forced wd2004 to match them by fix- +ing the hlum parameters at the appropriate values. Including this +constraint greatly improved the reliability of the results. +We find precise fractional radii for GL Car once our spec- +troscopic light ratio is included (Table 6). The uncertainties +Fig. 11: The light curves and best WD models for GL Car. Other +comments are the same as for Fig. 10. +are dominated by those from this light ratio, but are still be- +low 1% and a factor of three smaller than those from the uvby +data alone. They also agree well with the less precise results +from Giménez & Clausen (1986). The fitted orbital eccentric- +ity is in excellent agreement with that from its apsidal motion +(e = 0.1459 ± 0.0015 from Wolf et al. 2008). The orbital phase +of secondary eclipse has changed a lot between the uvby and +TESS datasets, and the different morphology of the light curve +is obvious (see Fig. 11). +7.3. V573 Car +V573 Car was studied by Freyhammer et al. (2001) using the +Dutch 0.9 m telescope at ESO La Silla. A total of 1910 observa- +tions were obtained through the Strömgren filters: 763 in y, 513 +in b, 350 in v and 284 in u. We fitted all four light curves simul- +taneously, using the ephemeris from Freyhammer et al. (2001). +We assumed a circular orbit in most cases, but did run a fit with e +and ω free to check if this led to a better fit to the data (it didn’t). +We also assumed no third light, after attempts to fit for it had a +negligible effect on the results and also led to a slightly negative +value for this parameter. The best fit is shown in Fig. 12. +The fitted parameters were the potentials of the two stars, the +orbital inclination, a phase shift, and the light contributions of +the two stars in each passband. To avoid very small values for the +light contributions we renormalised each light curve to be at ap- +proximately zero relative magnitude at quadrature. We adopted +Article number, page 13 of 25 + +A&A proofs: manuscript no. harps_high-mass_binaries +Fig. 12: The light curves and best WD models for V573 Car. +Other comments are the same as for Fig. 10. +the linear LD law as it gave results very similar to those for the +logarithmic and square-root laws; attempts to fit for the LD co- +efficients led to unphysical solutions. The rotational velocities of +the stars were held to the synchronous values. +We found that the solution of the light curves is degenerate +in that significantly different values of the ratio of the radii or +the light contributions of the stars led to almost indistinguish- +able fits. This was also found by Freyhammer et al. (2001), who +constrained their solution using a light ratio measured from their +spectra. We took the same approach. +For our final result (Table 6) we give the solution for fitting +all four light curves simultanously, constrained by the spectro- +scopic light ratio. The uncertainties in the parameters include +contributions from the uncertainty in the spectroscopic light ra- +tio, the effect of a change of 5% in the rotation velocities of the +stars, and the treatment of albedo and gravity darkening. Other +sources of uncertainty (see above) were checked and found to +be negligible. We were able to measure the fractional radii of +the stars to precisions of 0.8% (star A) and 1.1% (star B); the +main contribution to these uncertainties is the spectroscopic light +ratio (for star A) and the treatment of gravity darkening (for +star B). The values we find are in reasonable agreement with +those from Freyhammer et al. (2001), but our uncertainties are +slightly larger. +Fig. 13: The light curves and best WD models for V346 Cen. +Other comments are the same as for Fig. 10. +7.4. V346 Cen +Extensive photometry in the Strömgren uvby system was ob- +tained by Giménez et al. (1986b), comprising 1056 observations +made simultaneously through all four filters using the Strömgren +Automated Telescope (Grønbech et al. 1976). These data have +been analysed by Giménez et al. (1986a) using the wink model, +and by Mayer et al. (2016) using the phoebe code. The two stud- +ies agree on the values of the fractional radii to within the uncer- +tainties quoted by Giménez et al. (1986a) but not the uncertain- +ties quoted by Mayer et al. (2016). We therefore performed our +own analysis of these data in order to assess robust errorbars and +check the level of agreement with the previous studies. +Our WD code model for the uvby data provided a good fit +to the observations (Fig. 13) but required ω to be fixed at a suit- +able value to avoid the fit diverging to unphysical solutions. We +set the rotation rates to 2.49 and 2.70 based on the rotational ve- +locities of the stars measured from the disentangled spectra. The +logarithmic LD law was adopted, although the other two laws +gave almost identical results. Third light was fixed at zero be- +cause attempts to fit for it returned a small negative value that +was consistent with zero. Our results were in excellent agree- +ment with those of Giménez et al. (1986b). +After this work had been performed, light curves from sec- +tors 37 and 38 of the TESS satellite became available. These +are of much higher quality so we used them for our final anal- +ysis. We performed a preliminary fit with jktebop to obtain an +orbital ephemeris then phase-binned them into 500 bins to make +Article number, page 14 of 25 + +Pavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +Table 7: Physical properties measured for the four systems analysed in this work. +Parameter +V1034 Sco +GL Car +V573 Car +V346 Cen +Mass ratio +0.5628± 0.0021 +0.943± 0.012 +0.8182± 0.0037 +0.7119± 0.0038 +Mass of star A (MN +⊙) +17.01± 0.14 +15.86± 0.31 +15.11± 0.13 +11.74± 0.12 +Mass of star B (MN +⊙) +9.573± 0.053 +14.95± 0.30 +12.365± 0.096 +8.359± 0.089 +Semimajor axis (RN +⊙) +22.767± 0.053 +23.79± 0.15 +16.412± 0.044 +39.12± 0.13 +Radius of star A (RN +⊙) +7.513± 0.075 +5.242± 0.048 +5.429± 0.043 +8.278± 0.079 +Radius of star B (RN +⊙) +4.328± 0.051 +4.968± 0.051 +4.528± 0.049 +4.123± 0.072 +Surface gravity of star A (log[cgs]) +3.917± 0.009 +4.199± 0.007 +4.148± 0.007 +3.672± 0.008 +Surface gravity of star B (log[cgs]) +4.147± 0.010 +4.220± 0.008 +4.218± 0.009 +4.130± 0.015 +Synch. rotational velocity of star A ( km s−1) +155.7± 1.6 +109.5± 1.0 +186.9± 1.5 +66.25± 0.64 +Synch. rotational velocity of star B ( km s−1) +89.7 ± 1.1 +103.8± 1.1 +155.9± 1.7 +33.00± 0.57 +Teff of star A (K) +32200± 500 +30960± 500 +31900± 400 +26100± 300 +Teff of star B (K) +25800± 300 +30390± 500 +28700± 350 +22500± 300 +Luminosity of star A log(L/LN +⊙) +4.738± 0.028 +4.357± 0.029 +4.439± 0.023 +4.457± 0.022 +Luminosity of star B log(L/LN +⊙) +3.874± 0.028 +4.278± 0.030 +4.098± 0.023 +3.594± 0.028 +Absolute bolometric magnitude of star A +−7.104± 0.071 +−6.15 ± 0.073 +−6.331± 0.057 +−6.403± 0.054 +Absolute bolometric magnitude of star B +−4.944± 0.057 +−5.96 ± 0.075 +−5.505± 0.058 +−4.245± 0.069 +Interstellar extinction E(B − V) (mag) +0.75 ± 0.05 +0.55 ± 0.05 +0.40 ± 0.05 +0.56 ± 0.03 +Distance (pc) +1460± 50 +2278 ± 63 +2466 ± 78 +2290 ± 60 +Gaia DR3 parallax (mas) +0.6452± 0.0231 +0.4232± 0.0130 +0.4428± 0.0200 +0.4380± 0.0261 +Gaia DR3 distance (pc) +1550± 56 +2363 ± 73 +2260 ± 100 +2280 ± 140 +Notes. The units labelled with a superscripted ‘N’ are given in terms of the nominal solar quantities defined in IAU 2015 Resolution B3 (Prša et al. +2016). +the computations faster. Our approach was the same as for the +uvby data except that we were able to fit for ω and also needed +to fit for third light due to significant contamination of the TESS +light curve. We found the best fit to the TESS data to be highly +stable against changes in mass ratio, rotation rate, treatment of +LD, albedo, gravity darkening and numerical grid size. We had +to fix the LD coefficients as they diverged to unphysical values +when we attempted to fit for them. +The final parameters and uncertainties of the fit are given +in Table 6. The fits are shown in Fig. 13, and two things are +worth highlighting. First, the morphology of the light curve has +changed between the uvby and TESS epochs due to apsidal mo- +tion. The phase of secondary eclipse has changed and it is no +longer annular – the primary eclipse has become a transit in- +stead. Second, the TESS data show a clear pulsation signature. +This affected the quality of our solution and was probably why +we were unable to fit for LD coefficients. The pulsation almost +certainly arises from the EB itself and not from the contaminat- +ing light, because they are commensurate with the orbital pe- +riod (see Fig. A.2). V346 Cen is therefore another high-mass EB +showing pulsations (Southworth & Bowman 2022). Because our +light curve solution did not account for pulsations, we have con- +servatively doubled the uncertainties in the measured fractional +radii. +7.4.1. Pulsations +Following the binary analysis, an analysis of the residual light +curve (hereafter called the pulsation light curve) revealed the +presence of tidally excited pulsations, as illustrated in Fig. 14. +To measure this tidally induced variability, we fitted sine waves, +corresponding to the 20 lowest-order orbital harmonic frequen- +cies, to the out-of-eclipse part of the pulsation light curve. Fit- +ted orbital harmonics were accepted when the signal-to-noise +ratio S/N ≥ 4.0, where S/N was calculated as the ratio of the +amplitude of the fitted sine wave, and the average signal ampli- +tude of the Lomb-Scargle periodogram (Scargle 1982) in a 1 d−1 +window around the considered frequency. Finally, the measured +amplitudes, phases and orbital harmonic frequencies were opti- +mised simultaneously by nonlinearly fitting them to the pulsation +light curve. Their values are listed in Table 8. +From these results, we determined that the tidally induced +pulsation corresponds to the 9th orbital harmonic, in agreement +with what is shown in Fig. 14. The physical origin of the other +measured orbital harmonics is less clear. While they may also +partially correspond to tidally induced pulsations, this could not +be confirmed. At least part of it is likely caused by the non- +sinusoidal nature and orbital-phase dependent amplitude modu- +lations of the 9th orbital harmonic signal. Moreover, as shown in +the middle panel of Fig. 14, this pulsation has a minimum dur- +ing the primary eclipse and a maximum during the secondary +eclipse, which indicates that it belongs to the primary compo- +nent. +Finally, after the significant tidally excited variability was re- +moved from the pulsation light curve, we evaluated the residuals. +As illustrated in Fig. 15, the remaining data exhibit signatures of +stochastic low-frequency variability, as has been reported in the +literature for other high-mass stars (e.g., Bowman et al. 2019, +2020). +7.5. Physical properties +We have determined the physical properties of the systems using +the results from the spectroscopic and photometric analyses out- +lined above. For this we used the velocity amplitudes, Teff values, +e and ω from the spectroscopic analysis, and the fractional radii +and orbital inclination from the photometric analysis. To perform +the calculations we used the jktabsdim code (Southworth et al. +2005), which propagates the errorbar from each input parame- +ter using a perturbation analysis. We used a version of jktab- +sdim modified to use the IAU system of nominal solar values +(Prša et al. 2016) plus the NIST 2018 values for the Newtonian +Article number, page 15 of 25 + +A&A proofs: manuscript no. harps_high-mass_binaries +Fig. 14: Tidally excited pulsations of V346 Cen. Top: observed +light curve of V346 Cen for sectors 37 and 38, phase-folded +with the binary orbital period. The eclipses are indicated by the +grey bands. Middle: Pulsation light curve of V346 Cen for sec- +tors 37 and 38, phase-folded with the binary orbital period. Data +points taken during the eclipses again lie within the grey bands. +Bottom: Lomb-Scargle periodogram, calculated for the out-of- +eclipse data points of the pulsation light curve for sectors 37 and +38. The dashed vertical lines indicate harmonics of the orbital +frequency. +Fig. 15: Stochastic low-frequency variability of V346 Cen. Top: +part of the (out-of-eclipse) residual light curve of V346 Cen, +after fitting the orbital harmonics. Bottom: Lomb-Scargle pe- +riodogram, calculated for the out-of-eclipse data points of the +residual light curve for sectors 37 and 38. +Table 8: Values of the amplitudes A, frequencies ν, phases φ and +signal-to-noise ratios S/N of the orbital harmonics, calculated +for the out-of-eclipse data points in the pulsation light curve of +V346 Cen. +norb +A (mmag) +ν (d−1) +φ (2π rad) +S/N +1 +0.813 ± 0.023 0.15818130 +0.0811 ± 0.0005 +6.3 +5 +0.812 ± 0.026 0.79090648 +0.378 ± 0.005 +6.7 +6 +0.771 ± 0.025 0.94908778 +−0.240 ± 0.005 +6.5 +8 +0.483 ± 0.025 1.26545037 +0.237 ± 0.008 +4.0 +9 +3.938 ± 0.025 1.42363166 −0.2993 ± 0.0010 33.4 +Notes. The frequency values were fixed at the indicated integer multi- +ples of the measured orbital frequency νorb. +gravitational constant and the Stefan-Boltzmann constant. The +results of this analysis are given in Table 7. +Distances have been derived using the measured radii and +Teffs of the stars, apparent magnitudes of the system in the +Johnson-Cousins UBVRI and 2MASS JHKs bands, and the the- +oretical bolometric corrections tabulated by Girardi et al. (2002). +We adjusted the interstellar extinction E(B − V) to obtain con- +sistent distances in the optical and infrared passbands. These re- +sults are given in Table 7 alongside the Gaia EDR3 parallaxes +(Gaia Collaboration 2016, 2021) and the distance from simple +inversion of the parallax. We see agreement within the errorbars, +the most discrepant (1.6σ) being for V573 Car. Very similar con- +clusions are drawn if we use the geometric or photogeometric +distances from Bailer-Jones et al. (2021). We conclude that our +results for all four targets are independently verified by the Gaia +parallaxes. +8. The parent clusters +Knowledge of the properties of stars in dEBs allows the determi- +nation of their distance. Moreover, a comparison of the proper- +ties of dEBs to stellar evolutionary models constrains their age. +The age of stars in our sample, except for GL Car, were deter- +mined from isochrone fitting in Tkachenko et al. (2020) for two +cases: (i) as a single star; (ii) as a binary where the two compo- +nents have the same age. Three different interior structures were +assumed in these calculations, hence in Table 9 we give lower +and upper limits for the age. The distances to the binary systems +in our sample are given in Table 7. +8.1. V1034 Sco and NGC 6231 +The distance to V1034 Sco was evaluated by Sana et al. (2005) +who found d = 1528+117 +−109 pc, which is within 1σ of the distance +we calculated. The light curve solution in Bouzid et al. (2005) +suffers from an ambiguity in setting the primary’s Teff so the au- +thors calculated the distance for both cases. The larger one is +exactly the same as those reported by Sana et al. (2005), while +the shorter one is d = 1399+20 +−20 pc. Mayer et al. (2008) deter- +mined the distance to another dEB in this cluster, V1007 Sco, +as 1622 pc (no uncertainty given) which is somewhat larger than +the other distance estimates mentioned here. +The open cluster NGC 6231 belongs to the star-formation +complex Sco OB1 (Perry et al. 1991). The cluster is the oldest +and most massive in Sco OB1 (Damiani et al. 2016). The ages +of the cluster members have been estimated to be between 2 and +8 Myr (Sung et al. 2013; Damiani et al. 2016; Kuhn et al. 2017), +with OB stars being an older population in the cluster. The clus- +Article number, page 16 of 25 + +flux (mmag) +0.0 +2.5 +2324 +2325 +2326 +BJD-2457000 +amplitude (mmag) +0.4 +0.2 +0.0 +0 +2 +4 +6 +8 +10 +frequency (d-1)(mmag) +100 +flux ( +200 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +orbitalphase +10 +(mmag +flux +10 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +orbital phase +amplitude +1 +2 +3 +4 +5 +frequency (d-1)Pavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +Table 9: Distances and ages for the binary systems in the present sample compared to the parent clusters. +Binary +Distance (pc) +Age (Myr) +Cluster +Distance (pc) +Ref. +Age (Myr) +Ref. +V1034 Sco +1460 ± 50 +5.0–7.7 | 5.3–7.0 +NGC 6321 +1538 ± 20 +1 +6.3 +2 +GL Car +2278 ± 63 +2.0 ± 0.5 +NGC 3572a +2444 ± 33 +3 +1–4 +4 +V573 Car +2466 ± 78 +1.5–3.1 | 2.2–2.7 +Trumpler16 +2360 ± 505 +5 +2 ± 1 +6 +V346 Cen +2290 ± 60 +10.5–16.0 | 10.7-16.0 +Stock 14 +2439 ± 326 +7 +10 ± 2 +7 +Notes. The distances to the binary systems are from the present work (Table 7). The ages were calculated by Tkachenko et al. (2020) except +GL Car for which the age is adopted from Giménez & Clausen (1986). Tkachenko et al. (2020) estimated the age for two options: assuming the +components are individual stars, and constraining the age to be the same for both components. Both measurements are given, separated by a +vertical line. +References. (1) Banyard et al. (2022); (2) Kuhn et al. (2017); (3) Clariá (1976); (4) Garcia (1994); (5) Göppl & Preibisch (2022); (6) Hur et al. +(2012); (7) Paunzen & Netopil (2006). +ter is rich in spectroscopic binaries: García & Mermilliod (2001) +listed about 30 systems of which 16 are certain. Mayer et al. +(2008) did an exhaustive search of the cluster members, and +listed ten EBs. The most recent distance determinations to +NGC 6231 are based on Gaia parallaxes. Kuhn et al. (2019) +quoted d = 1710+13 +−100 pc using Gaia DR2, while Banyard et al. +(2022) found the median geometric and photogeometric dis- +tances for their sample of about 60 stars in the cluster using Gaia +EDR3 parallaxes to be 1579 and 1576 pc, respectively. +Rosu et al. (2022b) determined the age of V1034 Sco to be +τ = 6.8 ± 1.4 Myr, in perfect agreement with the result of +Tkachenko et al. (2020). Three other binary systems that are +members of this cluster were studied: HD 152248 (Rosu et al. +2020), HD 152219 (Rosu et al. 2022b) and HD 152218 +(Rosu et al. 2022a). Their ages were determined from the apsidal +motion rate and range from 5 to 9.5 Myr. +8.2. GL Car and NGC 3572/Collinder 240 +Giménez & Clausen (1986) found a distance to GL Car of d = +2100 pc. They did not give an uncertainty but quoted an error of +100 pc due to bolometric corrections and interstellar reddening. +This distance is smaller than our result and that from the Gaia +DR3 parallax. Giménez & Clausen (1986) extensively discussed +possible physical relationships to the open clusters in the vicin- +ity of GL Car, which is in a region crowded with young open +clusters and in the direction of the Sagittarius-Carina spiral arm. +Membership of GL Car in NGC 3572 was proposed by +Sahade & Berón Dàvila (1963). Giménez & Clausen (1986) re- +jected this association due to the shorter distance to the dEB than +the cluster, and because NGC 3572 is a compact cluster with +a radius of 5′ and GL Car is at an angular distance of 40′. It +was recognised that the open cluster NGC 3572 consists of two +overlapping clusters, one at 2.3 kpc and one at 3.0 kpc (Clariá +1976). The nearer cluster is also considered by Clariá (1976) to +be the probable nucleus of a scattered group of OB stars located +in the vicinity, identified as Collinder 240 and an extension of +Car OB2. This is a region in which the line of sight is tangential +to the molecular cloud ridge in the Carina Arm, and is projected +on a rather small area in the sky. It shows as a region with a +higher concentration of OB stars, but with a radial extension of +several kpc. +The age of GL Car was determined to be τ = 2.0 ± 0.5 Myr +(Giménez & Clausen 1986). This is compatible with age deter- +minations for Collinder 240, τ ∼ 1 Myr, and Car OB2, τ = 4 +Myr (Garcia 1994). +8.3. V573 Car and Trumpler16 +Freyhammer et al. (2001) determined a distance to V573 Car of +d = 2600 ± 120 pc, and an age of τ = 1.5 ± 1.0 Myr. Their dis- +tance determination is within 1σ of ours. Also, the very young +age is confirmed with extensive isochrone fitting to different stel- +lar interior structure models in Tkachenko et al. (2020), as sum- +marised in Table 9. +V573 Car is situated near the centre of the open cluster Trum- +pler16, close to η Carinae, the brightest star in the cluster, and +one of the most intriguing objects in the Galaxy. The cluster +itself, with its neighbouring clusters, Trumpler14, and Trum- +pler15, forms a chain of rich clusters in the prominent Carina +star-forming complex, a conspicous part of the Carina-Vela spi- +ral arm. The whole region is recognised as the young asso- +ciation Carina OB1, which also includes NGC 3293 and sev- +eral small open clusters, the H ii region and prominent neb- +ula NGC 3372 powered by η Car (Smith 2006; Wright 2020). +Pre-Gaia distance estimates relied mostly on multicolour pho- +tometry, and gave distances in the range 2.2–2.9 kpc and +young ages in the range 1–3 Myr (Hur et al. 2012). Using +Gaia EDR3 Shull et al. (2021), Maíz Apellániz et al. (2022) and +Göppl & Preibisch (2022) found distances to the cluster Trum- +pler16 at the lower end of the range: 2.32±0.12, 2.38±0.20 and +2.36±0.05 kpc, respectively, all within 1σ of our determination. +8.4. V346 Cen and Stock 14 +The distance and age of V346 Cen were also determined by +Giménez et al. (1986b), which allows a direct comparison with +our results. Giménez et al. (1986b) determined the distance d = +2.38±0.18 kpc, which is within 1σ of our determination. The age +of the binary system they found, τ = 10.0+5.8 +−3.6 Myr, also agrees +well with our result, τ = 10.7–16.0Myr (Table 9). +Stock 14, the parent cluster of V346 Cen, is described as +a loose but clearly defined open cluster (Moffat & Vogt 1975; +Eichendorf & Reipurth 1979). The most recent deep UBV pho- +tometry of Stock 14 was obtained by Drobek et al. (2013) pri- +marily in a search for new variable stars. Their photometry +allowed determination of the distance and an estimate of the +age for Stock 14, d = 2399+56 +−55 pc, and τ = 20 ± 10 Myr. +The authors confirmed the cluster membership of V346 Cen. +Re-evaluation of the photometric distance and age of Stock 14 +by Paunzen & Netopil (2006) also favoured a shorter distance +than previous determinations. They obtained a distance of d = +2439 ± 326 pc, and an age of τ = 10 ± 2 Myr, in fine agreement +with the extensive photometric study by Drobek et al. (2013) as +well as our results for V346 Cen. +Article number, page 17 of 25 + +A&A proofs: manuscript no. harps_high-mass_binaries +9. Discussion +The results of the analyses above are summarised in Table 4 for +elemental abundances and Table 7 for fundamental stellar quan- +tities. The stars in the present work cover a range of mass (8.4– +17.1 M⊙), radius (4.1–8.3 R⊙), Teff (22 500 to 32 200 K), sur- +face gravity (3.7–4.2dex) and v sin i (90–185km s−1) and are all +unevolved main sequence stars from late-O to early-B spectral +types. We have achieved a high accuracy in the fundamental stel- +lar properties, with uncertainties in mass of 0.6–2.0%, radius of +0.8–1.7%, and log g of 0.009–0.021dex. Having a precise log g +allows us to avoid its degeneracy with Teff in spectral analysis, +resulting in uncertainties of 1.7–2.5% in Teff. Since Teff and log g +are the principal quantities for specifying a model atmosphere, +precise values are a prerequisite in measuring chemical abun- +dances to a high precision. We now discuss the implications of +our results for two subjects: evolutionary models for high-mass +stars, and chemical evolution in high-mass binaries. +Studies of chemical abundances in high-mass stars mostly +concentrate on more advanced evolutionary stages, so it is dif- +ficult to perform a quantitative comparison between our results +and those published elsewhere. Martins et al. (2017) presented a +study of six short-period binary systems. Of them, two are con- +tact or overcontact systems so will have abundances altered by +mass transfer, one (DH Cep) has component stars considerably +more massive than our sample (38 M⊙ and 33 M⊙), while the re- +maining three (Y Cyg, AH Cep and V478 Cyg) are suitable for +the comparison. Of these, AH Cep and V478 Cyg were analysed +in our previous work (Pavlovski et al. 2018) so a direct compar- +ison is possible. Two studies agreed to within 2σ uncertainties +in the [N/C] and [N/O], but only because of the large uncertain- +ties quoted by Martins et al. (2017). It is hard to trace the reason +for this, but it may be related to the large uncertainties in the +atmospheric parameters in their study. +Results of a comprehensive analysis of a large sam- +ple of binary and/or multiple stars in the Tarantula Nebula +have recently been published (Almeida et al. 2017; Mahy et al. +2020b,a) based on medium-resolution (R = 6400) spectra from +VLT/FLAMES/GIRAFFE covering 3964–4567Å. A total of 51 +SB1 and SB2 systems were studied, of which 13 are eclips- +ing. The atmospheric parameters were determined using NLTE +methods, and He, C and N abundances derived. The objects +studied fall into five different groups: (1) long-period systems +(P > 20 d) with well-detached components; (2) eccentric short- +period (P < 10 d) detached binaries; (3) circular-orbit short- +period (P < 10 d) binaries with strong tidal effects; (4) semi- +detached systems; and (5) contact systems. No N enrichment +was found for binaries in the first two groups, despite the com- +ponents having v sin i values of 50–250km s−1. This finding is +in disagreement with evolutionary models with rotationally in- +duced mixing (Maeder & Meynet 2000; Heger & Langer 2000; +Heger et al. 2000). Furthermore, a large N abundance was found +for apparently slowly rotating stars in binaries. This agrees with +initial findings by Hunter et al. (2007, 2008, 2009) who detected +three distinctive groups in a diagram of [N/H] versus v sin i for +single OB stars (sometimes dubbed the “Hunter diagram”): (1) +stars showing N enrichment with v sin i; (2) rapidly rotating stars +with no sign of N enrichment; and (3) stars with low v sin i and +excessive N abundance. +In the third group of binaries from Mahy et al. (2020b), N +enrichment was found for the fast rotators. This is a group of +stars in which the strongest influence of tidal forces on rotation- +ally induced mixing is expected, following theoretical calcula- +tions by de Mink et al. (2009). By far the largest N enhancement +was found for stars with almost the lowest v sin i in this group +(∼50 km s−1), just as in the case of findings for stars in the first +two groups. Mahy et al. (2020b) concluded that stars in detached +binaries (groups 1 to 3) are evolving as single stars. A lack of a +clear relationship between N abundance and v sin i is in conflict +with theoretical models and makes it hard to understand the ef- +fect of rotationally induced turbulent mixing in stellar interiors. +A very recent comprehensive spectroscopic analysis of a +large set of B-type stars in the young open cluster NGC 3293 +(Morel et al. 2022) also corroborates these results: in the sam- +ple of almost 150 B-type stars of which the majority have high +v sin i, apparently no star with excess N abundance was detected. +Only two stars are found with mild N enhancement, and these +stars have a low v sin i. A lack of N enhancement in fast-rotating +B stars, and conversely, further evidence for N enhancement in +low-v sin i B stars, is in clear contradiction with theoretical evo- +lutionary models which incorporate rotationally-induced mix- +ing. +A state-of-the-art statistical analysis was carried out by +Aerts et al. (2014) to identify possible mechanism(s) that could +explain the distribution of stars in the Hunter diagram. The au- +thors collected a statistically significant sample of well-studied +Galactic single B stars for which seven observables were avail- +able (surface N abundance, rotational frequency, magnetic field +strength, and the amplitude and frequency of their dominant +acoustic and gravity modes of oscillation). A multivariate anal- +ysis indicated that the Teff and the frequency of the dominant +acoustic oscillation mode have the most predictive power of the +surface N abundance, whereas the rotational frequency of the +star does not have any predictive power at all. Up to now, no +follow-up studies have been undertaken to investigate these un- +expected results. +Strong support for rotationally induced mixing has come +from detailed abundance study for early B-type stars by +Przybilla et al. (2008) and Nieva & Przybilla (2012). The au- +thors selected 20 early-B stars with a low v sin i to allow a high +precision in determination of the atmospheric parameters and +chemical abundances. Przybilla et al. (2010) confirmed an obser- +vationally tight correlation in the plot of abundance ratios N/C +versus N/O, with a slope predicted via nuclear reactions in the +CNO process. The targets had a broad evolutionary range, from +dwarfs to supergiants, and their CNO abundances followed pre- +dictions of the nuclear reaction theory. +From our current and previous (see Section 6) studies, we +have a sample of 13 dEBs of which 25 components have +measured CNO abundances. We compared these to a sam- +ple of high-mass stars published in Przybilla et al. (2010) and +Nieva & Przybilla (2012) in the logarithmic N/C versus N/O +diagram (left panel in Fig. 16). This is a powerful diagnostic +tool in which the slope between [N/C] and [N/O] represents +changes in CNO abundances due to nuclear reactions as de- +rived in Przybilla et al. (2010). It is striking that the two sam- +ples cover the same mass range (8–20 M⊙) but do not fully over- +lap in the diagram. For binary components there is a cut-off +at [N/C] ∼ −0.4 dex and [N/O] ∼ −0.8 dex and they cluster +around values close to solar ([N/C]⊙ = −0.52 dex and [N/O]⊙ = +−1.00 dex), but a slope can be seen. The targets in the current +work strengthen our previous conclusion that properties of inte- +rior mixing in binary stars are different from and might be less +efficient than in single high-mass stars (Pavlovski et al. 2018). +This striking effect is also clearly seen in the diagram of +[N/C] versus log g (Fig. 16). Theoretical evolutionary tracks +are presented for a 15 M⊙ star and five values of the ini- +tial rotational velocity Ω/Ωcrit = 0.1, 0.3, 0.5, 0.7, and 0.9 +Article number, page 18 of 25 + +Pavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +Fig. 16: Left panel: Abundances of the CNO elements for high-mass stars in a diagram of [N/C] index versus [N/O] index. Stars in +binary systems (Section 6) are represented by solid blue circles. For comparison, abundance determinations for single early-B type +stars (Nieva & Przybilla 2012) are represented by green circles. Single stars obey a trend indicated by an analytical approximation +to the nuclear reactions path for the CNO cycle derived in Przybilla et al. (2010) and Maeder et al. (2014). The slope in this diagram +indicates a gradual enhancement of N at the expense of C. A slight decrease in O abundance is also predicted. Whilst the single +and binary stars span almost identical mass and Teff ranges, and are all main-sequence stars, it is clear they do not share the same +distribution (see Section 9). Right panel: The observed [N/C] index for 23 high-mass stars in binaries (solid blue circles), compared +to single B-type stars showing in solid green circles (Nieva & Przybilla 2012), as a function of surface gravity. Solid red lines in +the right panel show theoretical evolutionary tracks for a 15 M⊙ star and five values of the initial rotational velocity Ω/Ωcrit = 0.1, +0.3, 0.5, 0.7, and 0.9 (Georgy et al. 2013). Striking differences between single stars, and stars in binary systems are discussed in +Section 9. +(Georgy et al. 2013). The overall spread in [N/C] could be in- +terpreted as due to evolutionary changes or (very) high initial +rotational velocities. However, only single stars from the sample +of (Nieva & Przybilla 2012) tend to be consistent with the large +[N/C] ratio predicted by the models for large initial rotational ve- +locity values of Ω/Ωcrit ⪆ 0.5 and [N/C] ⪆ −0.4 dex. The main +issue with the interpretation of the observed distribution in the +context of the rotationally induced mixing alone is the generally +low projected rotational velocity values (v sin i < 30 km s−1) +found by Nieva & Przybilla (2012) for about half of their sample +stars. For the effect of rotational mixing being alone responsible +for the observed [N/C] and [N/O] abundance ratios, one would +require the majority of apparently slow rotators in the sample +of Nieva & Przybilla (2012) to be stars that are seen pole-on. +This is a highly improbable scenario, so we conclude that the +CNO abundances and their ratios observed in single high-mass +stars are altered by multiple processes rather than just a single +mechanism of rotational mixing. For example, high-mass stars +are know to possess magnetic fields, stellar winds, and pulsa- +tions. To this (strong) tidal effects in close high-mass binary sys- +tems should be added. All these mechanisms, in one way or an- +other, are expected to impact the efficiency of internal mixing, +and hence the surface chemical composition. +However, the comparison between these two sets of empir- +ical data, one with single high-mass stars and the other with +high-mass stars in binary systems, is not straightforward. First +and foremost, the sets differ in their distributions of v sin i. The +set of single stars were deliberately selected to be sharp-lined +stars, so contains a mix of intrinsically slowly-rotating stars and +ones with small inclinations and thus small sin i terms. The set +of binary stars, on the other hand, contains objects whose equa- +torial rotational velocities are accurately known, assuming their +rotational and orbital axes have been aligned during formation +or by tidal effects. Furthermore, even though the v sin i distribu- +tion could be statistically corrected to intrinsic rotational veloc- +ities for single stars, there is a substantial difference in the ro- +tational history between single and binary stars that one cannot +easily account for. Evolution of stellar rotational velocity from +its initial value at the zero-age main sequence, and its subse- +quent changes in the course of stellar evolution due primarily to +changes in radius, is substantially different due to tidal effects. +This is particularly important for short-period systems whose ro- +tation is synchronised with and thus governed by their orbital +period. Nevertheless, the non-detection of substantial changes in +the CNO abundances of stars in binaries contradicts the predic- +tions of single-star rotational evolutionary models. For the sam- +ple of Przybilla et al. (2010) and Nieva & Przybilla (2012), i.e. +single high-mass stars with low observed v sin i, the possibility +remains that they agree with the theoretical predictions. +Tidal forces in binary and/or multiple systems affect the ge- +ometry of the orbits and the shape and spin of the components +(Mazeh 2008). In order of increasing timescale, the stellar spin +axes are aligned first, then their rotation is synchronised, and fi- +nally the orbit is circularised. Later evolution is dominated by +mass transfer due to the increase in the sizes of the component +stars. Our hypothesis that tidal effects suppress the efficiency +Article number, page 19 of 25 + +A&A proofs: manuscript no. harps_high-mass_binaries +of rotational mixing is not supported by theoretical calculations +(de Mink et al. 2013), which predict precisely the opposite: that +short-period circularised binary systems should experience rota- +tionally induced turbulent mixing in stellar interiors. +In +looking +for possible +mechanisms which +diminish +turbulent mixing in the components of binary systems, +Koenigsberger et al. (2021) examined differential rotation in +asynchronous binary systems. If the components in a binary +system are not yet in synchronous rotation, tidally-induced and +time-variable differential rotation could develop. The calcula- +tions by Koenigsberger et al. (2021) clearly show the role of +asynchronism: the induced rotation structure and its temporal +variability depend on the degree of departure from synchro- +nism. The authors further speculated that, in this context, slowly- +rotating asynchronous binaries could have more efficient mixing +than the more rapidly-rotating but tidally locked systems. This +shows that processes triggered by asynchronous rotation in bi- +nary systems cannot be ignored, while a comparison between +samples of single and binary stars should be done with partic- +ular care, even when the latter are in a detached configuration. +We note that 12 of the 14 binaries in our sample have eccentric +orbits but that most of the component stars rotate synchronously. +10. Conclusion +Despite their astrophysical importance, high-precision funda- +mental stellar quantities (mass, radius, Teff) have been deter- +mined for only a few high-mass stars in binary systems in our +galaxy (Southworth 2015). Even fewer have measurements of +their surface chemical composition (Serenelli et al. 2021). In the +present work we have added four more binary systems to this list: +V1034 Sco, V346 Cen, GL Car and V573 Car, containing stars of +masses from 8.4 to 17.1 M⊙. Most of these stars are young, with +only two in the second half of their MS evolution. +We have presented high-quality HARPS spectra and anal- +ysed them using spectral disentangling to determine their spec- +troscopic orbits and the individual spectra of the component +stars. These were analysed using an NLTE approach. We have +modelled the available light curves for our systems, compris- +ing uvby photometry in all cases and TESS photometry in three +cases, to determine their photometric parameters. Combining +these analyses, we have determined high-precision masses, radii, +surface gravities, Teff values, v sin i values and C, N, O, Mg and +Si abundances for all eight stars in the four binary systems. Of +particular interest are the CNO abundances since these elements +are involved in core hydrogen burning through the CNO cy- +cle. During a star’s evolution its N abundance increases and its +C abundance decreases. Rotationally induced mixing of stellar +material, or some other mixing processes, could bring nuclear- +processed material from the stellar core to the surface. Therefore, +the [N/C] ratio is a sensitive probe of interior mixing processes +during the MS evolutionary stage. +The CNO abundances determined in this work corroborate +our previous findings (Pavlovski et al. 2018) that interior mixing +is different in binary stars to single stars. A tight correlation of +[N/C] with [N/O] versus the predicted evolutionary changes has +been found for single early B-type stars (Przybilla et al. 2010; +Nieva & Przybilla 2012), whereas binary systems in our sample +show much less variation in both [N/C] and [N/O]. However, +care is needed when comparing them with single stars due to the +differences in rotational velocity between these types of object. +It remains true that the binary sample does not reproduce the +results found for a sample of single low-v sin i B-type stars. +On other hand, recent spectroscopic analysis of large sam- +ples of high-mass stars in binaries (Mahy et al. 2020a), and +single B-type stars in the young open cluster NGC 3293 +(Morel et al. 2022) apparently confirmed the lack of substantial +changes in CNO abundances for high-v sin i stars, i.e. for intrin- +sically fast-rotating stars. +We speculate that proximity effect in binary systems some- +how suppress mixing and/or transport of chemical elements from +the interior to the surface. However, firmer conclusions will need +a substantial expansion of the binary stars sample and an exten- +sion to more massive and hotter stars, and/or wider long-period +binary systems. +Acknowledgements +Careful reading of the manuscript and useful suggestions pro- +vided by the referee are acknowledged. We are indebted to Keith +Butler and Norbert Przybilla for kindly sharing their codes, and +the model atoms used in the present work. KP and ET were +initially supported by the Croatian Science Foundation through +research grant IP-2014-09-8656. The research leading to these +results has (partially) received funding from the KU Leuven +Research Council (grant C16/18/005: PARADISE) and from +the BELgian federal Science Policy Office (BELSPO) through +PRODEX grant PLATO. TVR gratefully acknowledges sup- +port from the Research Foundation Flanders (FWO) under grant +agreement number 12ZB620N. +References +Aerts, C., Molenberghs, G., Kenward, M. 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The reduced photometric data are given +in Table A.1 only available in electronic form at the CDS (see article front page). +Article number, page 23 of 25 + +A&A proofs: manuscript no. harps_high-mass_binaries +Fig. A.2: Light curve of V346 Cen from the TESS satellite, plotted versus orbital phase but with a small magnitude offset linearly +dependent on time to shift successive cycles upward in the figure. The earliest points are coloured red and the latest points are +coloured blue. The repetition of the pulsation signature with orbital phase is easy to see. +Article number, page 24 of 25 + +Pavlovski et al.: High-mass eclipsing binaries: a testbed for models of interior structure and evolution +Fig. A.3: Portions of the disentangled spectra of the stars (labelled) studied in this work. +Article number, page 25 of 25 + diff --git a/8dE2T4oBgHgl3EQf8Ait/content/tmp_files/load_file.txt b/8dE2T4oBgHgl3EQf8Ait/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f16312372a2d165f4060e7af573eb5acd80d110 --- /dev/null +++ b/8dE2T4oBgHgl3EQf8Ait/content/tmp_files/load_file.txt @@ -0,0 +1,2486 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf,len=2485 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='04215v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='SR] 10 Jan 2023 Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries ©ESO 2023 January 12, 2023 High-mass eclipsing binaries: a testbed for models of interior structure and evolution⋆ ⋆⋆ Accurate fundamental properties and surface chemical composition for V1034 Sco, GL Car, V573 Car and V346 Cen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pavlovski1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Southworth2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Tkachenko3, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Van Reeth3, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Tamajo4 1 Department of Physics, Faculty of Science, University of Zagreb, 10 000 Zagreb, Croatia e-mail: pavlovski@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='hr 2 Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK 3 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium 4 University of Applied Sciences, 10 410 Velika Gorica, Croatia January 12, 2023 ABSTRACT Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The surface chemical compositions of stars are affected by physical processes which bring the products of thermonuclear burn- ing to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Despite their potential in understanding the structure and evolution of stars, elemental abundances are available for only a few high-mass binary stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We aim to enlarge this sample by determining the physical properties and photospheric abundances for four eclipsing binary systems containing high-mass stars: V1034 Sco, GL Car, V573 Car and V346 Cen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The components have masses 8–17 M⊙ and effective temperatures from 22 500 to 32 200 K, and are all on the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We present new high-resolution and high signal-to-noise spectroscopy from HARPS, and analyse them using spectral disentangling and NLTE spectral synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We model existing light curves and new photometry from the TESS satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We measure the stellar masses to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0% precision, radii to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7% precision, effective temperatures to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6% preci- sion, and abundances of C, N, O, Mg and Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The abundances are similar to those found in our previous studies of high-mass eclipsing binaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' our sample now comprises 25 high-mass stars in 13 binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We also find tidally-excited pulsations in V346 Cen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We reinforce our previous conclusions: interior chemical element transport is not as efficient in binary star components as in their single-star counterparts in the same mass regime and evolutionary stage, possibly due to the effects of tidal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our ulti- mate goal is to provide a larger sample of OB-type stars in binaries which would enable a thorough comparison to stellar evolutionary models, as well as to single high-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' stars: fundamental parameters – stars: evolution – binaries: spectroscopic – binaries: eclipsing – stars: abundances 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Introduction The interior structure and evolution of a star are largely deter- mined by its mass and chemical composition at formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pre- cise and accurate observational constraints on these fundamen- tal physical quantities are required for the validation, calibration and improvement of theoretical models of the interior structure and evolution of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Despite being much more complex than single stars, binary star systems are a treasure trove for testing stellar structure and evolution models and understanding how these might be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In the case of eclipsing binaries (EBs) where both components are detected spectroscopically, it is pos- sible to measure their masses and radii with high precision and accuracy using only orbital mechanics and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Detached systems are particularly valuable as they are expected to evolve as single stars without alteration of their evolution by mass trans- fer episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' ⋆ Based on observations made with the ESO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 m Telescope and the HARPS spectrograph, operated on La Silla, Chile by the European Southern Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' ⋆⋆ Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 is only available in electronic form at the CDS via anonymous ftp to cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='fr (130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5) or via https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='fr/viz-bin/cat/J/A+A/ The role of precise empirical mass measurements is difficult to overstate for validating and calibrating modern and sophisti- cated stellar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Herrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1992) presented a study of 25 luminous galactic OB-type stars and reported a discrepancy between the masses inferred from their spectra (via wind theory) and those predicted by evolutionary models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The authors termed the effect the “mass discrepancy” and emphasised the difficulty in attributing it to either of the two theories (wind and stellar evolution) involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Since then, many attempts have been made to diagnose the cause of the mass discrepancy in intermediate- to high-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Given the high precision and accuracy that SB2 detached eclipsing binaries (dEBs) allow us to achieve in measurements of mass and surface gravity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=', Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2010), these objects are important in studying the mass discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Burkholder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1997) studied seven early-type spectroscopic binaries with masses below 15 M⊙ and reported a good agreement between masses inferred from binary dynamics and those estimated with evolutionary models in all cases where the stars are non- interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Guinan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2000) and Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2009) pre- sented independent studies of the high-mass SB2 dEB V380 Cyg and reported a substantial mass discrepancy for the evolved pri- Article number, page 1 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries Table 1: Basic characteristics of binary systems studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Binary Other Orbital Vmax Spectral Age Cluster Apsidal system designation period (d) (mag) types (Myr) membership period (yr) V1034Sco CPD−41o7742 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='44 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 V + B1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 V 3–8 NGC 6231 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 GL Car HD 306168 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='42 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='74 B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 V + B1 V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 NGC 3572 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='02 V573 Car CPD−59o2628 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='47 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='47 O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 V + B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 V 2 Trumpler16 V346 Cen HD 101837 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='32 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='57 B1/3II/III 10 Stock 14 306 ± 4 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' References to the quantities are given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' mary component, in the sense that its dynamical mass is too low compared to the predictions of standard stellar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Whereas Guinan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2000) showed that the discrepancy could be re- solved by introducing extra near-core mixing in the form of con- vective core overshooting, Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2009) found that ro- tationally induced mixing in models was insufficient to explain the mass discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Indeed, Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2014b) demon- strated that only the combined effects of rotation and convec- tive core overshooting can account for the mass discrepancy ob- served in V380 Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Recently, Massey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2012), Morrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2014), Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2015), and Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2018) reported sys- tematic discrepancies between the Keplerian and evolutionary masses of stars less massive than 30 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020b) found a good agreement between the spectroscopic and dynam- ical masses for 26 early-type binary components whereas their evolutionary masses appear to be systematically overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' These results hint towards models of interior structure and evolution being the primary cause of the mass discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020) and Johnston (2021) demonstrated that the problem cannot be attributed to differences in observation and analysis methods between research groups, and instead showed that the mass discrepancy progressively increases with the evolutionary stage of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In particular, the authors found that higher convective core masses were required in models of stellar structure and evolution for stars that are born with a convective core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The effects of excess core mass can be efficiently mimicked with an enhanced mixing in the near-core regions, irrespective of the true cause(s) of the mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Connecting the treatment of interior mixing in stellar evo- lution models and the mass discrepancy requires extra obser- vational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Surface chemical composition measure- ments are ideal because chemical abundance patterns are ex- pected to be substantially altered by various mechanisms of in- terior mixing and chemical element transport in stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For ex- ample, Heap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2006) found surface nitrogen enrichment in 80% of their sample stars and speculated on the role of rotation in causing this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Overall, these observational find- ings are in good agreement with predictions from rotating stellar evolution models for high-mass stars (Meynet & Maeder 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Maeder & Meynet 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Heger & Langer 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Langer 2012) with the caveat that the observed nitrogen enrichments (Heap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2006) are larger than those predicted by the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Hunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2008, 2009) studied a large sample of intermediate- to high-mass stars in the Magellanic Clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Some of their findings corroborate the theory of rotationally-induced mixing while others contradict it (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=', rapidly and slowly ro- tating stars without and with substantial surface nitrogen en- richment, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Slowly-rotating, nitrogen-enriched stars were also found by Markova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2018) and the authors sug- gested that inadequacies of the models in these particular cases might be related to the efficiency of rotational mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' At the same time, Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2018) presented a detailed study of the surface chemical compositions in several high-mass SB2 dEBs and found no dependence of the abundances of carbon (C), nitrogen (N) and oxygen (O) on either the projected rotational velocity (v sin i) or surface gravity (log g) of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Whilst changes in the photospheric CNO abundances of high-mass B-type single stars have been found (Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Nieva & Przybilla 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Maeder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Cazorla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2017a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Markova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018), the role of rotationally-induced mixing in the formation of these chemical abundance patterns remains poorly quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' At the same time, Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2014) demonstrated that neither v sin i nor the rotational frequency of a star has significant predictive power for the surface N abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Instead, the latter correlates strongly with the effective tempera- ture (Teff) of the star and the frequency of its dominant acoustic oscillation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Furthermore, Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2013) showed that internal gravity waves (IGWs) excited at the convective-radiative boundary near the core in high-mass stars are efficient in trans- porting angular momentum and chemicals on short timescales and over large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pedersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2018) demonstrated that the IGW-driven functional form of the interior mixing pro- file is a good candidate to simultaneously explain the observed properties of gravity-mode oscillations and surface abundances in B-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' SB2 dEBs are at the forefront of efforts to resolve deficien- cies in theoretical stellar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The masses and radii of the component stars can be measured precisely and independently of models, and the requirement for the stars to have the same age and initial chemical composition at formation provides an addi- tional stringent constraint on theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Moreover, mea- sured masses and radii give a precise surface gravity which can be used to break the degeneracy between Teff and log g in spec- tral analysis, boosting the accuracy of measurements of the sur- face chemical compositions of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The DEBCat1 catalogue of dEBs (Southworth 2015) currently lists approximately 300 examples with precisely-measured masses and radii, but only a small fraction of high-mass systems have useful constraints on their photospheric chemical abundances (Serenelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In this study, we aim to enlarge the sample of high-mass SB2 dEBs with accurately determined surface chemical abundance patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In Sections 2 and 3, we present the sample and high- quality spectroscopic data used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Section 4 covers the determination of the spectroscopic orbits of the stars, Sections 5 and 6 the inference of their atmospheric parameters and chemical abundances, and Section 7 the light curve analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The chemical compositions, ages and distances to the binary stars analysed are compared in Section 8 to the properties of their parent clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We finish with a discussion (Section 9) and conclusions (Sec- tion 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' It is important to state that the various analyses pre- 1 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='keele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='uk/jkt/debcat/ Article number, page 2 of 25 Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution sented in this work were performed iteratively to ensure internal consistency in the derived results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Sample We selected four main-sequence (MS) dEBs for study, based on the masses of the components, their membership of open clusters or associations, and on their visibility during the telescope time we were allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Basic information on these targets is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' All eight stars have a spectral type of late-O or early-B and a surface gravity between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 dex, so are in a rela- tively early evolutionary phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' All systems except V573 Car re- side in eccentric orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' All are confirmed members of open clus- ters, although we did not impose any additional constraints on their ages and/or chemical compositions from the cluster mem- bership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' All of our targets except GL Car were included in the homogeneous sample of Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V1034Sco is located in the core of the open cluster NGC 6231, which in turn is near the centre of the Sco OB1 asso- ciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A detailed spectroscopic and X-ray study was presented by Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Light curves have been presented and anal- ysed by Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005) and Bouzid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The most re- cent analysis is that published by Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022b), who deter- mined physical properties of the components and measured ap- sidal motion from our spectra (retrieved from the ESO archive) and the light curves from Bouzid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' GL Car is a dEB studied by the Copenhagen group (Giménez & Clausen 1986) using uvby photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The system shows a significant orbital eccentricity (e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='157) and fast apsi- dal motion with U = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='02 yr (Giménez & Garcia-Pelayo 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Giménez & Clausen 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We know of no previous time-series spectroscopy of the system, so the cur- rent work provides the first measurements of its physical proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V573 Car is one of the brightest stars in very young open cluster Trumpler16, although its membership is disputed (Kaltcheva & Georgiev 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Its spectroscopic binary nature was found by Walborn (1982), and the discovery of eclipses was made by Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001) during a study of the nearby massive binary system η Carinae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001) ob- tained extensive uvby photometry and, combined with radial ve- locities (RVs) from Levato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1991), determined the physical properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V346 Cen contains early B-type components (Houk & Cowley 1975) in an orbit with a significant eccentric- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Apsidal motion is present with a period of U = 306 ± 4 yr (Giménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1986a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Drobek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' High-quality light curves in the Strömgren uvby system were obtained and analysed by the Copenhagen group (Giménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1986a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The only full spectroscopic dataset for this system is our own HARPS data, available through the ESO archive, and which were already analysed by Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Observations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Spectroscopy The spectra presented in this work were all taken in one observing run2 over the nights 2–7 April 2009 using the High-Accuracy Radial-velocity Planet Searcher (HARPS) cross- dispersed échelle spectrograph (Mayor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2003) at the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6-m 2 ESO proposal 083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='D-0040(A), PI J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Southworth telescope at ESO La Silla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' HARPS achieves extreme RV preci- sion due to a high mechanical stability, being fed by two opti- cal fibres, sited in a vacuum chamber, and calibrated by a Th- Ar emission lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We operated HARPS in the high-efficiency EGGS mode, which has a resolving power of R = 80 000, and used the second fibre to obtain the sky background during each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Each spectrum consists of 72 orders incident on two CCDs, covering 3780–6900Å with a gap at 5304–5337Å be- tween the CCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We reduced the spectra using semi-automatic IRAF3 scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Reduction of the spectra included the standard steps: bias sub- traction, flat-field correction, spectral order localisation, extrac- tion, and wavelength calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Normalisation of extracted spectral orders was performed by fitting ninth-order polynomial functions to selected continuum points in the blaze function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Since the Balmer lines cover up to three consecutive spectral or- ders, these were normalised by interpolating the blaze functions from adjacent orders as described by Kolbas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The HARPS blaze functions are very stable so the normalisation and merging of even these difficult orders produced very satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Photometry Our analysis below originally relied on published ground-based light curves, as will be discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In the course of this work, additional data became available from the NASA Transiting Exoplanet Survey Satellite (TESS), a space-based mission that has observed most of the celestial sphere in sec- tors of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 d duration (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The TESS datasets used in the current study are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Additional data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1, and may be useful in future for period or apsidal motion studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' TESS observed V1034 Sco in sectors 12 (1800 s cadence) and 39 (600 s cadence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We extracted the light curves using cus- tom aperture masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V1034 Sco is in a crowded field and the TESS pixels subtend a large angle (21′′) so the light curves con- tain a significant amount of third light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our analysis was based on sector 39 due to the better temporal sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V346 Cen was observed using TESS in sectors 10 and 11 (1800 s cadence), and 37 and 38 (600 s cadence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The light curves available for download from MAST4 are very affected by the field crowding but are nevertheless much better than the ground-based data for this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We based our analysis on the data from sectors 37 and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' TESS observed V573 Car in sectors 10, 36 and 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Because this object is very close to the extremely bright η Car binary sys- tem, the standard data products from TESS (Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2016) are unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We therefore extracted photometry from the halo of V573 Car by making a careful customised pixel selection for the aperture mask, with the aim to maximise the collected flux of V573 Car compared to the flux of η Car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The resulting light curves are of relatively low quality and suffer from a large and varying amount of third light, so we did not use these in our anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We note that V573 Car was just outside the field of view of TESS during sector 11 but we were still able to extract a light curve using halo photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' GL Car was observed by TESS in sectors 10 and 11 (1800 s cadence), and 37 (600 s cadence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The light curves available on 3 IRAF is distributed by the National Optical Astronomy Observatory, which are operated by the Association of the Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=', under cooperative agreement with the NSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4 https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='edu/portal/Mashup/Clients/Mast/Portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='html Article number, page 3 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1: Light curves used in the current study from our reduction of data from the TESS satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' They have been normalised to zero magnitude for display purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' MAST have eclipses that are too deep so we again extracted our own photometry from the TESS full-frame images using custom aperture masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Spectroscopic orbits The spectra of binary systems containing high-mass stars are dif- ficult to analyse for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' First, the v sin i values are typically large, smearing out the spectral lines and causing the lines from the two components to blend together even around the phases of maximum RV difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Second, there are relatively few spectral lines that are strong enough to provide useful RV in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We therefore determined the spectroscopic orbits of the stars using the method of spectral disentangling (SPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This method was introduced by Simon & Sturm (1994) in wavelength space and by Hadrava (1995) in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' It represents the observed composite spectra of a binary system as a sum of the in- dividual spectra of the two stars shifted in RV according to their orbital motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' SPD makes it possible to quantitatively anal- yse time-series spectra of SB2 systems even when line blending is strong (Hensberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pavlovski & Hensberge 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Note that no template spectra are needed for SPD, thus avoiding any biases due to template mismatch (Hensberge & Pavlovski 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We used the FDBinary5 code (Ilijic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2004) to perform SPD in Fourier space using Fast Fourier Transform (FFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For each object we analysed all spectra simultaneously to determine the disentangled spectra of the two stars and their spectroscopic orbital parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We fitted directly for the orbital parame- ters, without the intermediate step of calculating RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The or- bital parameters were the orbital period, P, time of periastron pasage, Tperi, eccentricity, e, argument of periastron, ω, and ve- locity semiamplitudes, KA and KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The orbital periods were held fixed as they are well determined from previous analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The orbital solutions are given in Table 2 in which the mass ratio (q = KA/KB) is also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We also disentangled individual short segments of spectra in order to concentrate on spectral lines of interest, avoid interstel- lar lines, and achieve reasonable computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The Balmer lines were not used in the determination of the spectroscopic orbits because they are much wider than the changes in RV of the stars over an orbital cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The best fits were obtained us- ing the downhill simplex algorithm (Press et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We found 100 runs with 1000 iterations each to be sufficient to ensure the 5 http://sail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='zpf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='fer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='hr/fd3 Article number, page 4 of 25 Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2: Visualisation of the spectroscopic orbits of our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The best-fitting orbits are shown with black lines and the RVs of the stars at the times of observation with red symbols for the primary component, and blue symbols for the secondary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Note that these are not measured RVs, hence the uncertainties in RVs are not assigned to individual symbols, because we calculate orbital parameters directly from all observed spectra for each system (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Table 2: Parameters of the spectroscopic orbits for the four targets determined by SPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Binary P Tperi e ω KA KB q system (d) (BJD) (deg) (kms1) (kms1) V1034Sco 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='440656 51934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='356±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='003 191 ± 12 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='563 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='002 GL Car 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='422238 54901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='182±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='146 (fixed) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='943 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='009 V573 Car 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='469332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 90 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='71 306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='818 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='011 V346 Cen 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='321835 50452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='543±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='289 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='006 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='712 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='038 global minimum was found whilst keeping the required com- putation time manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Convergence was achieved quickly because of the high quality of the HARPS spectra and the avail- ability of preliminary orbital parameters from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Un- certainties in the results were obtained using 10 000 bootstrap- ping simulations (Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2 is a visualisation of the spectroscopic orbits of the four targets and the phase dis- tribution of our spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V1034 Sco V1034 Sco has been found to have a small eccentricity (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Levato & Morrell 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We have been able to measure precise velocity amplitudes for the components (Ta- ble 2) which highlight the low mass ratio of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our results are in good agreement with those from Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2003, 2005) and agree within the errorbars with those from Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We conclude that the RV semi-amplitudes of the components of V1034Sco are now well-determined since the the accuracy achieved is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2% for the primary and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4% for the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The argument of periastron is quite uncertain due to the small eccentricity, and is much better determined from the photometric analysis in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Article number, page 5 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' GL Car To the best of our knowledge, our spectroscopic orbit for GL Car is the first one published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The mass ratio is in fairly good agree- ment with the photometric value of q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='943 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='009 found by Giménez & Clausen (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We fixed the eccentricity to a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='146, which is precisely known from analyses of its apsidal motion (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We fitted for the argument of perias- tron, which is well-determined when the eccentricity is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V537 Car This is the only binary system with a circular orbit in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The HARPS spectra densely cover both quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We also obtained spectra during the primary and secondary minimum, but did not use these in our analysis because the eclipses are not total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A spectroscopic orbit for V573 Car has previously been published by Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001) but based on only two newly obtained spectra and eight spectra taken from Levato & Malaroda (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The velocity amplitudes measured by these authors are quite uncertain but agree with ours to within the errorbars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V346 Cen V346 Cen has a significant eccentricity of e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='289 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our HARPS spectra have good phase coverage and SPD quickly converged to a stable solution (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our results are in rea- sonable agreement with the only previous spectroscopic analysis of this system (Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2016), as expected because they used the same spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' However, the velocity amplitudes we measured are both 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 km s−1 lower than those of Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We attribute this to differences in the methods employed in the two analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In particular, Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2016) employed cross-correlation (Zucker & Mazeh 1994) to determine RVs, using as templates the disentangled spectra themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This approach is mathemat- ically incorrect and extensive numerical experiments have shown that it is not reliable (Ilijic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' SPD has instead been shown to be the best approach to determining spectroscopic or- bits (Southworth & Clausen 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our approach yields masses that are smaller by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='26 M⊙ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='11 M⊙ than those found by Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2016) for the primary and secondary component of the system, respectively, which is larger than the quoted uncer- tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Atmospheric parameters For determination of the atmospheric parameters and individ- ual abundances of C, N, O, Mg and Si, we employed a hy- brid NLTE approach as described in detail in Nieva & Przybilla (2007, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A hybrid NLTE approach means that the mod- elling combines hydrostatic, plane-parallel, and line-blanketed model atmospheres in local thermodynamic equilibrium (LTE) with line formation calculated in NLTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We used the Atlas9 code (Kurucz 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Castelli & Kurucz 2003) for the calculations of model atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Then emergent fluxes and line profiles were calculated with the codes Detail and Surface (Giddings 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Butler & Giddings 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In Detail the coupled radia- tive transfer and statistical equilibrium equations are solved, while Surface was used for the calculations of NLTE syn- thetic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The following model atoms were used in these calculations: H i (Przybilla & Butler 2004), He i/ii (Przybilla Table 3: The atmospheric parameters derived from optimal fit- ting of disentangled spectra of the components to a grid of NLTE spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Star Teff v sin i ξt (K) (km s−1) (km s−1) V1034 Sco A 32 200 ± 500 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 5 ± 1 V1034 Sco B 25 800 ± 300 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 5 ± 1 GL Car A 30 950 ± 500 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 4 ± 1 GL Car B 30 400 ± 500 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 2 ± 1 V573 Car A 31 900 ± 400 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7 5 ± 1 V573 Car B 28 700 ± 350 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 3 ± 1 V346 Cen A 26 100 ± 300 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 5 ± 1 V346 Cen B 22 500 ± 300 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 5 ± 1 2005), C ii/iii (Nieva & Przybilla 2006), N ii (Przybilla & Butler 2001), O i/ii (Becker & Butler 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2000), Mg ii (Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2001), and Si ii/iii/iv (Becker & Butler 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We used the disentangled spectra generated in the previous section to determine the Teff, v sin i, and microturbulent veloc- ity (ξt) for each of the eight stars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This process was greatly helped by the availability of log g values from the measured masses and radii (see Section 7) so our analysis was performed iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The disentangled spectra were still in the common continuum of the binary system so needed to be renor- malised to the continuum of the individual component stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This was done iteratively alongside the light curve analysis, to arrive at light ratios that were consistent between the two types of the analysis (Ilijic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pavlovski & Hensberge 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The exception to the process above was GL Car, for which the light curve solutions suffered from a degeneracy which caused the light ratio to be highly uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We therefore fitted the disentangled spectra to obtain the Teff, v sin i and log g values and the light ratio directly, using the approach of Tamajo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2011) and Kolbas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' After iteration with the light curve solution, log g was fixed for the final measurements of the remaining parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We have found that such spectroscopi- cally determined light ratios can be competitive with those from light curve analysis (Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2009, 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Since we are dealing with late-O, and early-B type stars, the helium ionisation balance (He i/He ii) is a sensitive indicator of Teff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our spectra cover a broad spectral range and thus allowed us to use a large number of lines: 4009, 4026, 4388, 4437, 4471, 4713, 4921, 5015, 5047, 5875 and 6678 Å for He i and 4200, 4541, 4686, and 5411 Å for He ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Once a first set of parameters was obtained, we made the light ratio a free parameter to check its reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' As a further check we also fitted the Hδ, Hγ and Hβ lines, during which we excluded wavelengths affected by inter- stellar absorption (specifically the red wing of Hβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We did not base our Teff measurements on the Balmer lines because their large width makes them susceptible to errors due to continuum normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The He line strengths also depend on ξt, which for hot stars can be obtained by minimising the scatter in the O abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We started with the assumptions of solar He abundance and ξt = 2 km s−1, and subsequently relaxed each of them before refitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Convergence was fast, taking either one or two itera- tions for all eight stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Once this was achieved, we repeated our optimal fitting of the disentangled spectra described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The results of this process are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Below we compare our results to published determinations for each system, except Article number, page 6 of 25 Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 3: Fits to the He i 4388Å and He ii 4541 Å lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The ionisation balance of He i and He ii was used in the determination of Teff for the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The blue data are the disentangled spectra and the red lines the best fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The upper row is for the primary stars and the lower row is for the secondary stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The lower S/N for the secondary components arises because they are fainter than the primary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The absence of He ii 4541 Å absorption in V1034 Sco B and V346 Cen B is obvious and indicates that Teff < 23 000K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' for GL Car for which there is no other analysis based on modern spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V1034 Sco In the most recent study, Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022b) analysed disen- tangled spectra of the components obtained from the HARPS spectra obtained in our observing run, and available at the ESO archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The Teffs they derived are within 1σ uncertainty of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This is encouraging, especially as Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022b) used a different NLTE spectrum synthesis code to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022b) also fitted for surface gravity, using the wings of the Balmer and some He i lines, whereas we prefer the surface gravities determined with a high precision from the masses and radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The two analyses agree to within 2σ, but the uncertainties of the values from Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022b) are much larger (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='10 dex) than our own (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='01 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V573 Car Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001) studied V573 Car using two high- resolution spectra from the FEROS spectrograph, taken near op- posite quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' They fitted the spectra with NLTE synthetic spectra for the He i, He ii, Hδ and Hγ lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The helium lines of the components are not completely resolved at quadrature due to the high v sin i, and the Balmer lines are not resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The agreement between their and our Teff measurements is well within the 1σ errorbars for component A, but only within 2σ for component B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We attribute this to the very small num- ber of spectra available to Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001) compared to our own extensive dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We also find that fitting disentan- gled spectra is superior to fitting individual observed spectra be- cause it avoids problems with blending of lines from the two stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Moreover, disentangled spectra have a higher S/N than in- dividual observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V346 Cen The atmospheric parameters for both components of V346 Cen were determined by Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2016) from the same set of the HARPS spectra as we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In their analysis, Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2016) used optimal fitting of disentangled spectra in similar manner as we did, for fixed surface gravities and microturbulent velocities (fixed to ξt = 2 km s−1) and using a grid of synthetic spectra from Lanz & Hubeny (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' They found a large discrepancy for the secondary: their spectroscopic analysis gave 20 991 ± 190 K and Article number, page 7 of 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 He I 4541 He I 4388 lux ormalised 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 V346 Cen 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='P 6 4 2 0 2 4 Relative wavelength A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 He I 4541 He I 4388 ormalis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 V573 Car B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 6 4 2 0 2 4 Relative wavelength1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 He I 4541 ux He I 4388 p ormalis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 GL Car B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 6 4 2 0 2 4 Relative wavelength1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 He I 4541 He I 4388 1lx ormalised 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 V1034 Sc0 B 6 4 0 2 4 Relative wavelength1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 He I 4541 He I 4388 lux ormalised 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 V346 Cen A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='P 6 4 2 0 2 4 Relative wavelength A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 He I 4541 ux He I 4388 pi ormalis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 V573 Car A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 6 4 2 0 2 4 Relative wavelength1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 He I 4541 He I 4388 pa ormalis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 GL Car A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 6 4 2 0 2 4 Relative wavelength1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 He I 4541 He I 4388 xni ormalised 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 V1034 Sc0 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='P 6 4 2 0 2 4 Relative wavelength AA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries Table 4: Abundances determined for the stars in our sample of binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Star log ǫ(C) log ǫ(N) log ǫ(O) [N/C] [N/O] log ǫ(Mg) log ǫ(Si) V1034 Sco A 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='12 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='17 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='01 V1034 Sco B 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='08 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='12 −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='09 B starsb 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The Teff and log g values used for the construction of the model atmospheres are given in Tables 3 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (a) The abundances found for OB binaries in our previous work (Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018) (b) The ‘present-day cosmic abundances’ for B stars (Nieva & Przybilla 2012) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4: Example of fits to N lines for our target stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In this case the N ii 3995 Å line is shown for V1034 Sco (primary star on the left, secondary star on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The blue lines show the renormalised disentangled spectra of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The red lines show synthetic spectra from our precalculated grid for three dif- ferent abundances (labelled on the bottom right corner in each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' their light curve analysis gave 25 376 ± 18 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The former value is in much better agreement with our result (Table 3), and the complete absence of the He ii lines demands Teff < 23 000 K (based on a detailed examination of theoretical spectra for the He ii 4686 Å line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2016) did not discuss the He ii lines in the secondary’s spectrum at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' They also gave unre- alistically small uncertainties for Teff: ±25 K for the primary, ±190 K for the secondary from spectroscopy, and ±18 K for the secondary from the light curve analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Such low uncertainties are typical for formal errors of the fitting algorithms, but are un- realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Abundance analysis With the Teff, ξt and v sin i from Section 5, and log g from the masses and radii of the stars (Section 7), we have all quanti- ties needed for determining surface abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We calculated model atmospheres for the Teff and log g values of the compo- nents with the atlas9 code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Then a grid of synthetic spectra was calculated in NLTE with detail, and surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The follow- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 5: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4 but for O lines in the components of GL Car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The complex blend of O ii lines at 4941 and 4943 and the O ii line at 4955 Å are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' ing species were considered: C, N, O, Mg and Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Spectra for a broad range of elemental abundances were calculated, spanning ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 dex in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 dex, around the ‘present-day cosmic abundances’ determined in Nieva & Przybilla (2012) (log ǫ(C) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='25, log ǫ(N) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='69, log ǫ(O) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='71, log ǫ(Mg) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='56, and log ǫ(Si) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' These were broadened by the instrumental broadening, and a rotational kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The microturbulent velocity was taken into account in the line profile calculations with sur- face as determined from minimising the scatter in the O abun- dances and given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Abundances were determined by minimising the residuals (χ2 criterion) between the renormalised disentangled spectrum and the synthetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=" In the renor- malisation of the disentangled spectra to their individual contin- uum, the light ratio obtained in the light curve analysis (Table 6) Article number, page 8 of 25 ux ormalised 66'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='97 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70 GL Car B 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content="96 4930 4940 4950 4960 Wavelength [Axn ormalised 66'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='97 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70 G tar 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='96 4930 4940 4950 4960X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 pasi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='96 ormal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='92 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70 V1034 Sc0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 3992 3994 3996 3998 Wavelength Aux .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content="995 p含 ormal 66'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='985 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70 V1034 Sco 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 3992 3994 3996 3998 Wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' FAPavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 6: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4 but for lines in the components of V573 Car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The upper two panels show O ii lines at 4661, 4673 and 4676 Å (shown with green lines), which are blended with C ii lines at 4659, 4663, 4665 and 4673 Å (shown with red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The calculated synthetic spectrum is shown using a black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The lower two panels show the N ii 3995 Å lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' was used, except for GL Car where the spectroscopically deter- mined light ratio was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The number of lines available for the abundance determina- tion of a particular element varies due primarily to Teff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For the Teff range covered by our target stars, the spectral lines of CNO are quite varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The most numerous spectral lines are for O, which is why we used them to determine ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The spectral lines of C are the least numerous, and the broad wavelength coverage of HARPS spectra is of vital importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We show examples of the disentangled and synthetic spectra in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4 to 7 for selected CNO lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The results for all five elements are given in Table 4, as well as the indices [N/C] and [N/O].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Uncertainties were calculated including the standard deviation of the mean for available spec- tral lines, and the uncertainty due to uncertainties in Teff and ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 7: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 4 but for V346 Cen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The upper two panels show the C ii lines at 5133–5151Å, and the bottom two panels show the N ii 4630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5Å line for the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Uncertainty in the abundances due to uncertainties in the surface gravity are negligible since log g is determined to high precision from the masses and radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A fairly good agreement between the abundances in both components of the same binary system is seen from examina- tion of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The most notable difference is for the abundance of magnesium (Mg) which in three cases (V1034 Sco, V573 Car and V346 Cen) is modestly larger than the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The Mg abundances are inferred from a single line, Mg ii 4481 Å, so their uncertainties are larger than for other species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For other elements, the observed abundance differences are mostly below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 dex, well within the 1σ uncertainty interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Apart from Mg, the largest deviations are for the C abundance in V346 Cen (log ǫ(C)A − log ǫ(C)B = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='08 dex), the N abundance in V573 Car (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='12), and the O abundance in V346 Cen (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V346 Cen A has the lowest C abundance among the eight stars, with log ǫ(C) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05, almost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='20 dex less than the mean C abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Contrary to this, the N abun- dance for the same star is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' It is also worth noting that V346 Cen A is the most evolved in our sample of eight OB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Article number, page 9 of 25 Iux ised 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 ormali 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='96 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='90 V346 Cen B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='94 4626 4628 4630 4632 4634 4636 Wavelengthux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='99 ormalised 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='97 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 V346 Cen A 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='90 4626 4628 4630 4632 4634 4636 Wavelength Apast 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 ormal 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='96 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='20 V346 Cen B 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='40 5130 5135 5140 5145 5150 5155 Wavelength [AX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='995 ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='985 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='20 V346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 Cen 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='30 5130 5135 5140 5145 5150 5156xnl ormalised 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='96 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='90 0668 3992 3994 3996 3998 4000 Wavelengthxn ormalised 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 V573 Car 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='90 0668 3992 3994 3996 3998 4000 Wavelength Apa ormalis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='96 C V573 Car B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='94 0 4660 4665 4670 4675 4680 Wavelengthxn 1 pa STT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='98 ormal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='96 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='94 :ar 0 4660 4665 4670 4675 4680A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8: Individual abundances of carbon (left), nitrogen (middle), and oxygen (right) as a function of surface gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The stars in the present sample are indicated with filled blue circles while stars taken from our previous studies are shown with open blue circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The surface gravity (obtained from the binary solution) is used as a proxy for stellar evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Solid red lines show theoretical evolutionary tracks for a 15 M⊙ star and three values of the initial rotational velocity Ω/Ωcrit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 (Georgy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The cosmic standard abundance values of Nieva & Przybilla (2012) are indicated with horizontal dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 9: Same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8 but for the [N/C] and [N/O] abundance indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8 compares individual CNO abundances determined for the eight stars in this work to our previous abundance measurements in high-mass binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The new determinations are shown in solid blue circles, whilst our previous results are represented with open blue circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our previous deter- minations are for 17 high-mass stars in nine binary systems: V578 Mon (Pavlovski & Hensberge 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018), V453 Cyg (Pavlovski & Southworth 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018), V380 Cyg (Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Table 5: Comparison of abundances determined for targets in present work to their parent clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' NGC 6231 Element Kilian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Mathys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This work (1994) (2002) V1034 Sco log ǫ(C) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='29±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='17 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='08 log ǫ(N) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='08 log ǫ(O) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='42 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='12 log ǫ(Mg) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='04 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='11 log ǫ(Si) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='06 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='10 [N/C] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='11 [N/O] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='40 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='14 NGC 3293 Element Hunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Morel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V573 Car (2009) (2022) GL Car log ǫ(C) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='97±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='19 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='17 log ǫ(N) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='60±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='27 log ǫ(O) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='17 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='18 log ǫ(Mg) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='16 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='20 log ǫ(Si) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='09 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='26 [N/C] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='27 [N/O] −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='26 – −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='11 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V1034 Sco is member of the open cluster NGC 6231 for which abundance analyses were published by Kilian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1994) and Mathys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For the open clusters NGC 3572 and Trumpler 16, parent clusters of GL Car and V573 Car, respectively, no abundance studies are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We used abundance studies of the open clus- ter NGC 3293, since it is part of the large Car OB1 complex, as are NGC 3572 and Trumpler 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2014b), σ Sco (Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2014a), α Vir (Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2016), CW Cep (Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2019), AH Cep (Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018), V478 Cyg (Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018) and the primary component in V621 Per (Southworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' There are no discernable systematics between the new and previous sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This is expected because we are con- sistently using the same reduction and analysis tools, so the 25 stars (in 13 binary systems) represent an homogeneous sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Article number, page 10 of 25 [N/O] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 [dex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 0 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 3 log g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 [N/C] 0 xap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 3 log g [dexPavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution In the last three rows of Table 4 the mean values for abundances in the present sample, in OB binaries studied previously by us, and the ‘present cosmic abundance standard’ – an abundance pattern evaluated for B-type stars by Nieva & Przybilla (2012) – are given for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' As already mentioned, both our samples are in perfect agreement and there are no outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' However, it can be seen that the CNO abundances for OB stars in binary systems are below the cosmic abundance standard, with very few exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Nieva & Przybilla (2012) determined elemental abundances for sample of sharp-lined early B-type stars, enabling a very high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This was not an option for our work because our sample stars are all in short-period binary systems so are either moderate or fast rotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Thus the spectral lines are usually broad and overlapping, making the choice of suitable spectral lines for abundance determination more limited, and thus affecting the accuracy of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8 presents a comparison of the inferred CNO abun- dances with theoretical evolutionary tracks of a 15 M⊙ star com- puted for three values of the initial rotational velocity Ω/Ωcrit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 (Georgy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In their model calcula- tions Georgy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2013) used the following initial abundances: log ǫ(C) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='28, log ǫ(N) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='67, log ǫ(O) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The values for the abundances of C and N are in fair agreement with our present and preious findings (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Table 5, but differ by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='15 dex for the O abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Therefore, we empirically ‘corrected’ the initial O abundance in the theoretical models, and shift the O abundance upwards in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8, and accordingly for [N/O] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' One can see that the models predict significant depletion and enhancement of C and N, respectively, as the rotation rate of the star increases, while only a marginal depletion is predicted for O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Moreover, these abundance trends are substantial at the start of the main-sequence evolution already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' However, the individual abundances of CNO elements measured by us do not follow the relations predicted by the models: instead we observe a scatter of values around or slightly below the cosmic standard abundance values of Nieva & Przybilla (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Furthermore, the N to C abundance ratio index [N/C] is a sensitive probe of the stellar evolution model predictions, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The models suggest a noticeable in- crease of the surface N abundance with respect to the abun- dance of C (top panel) and O (bottom panel) as the rotation rate of the star Ω/Ωcrit increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Similar to the individual el- emental abundances discussed above, we do not observe the in- crease in the [N/C] and [N/O] indices as the surface gravity of the star decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Moreover, the bulk of our abundance measure- ments cluster around the mean [N/C] and [N/O] values found by Nieva & Przybilla (2012) in the solar neighbourhood, with the spread being significantly smaller than one would expect if the abundance ratios were altered substantially by the effect of stel- lar rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' No previous abundance determinations are available for any of the binary systems analysed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' However, we can check our results against published abundances for the open clusters our sample are members of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Photospheric abundances for B-type stars in the open cluster NGC 6231 were determined by Kilian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1994) and Mathys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Results from these studies are compared to our results for V1034Sco in Ta- ble 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The parent clusters for the other three systems have not (yet) been subject to a chemical composition study, but the open cluster NGC 3293 (which is part of young association Car OB1 Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1980) is well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Hunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2009) deter- mined abundances from 50 B-type stars, while in a recent publi- cation Morel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022) examined a large sample of about 150 B-type stars in the framework of the Gaia-ESO Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Since the dEBs V573 Car and GL Car belongs to the Car OB1 association, we use NGC 3293 as a proxy for the abundance pattern in Car OB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' It is interesting that the massive sample of B-type stars analysed in Morel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022), with a spread in v sin i values, show a pattern of under-abundances compared to the standard solar abundances (Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2009), in accordance with our general abundance pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Light curve analysis We assembled the available light curves of the four targets in this work and modelled them using a consistent approach in order to determine their physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The light curves were fit- ted using the Wilson-Devinney (WD) code (Wilson & Devinney 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Wilson 1979), which implements Roche geometry to determine the shapes of the stars and thus the brightness of binary systems as a function of orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We used the 2004 version of the WD code, driven with the jktwd wrapper (Southworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For each system we performed a series of tests to determine the best approach to modelling it with jktwd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Once we had ar- rived at the preferred solution, we performed further tests to determine the range of plausible solutions and thus the uncer- tainties in the fitted parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This step was taken because we have consistently found that the formal errorbars calculated by the WD code underestimate the true uncertainty of the fitted pa- rameters (Pavlovski & Southworth 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2009, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Southworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2020), as indicated in the user guide to the code (Wilson & Van Hamme 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Unless otherwise specified we used Mode 0 in the WD code, which is for detached binary systems where the light contribu- tions for each star are fitted individually, simple reflection, and the logarithmic limb darkening (LD) law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We fitted for the po- tentials and light contributions of the two stars, the orbital in- clination and a phase shift with respect to the adopted orbital ephemeris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The mass ratio was fixed at the spectroscopic value, bolometric albedos were set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0, synchronous rotation was as- sumed, and the gravity brightening exponents were set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A circular orbit was assumed for V573 Car but the possibility of an eccentric orbit was checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The input LD coefficients were obtained by bilinear interpolation in the tables of van Hamme (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For the purposes of determining the uncertainties in the fitted parameters, we ran a series of alternative solutions for differing choice of WD code mode of operation, choice of numerical reso- lution, treatment of reflection, choice of LD law, whether the LD coefficients were fixed or fitted, treatment of third light, variation of the mass ratio within the uncertainties, and the possibility of orbital eccentricity (for V573 Car).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We also considered the ef- fects of albedo, rotational velocity and gravity brightening, by fixing them at different values and also attempting to fit for them directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The net result of this process was a default solution for each system, accompanied by a measurement of how much each fit- ted parameter changed between this default solution and each of the alternative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' These changes were then added in quadrature to arrive at a final robust uncertainty value for each fitted parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The results for all four systems are summarised in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The fractional radii are volume-equivalent values ob- tained from the lc flavour of the WD code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Article number, page 11 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries Table 6: Summary of the parameters for the wd2004 solutions of the light curves of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Parameter wd2004 name V1034 Sco GL Car V573 Car V346 Cen Control and fixed parameters: wd2004 operation mode mode 0 0 0 0 Treatment of reflection mref 1 1 1 1 Number of reflections nref 1 1 1 1 Limb darkening law ld 2 (logarithmic) 2 (logarithmic) 1 (linear) 2 (logarithmic) Numerical grid size (normal) n1, n2 60 50 50 50 Numerical grid size (coarse) n1l, n2l 50 40 40 40 Fixed parameters: Orbital period (d) period 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='440646 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4222681 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4693316 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3220088 Primary eclipse time (HJD) hjd0 2451931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2652 2459321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2994 2450456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8164 2459335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5607 Mass ratio rm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='712 Teff star A (K) tavh 32 200 30 960 31 900 26 100 Teff star B (K) tavh 25 800 30 390 28 700 22 500 Rotation rates f1, f2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='67, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0, 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 Bolometric albedos alb1, alb2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0, 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0702 Star A potential phsv 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='670 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='044 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='736 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='033 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='913 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='024 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='933 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='015 Star B potential phsv 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='228 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='031 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='766 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='064 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='106 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='027 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='032 Orbital inclination (◦) xincl 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='32 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='17 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='14 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='12 Orbital eccentricity e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='027 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1465 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 (fixed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2750 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0006 Argument of periastron (◦) perr0 57 ± 18 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='36 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='28 Light from star A (u band) hlum 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='006 Light from star B (u band) clum 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='269 Light from star A (v band) hlum 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='861 Light from star B (v band) clum 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='449 Light from star A (b band) hlum 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='893 Light from star B(b band) clum 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='531 Light from star A (y band) hlum 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='888 Light from star B (y band) clum 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='551 Light from star A (TESS band) hlum 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='096 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='081 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='374 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='086 Light from star B (TESS band) clum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='204 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='967 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='006 Third light (TESS band) el3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='239 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='166 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='214 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='007 Fractional radius of star A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3300 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2203 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3308 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2088 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0014 Fractional radius of star B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1901 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2088 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2759 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1106 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0008 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Detailed descriptions of the control parameters can be found in the WD code user guide (Wilson & Van Hamme 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A and B refer to the primary and secondary stars, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Uncertainties are only quoted when they have been robustly assessed by comparison with a full set of alternative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V1034 Sco Two photometric studies of V1034 Sco have been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Bouzid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005) presented light curves taken in the Ström- gren uvby filters, with 409, 645, 1058 and 1036 datapoints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005) obtained light curves in two narrow-band filters, designated λ4685 and λ6051, containing 112 and 138 datapoints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For our exploratory so- lutions we used the Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005) data as the Bouzid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005) data are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In the course of this work a new light curve became avail- able from sector 39 of the TESS satellite (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' As the TESS data are of much higher quality than the other photome- try, we have based our final results for V1034 Sco on these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Before doing so, we performed a preliminary fit with jktebop (Southworth 2013) to obtain an orbital ephemeris then phase- binned these data down to 500 bins to decrease the computa- tion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our final solution is for an eccentric orbit, including third light, and the logarithmic LD law and the linear LD co- efficient fitted for each star and passband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The main contribu- tors to the uncertainty in the fractional radii are the treatment of albedo and gravity darkening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Uncertainties arising from the choice of numerical resolution, WD program mode, rotation rate and LD were all significantly smaller and therefore contributed negligibly when all uncertainties for each parameter were added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The parameters and their uncertainties are given in Table 6, and the best fits are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' GL Car Light curves of GL Car in the Strömgren uvby system were ob- tained by Giménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1985) using the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 m Strömgren Au- tomated Telescope at ESO La Silla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' They comprise 526 ob- servations through each filter, 234 in the 1982 observing sea- son and 308 in the 1983 season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' These data were analysed by Giménez & Clausen (1986) using the wink code (Wood 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The observations were obtained in electronic form from the archive of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Clausen and used in the current work to ob- Article number, page 12 of 25 Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 10: The light curves and best WD models for V1034 Sco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The differential magnitudes are plotted versus orbital phase and are colour-coded according to the central wavelengths of the passbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The source and passband of each light curve is la- belled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The residuals of the fit are shown at the base of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Offsets have been applied between the light curves and residuals for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' tain a preliminary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We found values and uncertainties for the fitted parameters in good agreement with those from Giménez & Clausen (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Subsequent to our analysis of the uvby data a new light curve of GL Car became available from TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We phase-binned this and modelled it using wd, fitting for an eccentric orbit and third light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The rotation rates (F1 and F2) were set to the ratios of the measured rotational velocities (Table 3) and the synchronous values, determined iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Unlike the uvby data, the TESS light curve shows a strong correlation between the light ratio and the amount of third light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We therefore applied the light ra- tio from our spectroscopic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Because there is no mech- anism to explicitly apply a spectroscopic light ratio in wd2004 we propagated the light ratio from our spectral interval (which corresponds closely to the Johnson B band) to the Strömgren uvby bands (see Southworth 2010) using atlas9 theoretical spec- tra (Castelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1997) and passband response functions from Maíz Apellániz (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We forced wd2004 to match them by fix- ing the hlum parameters at the appropriate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Including this constraint greatly improved the reliability of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We find precise fractional radii for GL Car once our spec- troscopic light ratio is included (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The uncertainties Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 11: The light curves and best WD models for GL Car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Other comments are the same as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' are dominated by those from this light ratio, but are still be- low 1% and a factor of three smaller than those from the uvby data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' They also agree well with the less precise results from Giménez & Clausen (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The fitted orbital eccentric- ity is in excellent agreement with that from its apsidal motion (e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1459 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0015 from Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The orbital phase of secondary eclipse has changed a lot between the uvby and TESS datasets, and the different morphology of the light curve is obvious (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V573 Car V573 Car was studied by Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001) using the Dutch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 m telescope at ESO La Silla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A total of 1910 observa- tions were obtained through the Strömgren filters: 763 in y, 513 in b, 350 in v and 284 in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We fitted all four light curves simul- taneously, using the ephemeris from Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We assumed a circular orbit in most cases, but did run a fit with e and ω free to check if this led to a better fit to the data (it didn’t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We also assumed no third light, after attempts to fit for it had a negligible effect on the results and also led to a slightly negative value for this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The best fit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The fitted parameters were the potentials of the two stars, the orbital inclination, a phase shift, and the light contributions of the two stars in each passband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' To avoid very small values for the light contributions we renormalised each light curve to be at ap- proximately zero relative magnitude at quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We adopted Article number, page 13 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 12: The light curves and best WD models for V573 Car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Other comments are the same as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' the linear LD law as it gave results very similar to those for the logarithmic and square-root laws;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' attempts to fit for the LD co- efficients led to unphysical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The rotational velocities of the stars were held to the synchronous values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We found that the solution of the light curves is degenerate in that significantly different values of the ratio of the radii or the light contributions of the stars led to almost indistinguish- able fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This was also found by Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001), who constrained their solution using a light ratio measured from their spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We took the same approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For our final result (Table 6) we give the solution for fitting all four light curves simultanously, constrained by the spectro- scopic light ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The uncertainties in the parameters include contributions from the uncertainty in the spectroscopic light ra- tio, the effect of a change of 5% in the rotation velocities of the stars, and the treatment of albedo and gravity darkening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Other sources of uncertainty (see above) were checked and found to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We were able to measure the fractional radii of the stars to precisions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8% (star A) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1% (star B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' the main contribution to these uncertainties is the spectroscopic light ratio (for star A) and the treatment of gravity darkening (for star B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The values we find are in reasonable agreement with those from Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001), but our uncertainties are slightly larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 13: The light curves and best WD models for V346 Cen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Other comments are the same as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V346 Cen Extensive photometry in the Strömgren uvby system was ob- tained by Giménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1986b), comprising 1056 observations made simultaneously through all four filters using the Strömgren Automated Telescope (Grønbech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' These data have been analysed by Giménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1986a) using the wink model, and by Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2016) using the phoebe code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The two stud- ies agree on the values of the fractional radii to within the uncer- tainties quoted by Giménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1986a) but not the uncertain- ties quoted by Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We therefore performed our own analysis of these data in order to assess robust errorbars and check the level of agreement with the previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our WD code model for the uvby data provided a good fit to the observations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 13) but required ω to be fixed at a suit- able value to avoid the fit diverging to unphysical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We set the rotation rates to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='49 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='70 based on the rotational ve- locities of the stars measured from the disentangled spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The logarithmic LD law was adopted, although the other two laws gave almost identical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Third light was fixed at zero be- cause attempts to fit for it returned a small negative value that was consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our results were in excellent agree- ment with those of Giménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1986b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' After this work had been performed, light curves from sec- tors 37 and 38 of the TESS satellite became available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' These are of much higher quality so we used them for our final anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We performed a preliminary fit with jktebop to obtain an orbital ephemeris then phase-binned them into 500 bins to make Article number, page 14 of 25 Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution Table 7: Physical properties measured for the four systems analysed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Parameter V1034 Sco GL Car V573 Car V346 Cen Mass ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5628± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='943± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8182± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7119± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0038 Mass of star A (MN ⊙) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='01± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='14 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='86± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='31 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='11± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='13 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='74± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='12 Mass of star B (MN ⊙) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='573± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='053 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='95± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='30 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='365± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='096 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='359± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='089 Semimajor axis (RN ⊙) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='767± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='053 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='79± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='15 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='412± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='044 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='12± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='13 Radius of star A (RN ⊙) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='513± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='075 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='242± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='048 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='429± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='043 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='278± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='079 Radius of star B (RN ⊙) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='328± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='051 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='968± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='051 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='528± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='049 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='123± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='072 Surface gravity of star A (log[cgs]) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='917± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='009 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='199± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='007 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='148± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='007 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='672± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='008 Surface gravity of star B (log[cgs]) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='147± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='010 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='220± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='008 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='218± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='009 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='130± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='015 Synch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' rotational velocity of star A ( km s−1) 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='25± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='64 Synch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' rotational velocity of star B ( km s−1) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='00± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='57 Teff of star A (K) 32200± 500 30960± 500 31900± 400 26100± 300 Teff of star B (K) 25800± 300 30390± 500 28700± 350 22500± 300 Luminosity of star A log(L/LN ⊙) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='738± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='028 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='357± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='029 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='439± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='023 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='457± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='022 Luminosity of star B log(L/LN ⊙) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='874± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='028 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='278± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='030 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='098± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='023 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='594± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='028 Absolute bolometric magnitude of star A −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='104± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='071 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='073 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='331± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='057 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='403± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='054 Absolute bolometric magnitude of star B −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='944± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='057 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='075 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='505± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='058 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='245± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='069 Interstellar extinction E(B − V) (mag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='03 Distance (pc) 1460± 50 2278 ± 63 2466 ± 78 2290 ± 60 Gaia DR3 parallax (mas) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6452± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4232± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4428± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4380± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0261 Gaia DR3 distance (pc) 1550± 56 2363 ± 73 2260 ± 100 2280 ± 140 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The units labelled with a superscripted ‘N’ are given in terms of the nominal solar quantities defined in IAU 2015 Resolution B3 (Prša et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' the computations faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our approach was the same as for the uvby data except that we were able to fit for ω and also needed to fit for third light due to significant contamination of the TESS light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We found the best fit to the TESS data to be highly stable against changes in mass ratio, rotation rate, treatment of LD, albedo, gravity darkening and numerical grid size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We had to fix the LD coefficients as they diverged to unphysical values when we attempted to fit for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The final parameters and uncertainties of the fit are given in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The fits are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 13, and two things are worth highlighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' First, the morphology of the light curve has changed between the uvby and TESS epochs due to apsidal mo- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The phase of secondary eclipse has changed and it is no longer annular – the primary eclipse has become a transit in- stead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Second, the TESS data show a clear pulsation signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This affected the quality of our solution and was probably why we were unable to fit for LD coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The pulsation almost certainly arises from the EB itself and not from the contaminat- ing light, because they are commensurate with the orbital pe- riod (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V346 Cen is therefore another high-mass EB showing pulsations (Southworth & Bowman 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Because our light curve solution did not account for pulsations, we have con- servatively doubled the uncertainties in the measured fractional radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pulsations Following the binary analysis, an analysis of the residual light curve (hereafter called the pulsation light curve) revealed the presence of tidally excited pulsations, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' To measure this tidally induced variability, we fitted sine waves, corresponding to the 20 lowest-order orbital harmonic frequen- cies, to the out-of-eclipse part of the pulsation light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Fit- ted orbital harmonics were accepted when the signal-to-noise ratio S/N ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0, where S/N was calculated as the ratio of the amplitude of the fitted sine wave, and the average signal ampli- tude of the Lomb-Scargle periodogram (Scargle 1982) in a 1 d−1 window around the considered frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Finally, the measured amplitudes, phases and orbital harmonic frequencies were opti- mised simultaneously by nonlinearly fitting them to the pulsation light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Their values are listed in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' From these results, we determined that the tidally induced pulsation corresponds to the 9th orbital harmonic, in agreement with what is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The physical origin of the other measured orbital harmonics is less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' While they may also partially correspond to tidally induced pulsations, this could not be confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' At least part of it is likely caused by the non- sinusoidal nature and orbital-phase dependent amplitude modu- lations of the 9th orbital harmonic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Moreover, as shown in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 14, this pulsation has a minimum dur- ing the primary eclipse and a maximum during the secondary eclipse, which indicates that it belongs to the primary compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Finally, after the significant tidally excited variability was re- moved from the pulsation light curve, we evaluated the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 15, the remaining data exhibit signatures of stochastic low-frequency variability, as has been reported in the literature for other high-mass stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=', Bowman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Physical properties We have determined the physical properties of the systems using the results from the spectroscopic and photometric analyses out- lined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For this we used the velocity amplitudes, Teff values, e and ω from the spectroscopic analysis, and the fractional radii and orbital inclination from the photometric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' To perform the calculations we used the jktabsdim code (Southworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2005), which propagates the errorbar from each input parame- ter using a perturbation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We used a version of jktab- sdim modified to use the IAU system of nominal solar values (Prša et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2016) plus the NIST 2018 values for the Newtonian Article number, page 15 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 14: Tidally excited pulsations of V346 Cen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Top: observed light curve of V346 Cen for sectors 37 and 38, phase-folded with the binary orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The eclipses are indicated by the grey bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Middle: Pulsation light curve of V346 Cen for sec- tors 37 and 38, phase-folded with the binary orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Data points taken during the eclipses again lie within the grey bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Bottom: Lomb-Scargle periodogram, calculated for the out-of- eclipse data points of the pulsation light curve for sectors 37 and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The dashed vertical lines indicate harmonics of the orbital frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 15: Stochastic low-frequency variability of V346 Cen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Top: part of the (out-of-eclipse) residual light curve of V346 Cen, after fitting the orbital harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Bottom: Lomb-Scargle pe- riodogram, calculated for the out-of-eclipse data points of the residual light curve for sectors 37 and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Table 8: Values of the amplitudes A, frequencies ν, phases φ and signal-to-noise ratios S/N of the orbital harmonics, calculated for the out-of-eclipse data points in the pulsation light curve of V346 Cen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' norb A (mmag) ν (d−1) φ (2π rad) S/N 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='813 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='15818130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0811 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0005 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='812 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='79090648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='378 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='005 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='771 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='94908778 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='240 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='005 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='483 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='26545037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='237 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='008 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='938 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='42363166 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2993 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0010 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The frequency values were fixed at the indicated integer multi- ples of the measured orbital frequency νorb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' gravitational constant and the Stefan-Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The results of this analysis are given in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Distances have been derived using the measured radii and Teffs of the stars, apparent magnitudes of the system in the Johnson-Cousins UBVRI and 2MASS JHKs bands, and the the- oretical bolometric corrections tabulated by Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We adjusted the interstellar extinction E(B − V) to obtain con- sistent distances in the optical and infrared passbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' These re- sults are given in Table 7 alongside the Gaia EDR3 parallaxes (Gaia Collaboration 2016, 2021) and the distance from simple inversion of the parallax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We see agreement within the errorbars, the most discrepant (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6σ) being for V573 Car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Very similar con- clusions are drawn if we use the geometric or photogeometric distances from Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We conclude that our results for all four targets are independently verified by the Gaia parallaxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The parent clusters Knowledge of the properties of stars in dEBs allows the determi- nation of their distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Moreover, a comparison of the proper- ties of dEBs to stellar evolutionary models constrains their age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The age of stars in our sample, except for GL Car, were deter- mined from isochrone fitting in Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020) for two cases: (i) as a single star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (ii) as a binary where the two compo- nents have the same age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Three different interior structures were assumed in these calculations, hence in Table 9 we give lower and upper limits for the age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The distances to the binary systems in our sample are given in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V1034 Sco and NGC 6231 The distance to V1034 Sco was evaluated by Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005) who found d = 1528+117 −109 pc, which is within 1σ of the distance we calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The light curve solution in Bouzid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005) suffers from an ambiguity in setting the primary’s Teff so the au- thors calculated the distance for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The larger one is exactly the same as those reported by Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2005), while the shorter one is d = 1399+20 −20 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2008) deter- mined the distance to another dEB in this cluster, V1007 Sco, as 1622 pc (no uncertainty given) which is somewhat larger than the other distance estimates mentioned here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The open cluster NGC 6231 belongs to the star-formation complex Sco OB1 (Perry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The cluster is the oldest and most massive in Sco OB1 (Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The ages of the cluster members have been estimated to be between 2 and 8 Myr (Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Damiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2017), with OB stars being an older population in the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The clus- Article number, page 16 of 25 flux (mmag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 2324 2325 2326 BJD-2457000 amplitude (mmag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 0 2 4 6 8 10 frequency (d-1)(mmag) 100 flux ( 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 orbitalphase 10 (mmag flux 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2 orbital phase amplitude 1 2 3 4 5 frequency (d-1)Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution Table 9: Distances and ages for the binary systems in the present sample compared to the parent clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Binary Distance (pc) Age (Myr) Cluster Distance (pc) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Age (Myr) Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V1034 Sco 1460 ± 50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7 | 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 NGC 6321 1538 ± 20 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 2 GL Car 2278 ± 63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 NGC 3572a 2444 ± 33 3 1–4 4 V573 Car 2466 ± 78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 | 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7 Trumpler16 2360 ± 505 5 2 ± 1 6 V346 Cen 2290 ± 60 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 | 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 Stock 14 2439 ± 326 7 10 ± 2 7 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The distances to the binary systems are from the present work (Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The ages were calculated by Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020) except GL Car for which the age is adopted from Giménez & Clausen (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020) estimated the age for two options: assuming the components are individual stars, and constraining the age to be the same for both components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Both measurements are given, separated by a vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' References.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1) Banyard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2) Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (3) Clariá (1976);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (4) Garcia (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (5) Göppl & Preibisch (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (6) Hur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (7) Paunzen & Netopil (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' ter is rich in spectroscopic binaries: García & Mermilliod (2001) listed about 30 systems of which 16 are certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2008) did an exhaustive search of the cluster members, and listed ten EBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The most recent distance determinations to NGC 6231 are based on Gaia parallaxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2019) quoted d = 1710+13 −100 pc using Gaia DR2, while Banyard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022) found the median geometric and photogeometric dis- tances for their sample of about 60 stars in the cluster using Gaia EDR3 parallaxes to be 1579 and 1576 pc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022b) determined the age of V1034 Sco to be τ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 Myr, in perfect agreement with the result of Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Three other binary systems that are members of this cluster were studied: HD 152248 (Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2020), HD 152219 (Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2022b) and HD 152218 (Rosu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Their ages were determined from the apsidal motion rate and range from 5 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' GL Car and NGC 3572/Collinder 240 Giménez & Clausen (1986) found a distance to GL Car of d = 2100 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' They did not give an uncertainty but quoted an error of 100 pc due to bolometric corrections and interstellar reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This distance is smaller than our result and that from the Gaia DR3 parallax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Giménez & Clausen (1986) extensively discussed possible physical relationships to the open clusters in the vicin- ity of GL Car, which is in a region crowded with young open clusters and in the direction of the Sagittarius-Carina spiral arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Membership of GL Car in NGC 3572 was proposed by Sahade & Berón Dàvila (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Giménez & Clausen (1986) re- jected this association due to the shorter distance to the dEB than the cluster, and because NGC 3572 is a compact cluster with a radius of 5′ and GL Car is at an angular distance of 40′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' It was recognised that the open cluster NGC 3572 consists of two overlapping clusters, one at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 kpc and one at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 kpc (Clariá 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The nearer cluster is also considered by Clariá (1976) to be the probable nucleus of a scattered group of OB stars located in the vicinity, identified as Collinder 240 and an extension of Car OB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This is a region in which the line of sight is tangential to the molecular cloud ridge in the Carina Arm, and is projected on a rather small area in the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' It shows as a region with a higher concentration of OB stars, but with a radial extension of several kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The age of GL Car was determined to be τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 Myr (Giménez & Clausen 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This is compatible with age deter- minations for Collinder 240, τ ∼ 1 Myr, and Car OB2, τ = 4 Myr (Garcia 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V573 Car and Trumpler16 Freyhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2001) determined a distance to V573 Car of d = 2600 ± 120 pc, and an age of τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Their dis- tance determination is within 1σ of ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Also, the very young age is confirmed with extensive isochrone fitting to different stel- lar interior structure models in Tkachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020), as sum- marised in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V573 Car is situated near the centre of the open cluster Trum- pler16, close to η Carinae, the brightest star in the cluster, and one of the most intriguing objects in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The cluster itself, with its neighbouring clusters, Trumpler14, and Trum- pler15, forms a chain of rich clusters in the prominent Carina star-forming complex, a conspicous part of the Carina-Vela spi- ral arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The whole region is recognised as the young asso- ciation Carina OB1, which also includes NGC 3293 and sev- eral small open clusters, the H ii region and prominent neb- ula NGC 3372 powered by η Car (Smith 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Wright 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Pre-Gaia distance estimates relied mostly on multicolour pho- tometry, and gave distances in the range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 kpc and young ages in the range 1–3 Myr (Hur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Using Gaia EDR3 Shull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2021), Maíz Apellániz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2022) and Göppl & Preibisch (2022) found distances to the cluster Trum- pler16 at the lower end of the range: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='12, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='20 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='05 kpc, respectively, all within 1σ of our determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' V346 Cen and Stock 14 The distance and age of V346 Cen were also determined by Giménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1986b), which allows a direct comparison with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Giménez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (1986b) determined the distance d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='18 kpc, which is within 1σ of our determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The age of the binary system they found, τ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6 Myr, also agrees well with our result, τ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0Myr (Table 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Stock 14, the parent cluster of V346 Cen, is described as a loose but clearly defined open cluster (Moffat & Vogt 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Eichendorf & Reipurth 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The most recent deep UBV pho- tometry of Stock 14 was obtained by Drobek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2013) pri- marily in a search for new variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Their photometry allowed determination of the distance and an estimate of the age for Stock 14, d = 2399+56 −55 pc, and τ = 20 ± 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The authors confirmed the cluster membership of V346 Cen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Re-evaluation of the photometric distance and age of Stock 14 by Paunzen & Netopil (2006) also favoured a shorter distance than previous determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' They obtained a distance of d = 2439 ± 326 pc, and an age of τ = 10 ± 2 Myr, in fine agreement with the extensive photometric study by Drobek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2013) as well as our results for V346 Cen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Article number, page 17 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Discussion The results of the analyses above are summarised in Table 4 for elemental abundances and Table 7 for fundamental stellar quan- tities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The stars in the present work cover a range of mass (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4– 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 M⊙), radius (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3 R⊙), Teff (22 500 to 32 200 K), sur- face gravity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2dex) and v sin i (90–185km s−1) and are all unevolved main sequence stars from late-O to early-B spectral types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We have achieved a high accuracy in the fundamental stel- lar properties, with uncertainties in mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='6–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='0%, radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7%, and log g of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='009–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='021dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Having a precise log g allows us to avoid its degeneracy with Teff in spectral analysis, resulting in uncertainties of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5% in Teff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Since Teff and log g are the principal quantities for specifying a model atmosphere, precise values are a prerequisite in measuring chemical abun- dances to a high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We now discuss the implications of our results for two subjects: evolutionary models for high-mass stars, and chemical evolution in high-mass binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Studies of chemical abundances in high-mass stars mostly concentrate on more advanced evolutionary stages, so it is dif- ficult to perform a quantitative comparison between our results and those published elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Martins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2017) presented a study of six short-period binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Of them, two are con- tact or overcontact systems so will have abundances altered by mass transfer, one (DH Cep) has component stars considerably more massive than our sample (38 M⊙ and 33 M⊙), while the re- maining three (Y Cyg, AH Cep and V478 Cyg) are suitable for the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Of these, AH Cep and V478 Cyg were analysed in our previous work (Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018) so a direct compar- ison is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Two studies agreed to within 2σ uncertainties in the [N/C] and [N/O], but only because of the large uncertain- ties quoted by Martins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' It is hard to trace the reason for this, but it may be related to the large uncertainties in the atmospheric parameters in their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Results of a comprehensive analysis of a large sam- ple of binary and/or multiple stars in the Tarantula Nebula have recently been published (Almeida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2020b,a) based on medium-resolution (R = 6400) spectra from VLT/FLAMES/GIRAFFE covering 3964–4567Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A total of 51 SB1 and SB2 systems were studied, of which 13 are eclips- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The atmospheric parameters were determined using NLTE methods, and He, C and N abundances derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The objects studied fall into five different groups: (1) long-period systems (P > 20 d) with well-detached components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2) eccentric short- period (P < 10 d) detached binaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (3) circular-orbit short- period (P < 10 d) binaries with strong tidal effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (4) semi- detached systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' and (5) contact systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' No N enrichment was found for binaries in the first two groups, despite the com- ponents having v sin i values of 50–250km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This finding is in disagreement with evolutionary models with rotationally in- duced mixing (Maeder & Meynet 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Heger & Langer 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Furthermore, a large N abundance was found for apparently slowly rotating stars in binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This agrees with initial findings by Hunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2007, 2008, 2009) who detected three distinctive groups in a diagram of [N/H] versus v sin i for single OB stars (sometimes dubbed the “Hunter diagram”): (1) stars showing N enrichment with v sin i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2) rapidly rotating stars with no sign of N enrichment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' and (3) stars with low v sin i and excessive N abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In the third group of binaries from Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020b), N enrichment was found for the fast rotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This is a group of stars in which the strongest influence of tidal forces on rotation- ally induced mixing is expected, following theoretical calcula- tions by de Mink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' By far the largest N enhancement was found for stars with almost the lowest v sin i in this group (∼50 km s−1), just as in the case of findings for stars in the first two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2020b) concluded that stars in detached binaries (groups 1 to 3) are evolving as single stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A lack of a clear relationship between N abundance and v sin i is in conflict with theoretical models and makes it hard to understand the ef- fect of rotationally induced turbulent mixing in stellar interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A very recent comprehensive spectroscopic analysis of a large set of B-type stars in the young open cluster NGC 3293 (Morel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2022) also corroborates these results: in the sam- ple of almost 150 B-type stars of which the majority have high v sin i, apparently no star with excess N abundance was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Only two stars are found with mild N enhancement, and these stars have a low v sin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A lack of N enhancement in fast-rotating B stars, and conversely, further evidence for N enhancement in low-v sin i B stars, is in clear contradiction with theoretical evo- lutionary models which incorporate rotationally-induced mix- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A state-of-the-art statistical analysis was carried out by Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2014) to identify possible mechanism(s) that could explain the distribution of stars in the Hunter diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The au- thors collected a statistically significant sample of well-studied Galactic single B stars for which seven observables were avail- able (surface N abundance, rotational frequency, magnetic field strength, and the amplitude and frequency of their dominant acoustic and gravity modes of oscillation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A multivariate anal- ysis indicated that the Teff and the frequency of the dominant acoustic oscillation mode have the most predictive power of the surface N abundance, whereas the rotational frequency of the star does not have any predictive power at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Up to now, no follow-up studies have been undertaken to investigate these un- expected results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Strong support for rotationally induced mixing has come from detailed abundance study for early B-type stars by Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2008) and Nieva & Przybilla (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The au- thors selected 20 early-B stars with a low v sin i to allow a high precision in determination of the atmospheric parameters and chemical abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2010) confirmed an obser- vationally tight correlation in the plot of abundance ratios N/C versus N/O, with a slope predicted via nuclear reactions in the CNO process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The targets had a broad evolutionary range, from dwarfs to supergiants, and their CNO abundances followed pre- dictions of the nuclear reaction theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' From our current and previous (see Section 6) studies, we have a sample of 13 dEBs of which 25 components have measured CNO abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We compared these to a sam- ple of high-mass stars published in Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2010) and Nieva & Przybilla (2012) in the logarithmic N/C versus N/O diagram (left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This is a powerful diagnostic tool in which the slope between [N/C] and [N/O] represents changes in CNO abundances due to nuclear reactions as de- rived in Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' It is striking that the two sam- ples cover the same mass range (8–20 M⊙) but do not fully over- lap in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For binary components there is a cut-off at [N/C] ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 dex and [N/O] ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='8 dex and they cluster around values close to solar ([N/C]⊙ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='52 dex and [N/O]⊙ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='00 dex), but a slope can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The targets in the current work strengthen our previous conclusion that properties of inte- rior mixing in binary stars are different from and might be less efficient than in single high-mass stars (Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This striking effect is also clearly seen in the diagram of [N/C] versus log g (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Theoretical evolutionary tracks are presented for a 15 M⊙ star and five values of the ini- tial rotational velocity Ω/Ωcrit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 Article number, page 18 of 25 Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 16: Left panel: Abundances of the CNO elements for high-mass stars in a diagram of [N/C] index versus [N/O] index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Stars in binary systems (Section 6) are represented by solid blue circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For comparison, abundance determinations for single early-B type stars (Nieva & Przybilla 2012) are represented by green circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Single stars obey a trend indicated by an analytical approximation to the nuclear reactions path for the CNO cycle derived in Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2010) and Maeder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The slope in this diagram indicates a gradual enhancement of N at the expense of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A slight decrease in O abundance is also predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Whilst the single and binary stars span almost identical mass and Teff ranges, and are all main-sequence stars, it is clear they do not share the same distribution (see Section 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Right panel: The observed [N/C] index for 23 high-mass stars in binaries (solid blue circles), compared to single B-type stars showing in solid green circles (Nieva & Przybilla 2012), as a function of surface gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Solid red lines in the right panel show theoretical evolutionary tracks for a 15 M⊙ star and five values of the initial rotational velocity Ω/Ωcrit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='7, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='9 (Georgy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Striking differences between single stars, and stars in binary systems are discussed in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (Georgy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The overall spread in [N/C] could be in- terpreted as due to evolutionary changes or (very) high initial rotational velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' However, only single stars from the sample of (Nieva & Przybilla 2012) tend to be consistent with the large [N/C] ratio predicted by the models for large initial rotational ve- locity values of Ω/Ωcrit ⪆ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='5 and [N/C] ⪆ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The main issue with the interpretation of the observed distribution in the context of the rotationally induced mixing alone is the generally low projected rotational velocity values (v sin i < 30 km s−1) found by Nieva & Przybilla (2012) for about half of their sample stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For the effect of rotational mixing being alone responsible for the observed [N/C] and [N/O] abundance ratios, one would require the majority of apparently slow rotators in the sample of Nieva & Przybilla (2012) to be stars that are seen pole-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This is a highly improbable scenario, so we conclude that the CNO abundances and their ratios observed in single high-mass stars are altered by multiple processes rather than just a single mechanism of rotational mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For example, high-mass stars are know to possess magnetic fields, stellar winds, and pulsa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' To this (strong) tidal effects in close high-mass binary sys- tems should be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' All these mechanisms, in one way or an- other, are expected to impact the efficiency of internal mixing, and hence the surface chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' However, the comparison between these two sets of empir- ical data, one with single high-mass stars and the other with high-mass stars in binary systems, is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' First and foremost, the sets differ in their distributions of v sin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The set of single stars were deliberately selected to be sharp-lined stars, so contains a mix of intrinsically slowly-rotating stars and ones with small inclinations and thus small sin i terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The set of binary stars, on the other hand, contains objects whose equa- torial rotational velocities are accurately known, assuming their rotational and orbital axes have been aligned during formation or by tidal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Furthermore, even though the v sin i distribu- tion could be statistically corrected to intrinsic rotational veloc- ities for single stars, there is a substantial difference in the ro- tational history between single and binary stars that one cannot easily account for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Evolution of stellar rotational velocity from its initial value at the zero-age main sequence, and its subse- quent changes in the course of stellar evolution due primarily to changes in radius, is substantially different due to tidal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This is particularly important for short-period systems whose ro- tation is synchronised with and thus governed by their orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Nevertheless, the non-detection of substantial changes in the CNO abundances of stars in binaries contradicts the predic- tions of single-star rotational evolutionary models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' For the sam- ple of Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2010) and Nieva & Przybilla (2012), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' single high-mass stars with low observed v sin i, the possibility remains that they agree with the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Tidal forces in binary and/or multiple systems affect the ge- ometry of the orbits and the shape and spin of the components (Mazeh 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In order of increasing timescale, the stellar spin axes are aligned first, then their rotation is synchronised, and fi- nally the orbit is circularised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Later evolution is dominated by mass transfer due to the increase in the sizes of the component stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Our hypothesis that tidal effects suppress the efficiency Article number, page 19 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries of rotational mixing is not supported by theoretical calculations (de Mink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2013), which predict precisely the opposite: that short-period circularised binary systems should experience rota- tionally induced turbulent mixing in stellar interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In looking for possible mechanisms which diminish turbulent mixing in the components of binary systems, Koenigsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2021) examined differential rotation in asynchronous binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' If the components in a binary system are not yet in synchronous rotation, tidally-induced and time-variable differential rotation could develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The calcula- tions by Koenigsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' (2021) clearly show the role of asynchronism: the induced rotation structure and its temporal variability depend on the degree of departure from synchro- nism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The authors further speculated that, in this context, slowly- rotating asynchronous binaries could have more efficient mixing than the more rapidly-rotating but tidally locked systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' This shows that processes triggered by asynchronous rotation in bi- nary systems cannot be ignored, while a comparison between samples of single and binary stars should be done with partic- ular care, even when the latter are in a detached configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We note that 12 of the 14 binaries in our sample have eccentric orbits but that most of the component stars rotate synchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Conclusion Despite their astrophysical importance, high-precision funda- mental stellar quantities (mass, radius, Teff) have been deter- mined for only a few high-mass stars in binary systems in our galaxy (Southworth 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Even fewer have measurements of their surface chemical composition (Serenelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' In the present work we have added four more binary systems to this list: V1034 Sco, V346 Cen, GL Car and V573 Car, containing stars of masses from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='4 to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Most of these stars are young, with only two in the second half of their MS evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We have presented high-quality HARPS spectra and anal- ysed them using spectral disentangling to determine their spec- troscopic orbits and the individual spectra of the component stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' These were analysed using an NLTE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We have modelled the available light curves for our systems, compris- ing uvby photometry in all cases and TESS photometry in three cases, to determine their photometric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Combining these analyses, we have determined high-precision masses, radii, surface gravities, Teff values, v sin i values and C, N, O, Mg and Si abundances for all eight stars in the four binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Of particular interest are the CNO abundances since these elements are involved in core hydrogen burning through the CNO cy- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' During a star’s evolution its N abundance increases and its C abundance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Rotationally induced mixing of stellar material, or some other mixing processes, could bring nuclear- processed material from the stellar core to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Therefore, the [N/C] ratio is a sensitive probe of interior mixing processes during the MS evolutionary stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The CNO abundances determined in this work corroborate our previous findings (Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2018) that interior mixing is different in binary stars to single stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A tight correlation of [N/C] with [N/O] versus the predicted evolutionary changes has been found for single early B-type stars (Przybilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Nieva & Przybilla 2012), whereas binary systems in our sample show much less variation in both [N/C] and [N/O].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' However, care is needed when comparing them with single stars due to the differences in rotational velocity between these types of object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' It remains true that the binary sample does not reproduce the results found for a sample of single low-v sin i B-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' On other hand, recent spectroscopic analysis of large sam- ples of high-mass stars in binaries (Mahy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2020a), and single B-type stars in the young open cluster NGC 3293 (Morel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' 2022) apparently confirmed the lack of substantial changes in CNO abundances for high-v sin i stars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' for intrin- sically fast-rotating stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We speculate that proximity effect in binary systems some- how suppress mixing and/or transport of chemical elements from the interior to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' However, firmer conclusions will need a substantial expansion of the binary stars sample and an exten- sion to more massive and hotter stars, and/or wider long-period binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Acknowledgements Careful reading of the manuscript and useful suggestions pro- vided by the referee are acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' We are indebted to Keith Butler and Norbert Przybilla for kindly sharing their codes, and the model atoms used in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' KP and ET were initially supported by the Croatian Science Foundation through research grant IP-2014-09-8656.' metadata={'source': 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+page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1: Light curves of our target stars, from our own reduction of data from the TESS satellite, that were not included in the work in this paper, but could useful for studies of the period changes, and apsidal motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The reduced photometric data are given in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='1 only available in electronic form at the CDS (see article front page).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Article number, page 23 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' harps_high-mass_binaries Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='2: Light curve of V346 Cen from the TESS satellite, plotted versus orbital phase but with a small magnitude offset linearly dependent on time to shift successive cycles upward in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The earliest points are coloured red and the latest points are coloured blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' The repetition of the pulsation signature with orbital phase is easy to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Article number, page 24 of 25 Pavlovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' : High-mass eclipsing binaries: a testbed for models of interior structure and evolution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content='3: Portions of the disentangled spectra of the stars (labelled) studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} +page_content=' Article number, page 25 of 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf'} diff --git a/9dAyT4oBgHgl3EQfdPeW/vector_store/index.pkl b/9dAyT4oBgHgl3EQfdPeW/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e184d6c3534a5adafa13229c584fdecc092e71fa --- /dev/null +++ b/9dAyT4oBgHgl3EQfdPeW/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:478414d9a786eb50068fda9202f31cebdc3c2dec2b2604c1c5e7852824a11d23 +size 113297 diff --git a/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf b/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1ae14e30527bfaa3058bbc95a59d158e56d14399 --- /dev/null +++ b/AtAyT4oBgHgl3EQfq_mZ/content/2301.00553v1.pdf @@ 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+Abstract +Multi-hop QA (Question Answering) is the +task of finding the answer to a question across +multiple documents. In recent years, a number +of Deep Learning-based approaches have been +proposed to tackle this complex task, as well +as a few standard benchmarks to assess mod- +els’ Multi-hop QA capabilities. In this paper, +we focus on the well-established HotpotQA +benchmark dataset, which requires models to +perform answer span extraction as well as sup- +port sentence prediction. We present two ex- +tensions to the state-of-the-art Graph Neural +Network (GNN) based model for HotpotQA, +Hierarchical Graph Network (HGN): (i) we +complete the original hierarchical structure by +introducing new edges between the query and +context sentence nodes; (ii) in the graph prop- +agation step, we propose a novel extension +to Hierarchical Graph Attention Network – +GATH (Graph ATtention with Hierarchies) – +that makes use of the graph hierarchy to up- +date the node representations in a sequential +fashion. +Experiments on HotpotQA demon- +strate the efficiency of the proposed modifica- +tions and support our assumptions about the +effects of model-related variables. +1 +Introduction +Question Answering (QA) tasks can be classified +into single-hop and multi-hop ones, depending on +the complexity of the underlying reasoning. Dif- +ferent from single-hop QA (Rajpurkar et al., 2016; +Trischler et al., 2017; Lai et al., 2017), where ques- +tions can be answered given a single paragraph or +single sentence in the context, multi-hop QA re- +quires us to retrieve and reason over scattered infor- +mation from multiple documents, as demonstrated +in Figure 1. There are many methods proposed for +addressing the multi-hop QA problem. One type of +∗Work carried out as part of MSc thesis supervised by +Huawei Noah’s Ark Lab, London +†Work carried out while working at Huawei Noah’s Ark +Lab, London +Question: +Where did the form of music played by Die +Rhöner Säuwäntzt originate? +Answer: +United States +Supports: +Document 9 +s1: +Die Rhöner Säuwäntzt are a Skiffle- +Bluesband from Eichenzell-Lütter in Hessen, +Germany. +Document 4 +s1: Skiffle is a music genre with jazz, blues, +folk and American folk influences [...] +s2: Originating as a term in the United States +in the first half of the 20th century [...] +Figure 1: Example of a multi-hop answer and support +prediction, as found in HotpotQA. +these recent approaches extends well-performing +single-hop machine reading comprehension mod- +els to be multi-hop, such as DecompRC (Min et al., +2019) and QFE (Nishida et al., 2019). +The other avenue is to develop models specifi- +cally aimed at multi-hop QA. Among those, Graph +Neural Networks (GNNs) have recently garnered a +lot of attention. In GNN-based approaches, gaphs +are employed to represent query and context con- +tents (nodes) and the underlying relationships be- +tween them (edges). Information between nodes is +simultaneously propagated via the edges with the +help of a variety of GNNs, such as Graph Convolu- +tional Network (GCN) (Kipf and Welling, 2017), +Graph Attention Network (GAT) (Veliˇckovi´c et al., +2017), or Graph Recurrent Network (GRN) (Song +et al., 2018b). With these GNNs, node representa- +tions are obtained conditioned on the question and +context documents, and used for the QA task. +In this paper, we focus on one particular GNN ap- +proach designed for the Hotpot QA benchmark, the +Hierarchical Graph Network (HGN) introduced in +Fang et al. (2020). HGN constructs a hierarchical +graph that integrates nodes from different granu- +larity levels (question/paragraph/sentence/entity). +The edges in the graph capture the interactions be- +tween the information from heterogeneous levels +of the hierarchy. This hierarchical graph structure +arXiv:2301.11792v1 [cs.CL] 27 Jan 2023 + +has been shown to be crucial to the model’s remark- +able performance1 on both finding scattered pieces +of supporting information across documents and +the answer span prediction. +The contribution of this work is three-fold: (i) +we extend the edges of HGN with a new edge type +between the query and sentences, completing its +original structure; (ii) we introduce a novel exten- +sion of the Graph Attention Network – Graph At- +tention with Hierarchies (GATH). GATH allows for +making use of the explicit hierarchical graph struc- +ture, by propagating information through the graph +in a sequential fashion based on the hierarchy’s +levels, rather than updating all nodes simultane- +ously. (iii) We perform initial experiments on the +HotpotQA benchmark, providing evidence of the +effectiveness of our proposed extensions. +Code related to graph completion and GATH +will be made publicly available at redacted. +2 +Background +To solve the multi-hop QA problem, two general +research paths have been studied. The first direc- +tion focuses on extending the successful single- +hop machine reading comprehension method to the +multi-hop QA. DecompRC (Min et al., 2019) de- +composes the multi-hop reasoning problem into +multiple single-hop sub-questions based on span +predictions and applied traditional machine reading +comprehension techniques on these sub-questions +to obtain answers to the question. Query-Focused +Extractor (QFE) (Nishida et al., 2019) reformulates +the multi-hop QA task as a query-focused summa- +rization task based on the extractive summarization +model (Chen and Bansal, 2018). +The second research direction natively addresses +the task as a multi-hop setting, and directly tries +to gather the information from all context doc- +uments in order to answer the question. Many +approaches based on the transformer architecture +(Vaswani et al., 2017) address the multi-hop QA +task as simply one of attention between all words +in all available documents. In such approaches, +the problem quickly becomes intractable due to +the long inputs involved, and they thus typically +focus on alleviating the problems of using a full +attention mechanism. The Longformer (Beltagy +et al., 2020), for example, introduces a windowed +attention mechanism to localise the problem, allow- +1At the time of writing, HGN achieves SOTA results on +HotpotQA, for GNN-based approaches. +ing for much longer input sequences to be handled +than with standard BERT-based language models +(Devlin et al., 2018). +However, recently more research effort has been +put toward approaches that employ Graph Neural +Networks, which allow for organising information +from various sources into a graph structure before +addressing the core task of Question Answering, +mitigating the need for very-long-distance attention +functions. +Coref-GRU (Dhingra et al., 2018) integrates mul- +tiple evidence associated with each entity mention +by incorporating co-reference information using +a collection of GRU layers of a gated-attention +reader (Dhingra et al., 2017). However, Coref- +GRU only leverages co-references local to a sen- +tence but ignores other useful global information. +To address this problem, MHQA-GRN and MHQA- +GCN (Song et al., 2018a) integrate evidence in a +more complex entity graph, with edges that also +connect global evidence. Similarly, De Cao et al. +(2019) also encode different relations between en- +tity mentions in the documents and perform the +graph reasoning via Graph Convolutional Network +(GCN) (Kipf and Welling, 2017). +All of the above methods which involve Graph +Neural Networks only consider entity nodes and the +relations between them. The HDE-Graph (Tu et al., +2019) extends these works by creating a new type +of graph with nodes corresponding to answer candi- +dates, documents and entities. Different edges are +included into the graph to capture the interaction +between these nodes. DFGN (Qiu et al., 2019) con- +structs a dynamic entity graph and performs graph +reasoning with a fusion block. This fusion block in- +cludes iterative interactions between the graph and +the documents (Doc2Graph and Graph2Doc flows) +in the graph construction process. Hierarchical +Graph Network (Fang et al., 2020) proposes a hier- +archical graph that incorporates nodes on different +levels of a hierarchy, including query, paragraph, +sentence, and entity nodes. This hierarchical graph +allows the model to aggregate query-related data +from many sources at various granularities. +One limitation that all of the above conventional +QA graph neural networks share is that their in- +formation propagation mechanisms do not directly +utilise the (explicit or implicit) hierarchical prop- +erty of the graph structure. In fields outside of +Natural Language Processing, recent studies on hi- +erarchical graph neural networks focus on passing + +information on each hierarchical level to the node +at different attention weights. +In multi-agent reinforcement learning, HGAT +(Ryu et al., 2020) generates hierarchical state- +embedding of agents. This HGAT model stacks +inter-agent and inter-group graph attention net- +works hierarchically to capture inter-group node +interaction. A two-level graph attention mechanism +(Zhang et al., 2020) was developed for propagating +information in the close neighborhood of each node +in the constructed hierarchical graph. HATS (Kim +et al., 2019) predicts stock trends using relational +data on companies in the stock market. HATS +selectively aggregates information from different +relation types with a hierarchically designed atten- +tion mechanism. By maintaining only important +information at each level, HATS efficiently filters +out relations (edges) not useful for trend prediction. +However, all previous studies on hierarchical +graph neural networks only exploit the possible +hierarchical structure on the graph node itself. Dif- +ferent from the above methods, our proposed hi- +erarchical graph attention mechanism allows the +graph node embeddings to be updated in the or- +der of the hierarchical granularity level, instead of +simultaneously. +3 +Model +As our proposed improvements are largely aimed at +the established Hierarchical Graph Network (HGN) +model (Fang et al., 2020) for HotpotQA, we briefly +describe the original architecture. HGN builds a +hierarchical graph with nodes from several granu- +larity levels (question/paragraph/sentence/entity). +This hierarchical graph structure is good at captur- +ing the interaction between nodes from different +granularity levels and has been shown beneficial +to the model’s remarkable performance on both +finding scattered pieces of supporting information +across documents, and to answer span prediction. +The full HGN model pipeline consists of four +modules: (i) the Graph Construction Module se- +lects query-related paragraphs and builds a hier- +archical graph that contains edges between nodes +from different granularity levels within the para- +graphs; (ii) the Context Encoding Module gives +an initial representation/embeddings for nodes in +the graph via encoding layers that consist of a +RoBERTa (Liu et al., 2019) encoder and a bi- +attention layer; (iii) the Graph Reasoning Mod- +ule updates the initial representation of all nodes +via reasoning over the hierarchical graph; (iv) the +Multi-task Prediction Module performs multiple +sub-tasks including paragraph selection, support- +ing facts prediction, entity prediction and answer +span extraction, based on the representation of all +nodes. This process is summarized in Figure 2, +as presented by the original authors of the HGN +model. +We note that HGN still has limitations on its +graph structure and the graph reasoning step, and +in this work introduce according changes. Our +proposed extensions aim to further improve HGN +through a more complete graph structure, and a +novel hierarchical graph nodes update mechanism. +As such, our method mainly targets the Graph Con- +struction and Graph Reasoning Modules, described +in more detail below, while we leave the Context +Encoding and Multi-task Prediction Modules un- +changed. +Graph Construction Module +The Hierarchical Graph is built based on the given +HotpotQA question-context pair. This construc- +tion process consists of two steps: (i) multi-hop +reasoning paragraph retrieval from Wikipedia, i.e. +selecting candidate paragraphs with potential multi- +hop relationship to the question as paragraph nodes; +(ii) adding edges between question, sentence and +entity nodes within the retrieved paragraphs. +In particular, the first step consists of retriev- +ing “first-hop” paragraphs, that is, paragraphs of +Wikipedia entries that belong to entities mentioned +in the question. After this, a number of “second- +hop” paragraphs is selected, from Wikipedia arti- +cles that are hyper-linked from these first hops. +Our work keeps the original paragraph selection +method, but introduces novel meaningful edges +between graph nodes. +Context Encoding Module +With the hierarchical graph structure in place, rep- +resentations of the nodes within the graph are ob- +tained via the Context Encoding Module. In this +encoder, query and context are concatenated and +fed into a pretrained RoBERTa (Liu et al., 2019). +The obtained representations are further passed into +a bi-attention layer (Seo et al., 2018) to enhance +the cross interactions between the question and the +context. Through this encoding mechanism, the +question node is finally represented as q ∈ Rd and +the i-th paragraph/sentence/entity nodes are repre- +sented by pi, si and ei ∈ Rd respectively. + +Figure 2: Model architecture of Hierarchical Graph Network (HGN). This illustration was originally introduced in +Fang et al. (2020). We include it here for completion, to provide an overview of HGN. +Graph Reasoning Module +Intuitively, the initial representations of the graph +nodes only carry the contextualized information +contained within their local contexts. To benefit +from the hierarchy and information across differ- +ent contexts, the Graph Reasoning Module further +propagates information between the graph nodes +using a single-layered Multi-head Graph Attention +Network (GAT) (Veliˇckovi´c et al., 2017). How- +ever, we believe the simultaneous node-update per- +formed by standard GAT can be improved, in the +presence of the explicitly given hierarchical prop- +erty of the graph. We therefore propose a novel +hierarchical graph reasoning method that performs +node updates sequentially, for different levels of +the hierarchy. In this manner, nodes on certain +granularity levels of the graph are allowed to first +aggregate some information, before passing it on +to their neighbours on other levels. We speculate +that this staggered information passing paradigm +can be beneficial to the multi-hop Question An- +swering task, by passing on more question-specific +contextualized information to relevant nodes. +Multi-task Prediction Module +The final step of the HGN model is to jointly pre- +dict answer and supporting facts for the question +via multi-task learning based on the updated graph +node representations. This is decomposed into five +sub-tasks: (i) paragraph selection determines if a +paragraph contains the ground truth; (ii) sentence +selection determines if a sentence from the selected +paragraph is a supporting fact; (iii) answer span +prediction finds the start and end indices of the +ground-truth span; (iv) answer type prediction pre- +dicts the type of the question; (v) entity prediction +determines if the answer can be found among the +selected entities. The above sub-tasks are jointly +trained through multi-task learning with the final +objective of the total loss from these sub-tasks: +Ljoint =Lstart + Lend + λ1Lpara+ +λ2Lsent + λ3Lentity + λ4Ltype +(1) +With HGN re-introduced for completeness, we +describe our proposed extensions to the original +architecture in the subsequent sections. +3.1 +Completion of the graph structure +HGN constructs a hierarchical graph connecting +the query node with the selected multi-hop para- +graphs. Each selected paragraph contains sentences +and entities which are also encoded as nodes in the +hierarchical graph. The graph not only incorpo- +rates the natural hierarchy existing in paragraphs, +sentences and entities, but also includes helpful +connections between them to faciliate the structual +information propagation within the graph. Specif- +ically, the graph consists of seven types of edges, +which link the nodes in the graph. These edges +are (i) edges between the question node and first- +hop paragraph nodes; (ii) edges between paragraph +nodes; (iii) edges between sentences in the same +paragraph; (iv) edges between paragraph nodes +and the corresponding within-paragraph sentence +nodes; (v) edges between second-hop paragraphs +and the hyperlinked sentences; (vi) edges between + +Multi-task Prediction Module +Graph Construction Module +Paragraph +Supporting Facts +Entity +Answer Span +Q +Selection +Prediction +Prediction +Extraction +↑ +↑ +↑ +介 +Paragraph +(P1 +P2 +Updated: +Gated Attention +Level +hyperlink +介 +Sentence +S1 +S2 +S3 +S4 +S5 +Graph Reasoning Module +Level +Initial Representations: +Entity +E2 +E3 +E4 +E1 +Level +Adriana +个 +New York +Virginia +Greenwich +Trigiani +Village +City +Context Encoding Module +↑ +Q +P1 +P2Figure 3: Hierarchical Graph with (orange-colored) +new question_sentence edges added. +the question node and its matching entity nodes; +(vii) edges between sentence nodes and their corre- +sponding within-sentence entity nodes. +We note that the only type of edge that seems to +be missing from the graph are question-sentence +edges. +Hence, we first complete the hierarchi- +cal graph by introducing novel question_sentence +edges which connect the question node with all +sentence nodes of selected paragraphs. Such new +connections are introduced as edge (viii) in the hier- +archical graph. The constructed hierarchical graph +with novel edges added is illustrated in Figure 3. +We reason that this more complete graph might +help the model to learn more useful embedding +because of the modification in the graph topology, +which facilitates the information transmission be- +tween the question and sentences. +3.2 +Graph Attention with Hierarchies +The Graph Reasoning Module updates the contex- +tualized representations of graph nodes to capture +the information aggregated from topological neigh- +bours such that the local structures of these nodes +can be included. In HGN, this process is realized +by the Graph Attention Network (GAT) (Veliˇckovi´c +et al., 2017), a well-established GNN approach. +However, we note that in the specific setting +of Multi-hop QA with the presence of an explicit +hierarchical graph structure, GAT might not be +able to make full use of the information encoded +in the graph, as it will not directly capture the +crucial dependencies between “levels” of the hi- +erarchical. To address this problem, we propose a +novel Graph Attention Network with Hierarchies +(GATH) which updates nodes sequentially condi- +tioned on an imposed order over the hierarchy lev- +els. This is expected to help the model more effec- +tively processes the local observation of each node +into an information-condensed and contextualized +state representation for individual nodes on specific +levels, e.g. for paragraphs, before passing their in- +formation on to their neighbours on other levels, +such as to entity nodes. We expect this staggered +flow of information might help the model aggregate +information that is more useful and conditioned on +the task at hand. +The nodes in the graph are split into four cate- +gories, and can be represented by q, P, S and E: +P = {pi}np +i=1 +S = {si}ns +i=1 +E = {ei}ne +i=1 +with each node embedded with an embedding func- +tion as described above, into a d-dimensional vec- +tor. These node representations are jointly repre- +sent the graph nodes as +H = {q, P, S, E} ∈ Rg×d, g = 1 + np + ns + ne +GATH updates all initial node embedding H to +H +′ through hierarchical graph updates. Different +from GAT, GATH updates the nodes representation +sequentially, according to a pre-determined order +of hierarchical levels, instead of simultaneously. It +takes the initial node representations H as input, +but first only updates information of node features +of the first hierarchical level while keeping other +node embeddings unchanged. +For example, if the first level to be updated is +the paragraph level, we obtain the updated graph +representation +Hpara = {h1, h +′ +2, h +′ +3, . . . h +′ +1+np, h2+np, . . . , hg} +Specifically, +h +′ +i = ∥K +k=1LeakyRelu( +� +j∈Ni +αk +ijhjWk) +(2) +where ∥K +k=1 represents concatenation of K heads, +Wk is the weight matrix to be learned, Ni repre- +sents the set of neighbouring nodes of node i and +αk +ij is the attention coefficient calculated by: +αk +ij = +exp(LeakyRelu([hi; hj]wk +eij)) +� +t∈Ni exp(LeakyRelu([hi; ht]wkeit)) (3) +where [hi; hj] denotes the concatenation of hi and +hj, and wk +eij is the weight vector corresponding to +the edge between node i and j. +Based on the updated embeddings on the para- +graph level Hpara, we might next consider updating + +Q +edge (vi) +edge (i) +Paragraph +edge (ii) +Level +P1 +P2 +édge (vili) +edge (v) +edge (iv) +Sentence +Level +S1 +S2 +S3 +S4 +S5 +edge (ii) +edge (vii) +Entity Level +E1 +E2 +E3 +E4the information on the sentence level. GATH propa- +gates information to all nodes on the sentence level +based on Hpara. This will output a further updated +graph representation +Hsent = {h1, h +′ +2, . . . h +′ +1+np+ns, h2+np+ns, . . . , hg} +with all nodes in P and S updated. +Continuing the process in this manner, we even- +tually will have updated all node representations +to obtain H +′ {h +′ +1, h +′ +2, ..., h +′ +g}. Algorithm 1 summa- +rizes the above procedures in pseudo code. Ad- +ditionally, these updating steps are combined and +illustrated in Figure 4. +4 +Experiments +In this section, we present experiments comparing +our extended HGN models with GATH with the +original one employing GAT, and provide a detailed +analysis of the proposed improvements and results. +For all experiments, we use RoBERTalarge as +the base embedding model. We train with a batch +size of 16 and a learning rate of 1e−5 over 5 epochs, +with λ1, λ3, λ4 = 1 and λ2 = 2, and we employ a +dropout rate of 0.2 on the transformer outputs, and +0.3 throughout the rest of the model. +4.1 +Dataset +The effects of the above proposed improvements +are assessed based on HotpotQA (Yang et al., 2018). +It is a dataset with 113k English Wikipedia-based +question-answer pairs with two main features: (i) It +requires reasoning over multiple documents with- +out constraining itself to an existing knowledge +base or knowledge schema; (ii) Sentence-level sup- +porting facts are given for the answer to each ques- +tion, which explain the information sources that the +answer comes from. The performance of models +on HotpotQA is mainly assessed on two metrics, +exact match (EM) and F1 score. The model is ex- +pected to not only provide an accurate answer to the +question, but also to give supporting evidences for +its solution. Thus, EM and F1 score are calculated +for both answer spans and supporting facts. +HotpotQA has two settings: Distractor and Full- +wiki. In the distractor setting, context paragraphs +consist of 2 gold truth paragraphs containing in- +formation that is needed to solve the question, and +8 paragraphs retrieved from Wikipedia based on +the question, serving as related yet uninformative +distractors for the question-answer pair. In the +Fullwiki setting, all context paragraphs come from +Wikipedia’s top search results, and they need to be +pre-ranked and selected in a first step. Compared +with the distractor setting, this setting requires us +to propose an additional paragraph selection model +concerned with information retrieval, before we +address multi-hop reasoning task. As all our pro- +posed extensions aim at the graph construction and +reasoning steps, we only perform these initial ex- +periments to assess the impact of our approach in +the distractor setting, where we are independent +from the influence of such a retrieval system. +4.2 +Experimental Results +Using the HotpotQA dataset, the models with our +extensions of graph completion and GATH are com- +pared against the baseline model of HGN with stan- +dard GAT. Since it could reasonably be argued +that GATH “simulates” a (partially) multi-layered +GAT in the sense that some nodes are updated only +after others have already been able to incorporate +neighbouring information – which in standard GAT +requires at least two full layers – we also include +an HGN trained with a two-layer network rather +than the single layer used in the original paper. +Table 1 summarizes the results on the dev set of +HotpotQA2. +2Authors’ note: unfortunately, despite our best efforts we +were not able to reproduce the numbers reported for HGN +in Fang et al. (2020), even with their original, open-sourced +code. We tried both the hyper-parameters as published in the +paper, and the ones shipped with the code release however, the +RoBERTalarge performance when training from scratch was +consistently much lower than expected on dev (∼ 74 vs ∼ 70 +joint F1). We contacted the original authors, who were not +able to help out with this. In light of these discrepancies, we + +Algorithm 1 +Graph Attention Network with Hierarchies +(GATH) +Input: H = hi, h2, .., hg} +Output: H' = {hi, h2, ..,h.] +for t in 1 : total number of levels do +create Ht = [ ] +foriin 1 : g do +if node i belongs to level t then +ht ← IIK=LeakyRelu(ZjeN; Qtit) +k;(t)ht-1wk,(t) +k,(t) +EsE N, exp( LeakyRelu([ht-1;h-1]we;(t)) +else +h ←ht-1 +end if +Append h, to Ht +end for +end for +return H' - Htotal number of levelsFigure 4: Hierarchical node representation update process. The grey-colored graph nodes are initial contextualized +embedding given by the Context Encoding Layer. Through the paragraph level message passing layer, only the +neighboring information of all paragraph nodes can be passed and renewed on them. Similar steps repeat for +sentence level and entity level. For convenience of labeling indices, we set np = 2, ns = 5, ne = 4 +Completion of graph structure +The HGN with +new query-sentence edges improves over the base- +line by 0.7/0.4 on Joint EM and F1 scores. This +supports our intuition that the the missing question- +to-sentence edges can indeed bring advantages to +the model’s abilities of both answer span extraction +and supporting facts prediction. +Graph Attention with Hierarchies +GATH al- +lows for pre-defining the order of level updates in +the model. Given that the order in which the hierar- +chy levels are updated is likely to affect the model’s +performance, we perform experiments with dif- +ferent orders (P/S/E3,4, E/S/P, S/E/P and S/P/E) +and compare them to the baseline models with +one and two-layer GAT. All the GATH-based ex- +tended models outperform the baseline model on +the answer-span extraction by an absolute gain of +1.6 to 2.4 points on the answer extraction metrics. +On the other hand, the order of hierarchical lev- +els does show an influence on the model’s evi- +decided to focus only on dev set performance when assessing +the impact of our extensions against re-trained vanilla HGN, +as a fair comparison to the original model on test was not +possible at this time. +3P/S/E abbreviates Paragraph/Sentence/Entity +4We exclude the query level update to make it more com- +parable to the baseline model, which also excludes this update. +dence collection ability. The “wrong” order leads +to worse performance of the extended model, such +as in the E/S/P and S/P/E cases. +On most metrics, but specifically on joint F1 +score, the extended GATH-based model with the or- +der S/E/P outperforms not only the baseline model, +but also the other GATH models. It achieves a Joint +EM/F1 score of 43.9/71.5, exceeding the baseline +model’s performance by 1.2 each. +Interestingly, the 2-layer GAT version of HGN +slightly under-performs when compared to the orig- +inal HGN setup. While gaining 0.3 points in sup- +port prediction F1, it loses the same amount of +performance in answer prediction and joint scores. +We assume this is why the original HGN calls for +only one layer, when we could intuitively have ex- +pected multi-layered networks to perform better. +Combined query-sentence edges and GATH +The above experimental results demonstrate the +individual effectiveness of these two proposed im- +provements of graph completion and GATH. Nat- +urally, we are also interested in the performance +resulting from combining both. The “HGN (Com- +bined)” row in Table 1 represents the model com- +bining graph completion and GATH-S/E/P. This +combined model brings slight improvement over + +H +Hsent + Hpara +h1 +h1 +edge (i) +Paragraph +edge (i) +Paragraph +Level +P1 +P2 +Level +P1 +h2 +h2 +h2 +edge (iv) +edge (v) +edge (iv) - +dge (v) +Sentence +Sentence +h's +Level +S2 +63 +h? +h3 +Level +S1 +edge (ii) +edge (vii)i +h'4 +Entity Level +h4 +h4 +E1 +EntityLevel +hs +Paragraph level nodes updating +.. +Sentence level nodes updating +hg +hg +hg +H' +Hent +edge(vi) +hi +edge (vi), +h1 +Paragraph +Paragraph +Level +P1 +P2 +Level +P1 +h2 +h2 +Sentence +hs +Level +S2 +S3 + $4 +S5 +Sentence +S1 +hs +Level +S3 +SA +edge (vii) : +h4 +h4 +edge (vi) i +Entity Level +E2 +E3 +E4 +EntityLevel +hs +E3 +hs +1 +.. +Question level nodes updating +hg +hg +Entity level nodes updatingAnswer +Support +Joint +Model +EM +F1 +P +R +EM +F1 +P +R +EM +F1 +P +R +HGN (baseline) +64.5 +78.3 +81.6 +79.0 +60.4 +87.4 +89.6 +87.5 +42.7 +70.3 +75.0 +70.9 +HGN (2-layer GAT) +64.1 +78.0 +81.4 +78.9 +59.9 +87.7 89.5 +88.4 +41.6 +70.0 +74.5 +71.3 +HGN (que_sent edge) +65.0 +79.1 +82.2 +80.1 +60.9 +87.0 +89.9 +86.9 +43.4 +70.7 +75.7 +71.5 +HGN with GATH(P/S/E) +66.1 +80.1 +83.1 +81.1 +54.2 +80.1 +86.1 +79.5 +38.7 +66.5 +73.6 +66.9 +HGN-GATH(E/S/P) +66.4 +80.3 +83.6 +81.2 +38.3 +74.4 +70.0 90.0 +27.2 +61.2 +59.8 +74.1 +HGN-GATH(S/E/P) +67.0 +80.6 83.7 81.4 +60.5 +86.3 92.3 83.7 +43.9 +71.5 78.8 70.2 +HGN-GATH(S/P/E) +66.8 +80.7 83.8 +81.6 +38.7 +74.6 +70.3 90.0 +27.3 +61.5 +60.1 74.5 +HGN (Combined) +66.7 +80.7 83.7 81.7 61.4 +87.0 +91.2 +85.6 +43.9 +71.9 77.8 +71.8 +Table 1: Performance of the proposed HGN with completed edges (HGN que_sent), GATH, and both extensions +combined on the development set of HotpotQA in distractor setting, against the baseline model HGN with GAT. +the other models on most metrics. This final model +sees further improvements, particularly in the an- +swer span prediction task, and achieves the overall +highest joint F1 score at 71.9, indicating that the +contributions of graph completion and GATH are +mutually benefitial. +4.3 +Error Analysis +In this section, we perform an error analysis on +the concrete influence of the proposed HGN (com- +bined) model based on question types. The major- +ity of questions in HotpotQA fall under the bridge5 +and comparison reasoning categories. +As sug- +gested by Fang et al. (2020), we split comparison +questions into comp-yn and comp-span. The former +represents questions that should answer the compar- +ison between two entities with “yes” or “no”, e.g. +“Is Obama younger than Trump?”, while the latter +requires an answer span, e.g. “Who is younger, +Obama or Trump?”. +Table 2 shows the performance of the original +HGN model and the proposed model HGN-GATH +(combined) on various types of reasoning questions. +Results indicate that comp-yn questions are easiest +for both models, and the bridge type is the hardest +to solve. The analysis table shows that HGN (com- +bined) is more effective than the original model +on all of these reasoning kinds except support EM +for comp-yn, though even here the much improved +answer prediction leads to an overall improvement +of 2.42 on Joint EM. +5requiring a bridging entity between support sentences, +needed to arrive at the answer +HGN-GAT +Question +Ans EM +Sup EM +Joint EM +Pct(%) +comp-yn +81.22 +81.44 +68.34 +6.19 +comp-span +65.50 +71.04 +48.49 +13.90 +bridge +63.08 +57.01 +39.73 +79.91 +HGN-GATH (Combined) +Question +Ans EM +Sup EM +Joint EM +Pct(%) +comp-yn +85.81 +80.79 +70.96 +6.19 +comp-span +68.42 +71.53 +50.34 +13.90 +bridge +64.95 +58.08 +40.71 +79.91 +Table 2: Original HGN (top) and HGN-GATH com- +bined (bottom) model results for various reasoning +types. ‘Pct’ signifies percentage of all questions per +category. +5 +Conclusions and Future Work +In this paper, we proposed two extensions to Hi- +erarchical Graph Network (HGN) for the multi- +hop Question Answering task on HotpotQA. First, +we completed the hierarchical graph structure by +adding new edges between the query and context +sentence nodes. Second, we introduced GATH as +the mechanism for neural node updates, a novel +extension to GAT that can update node representa- +tions sequentially, based on hierarchical levels. To +the best of our knowledge, this is the first time the +hierarchical graph structure is directly exploited in +the update mechanism for information propagation. +Experimental results indicate the validity of our +approaches individually, as well as when used +jointly for the multi-hop QA problem, outperform- +ing the currently best performing graph neural net- +work based model, HGN, on HotpotQA. +In the future, we would particularly like to in- +tegrate hierarchical graph attention weights into + +GATH, as motivated by related research in Rein- +forcement Learning. +References +Iz Beltagy, Matthew E Peters, and Arman Cohan. +2020. 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Hotpotqa: A dataset +for diverse, explainable multi-hop question answer- +ing. +Zhao Zhang, Fuzhen Zhuang, Hengshu Zhu, Zhiping +Shi, Hui Xiong, and Qing He. 2020. +Relational +graph neural network with hierarchical attention for +knowledge graph completion. +In Proceedings of +the AAAI Conference on Artificial Intelligence, vol- +ume 34, pages 9612–9619. + diff --git a/BNFKT4oBgHgl3EQfWi5i/content/tmp_files/load_file.txt b/BNFKT4oBgHgl3EQfWi5i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac0e0312fa6de02709b2c619ce2dadf35d5dbb94 --- /dev/null +++ b/BNFKT4oBgHgl3EQfWi5i/content/tmp_files/load_file.txt @@ -0,0 +1,585 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf,len=584 +page_content='Graph Attention with Hierarchies for Multi-hop Question Answering Yunjie He∗ University College London yunjie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='he.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='17@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='uk Ieva Stali¯unait˙e† Accelex Technology ieva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='staliunaite@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='com Philip John Gorinski Huawei Noah’s Ark Lab, London philip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='john.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='gorinski@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='com Pontus Stenetorp University College London pontus@stenetorp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='se Abstract Multi-hop QA (Question Answering) is the task of finding the answer to a question across multiple documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In recent years, a number of Deep Learning-based approaches have been proposed to tackle this complex task, as well as a few standard benchmarks to assess mod- els’ Multi-hop QA capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In this paper, we focus on the well-established HotpotQA benchmark dataset, which requires models to perform answer span extraction as well as sup- port sentence prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We present two ex- tensions to the state-of-the-art Graph Neural Network (GNN) based model for HotpotQA, Hierarchical Graph Network (HGN): (i) we complete the original hierarchical structure by introducing new edges between the query and context sentence nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (ii) in the graph prop- agation step, we propose a novel extension to Hierarchical Graph Attention Network – GATH (Graph ATtention with Hierarchies) – that makes use of the graph hierarchy to up- date the node representations in a sequential fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Experiments on HotpotQA demon- strate the efficiency of the proposed modifica- tions and support our assumptions about the effects of model-related variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 1 Introduction Question Answering (QA) tasks can be classified into single-hop and multi-hop ones, depending on the complexity of the underlying reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Dif- ferent from single-hop QA (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Trischler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2017), where ques- tions can be answered given a single paragraph or single sentence in the context, multi-hop QA re- quires us to retrieve and reason over scattered infor- mation from multiple documents, as demonstrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' There are many methods proposed for addressing the multi-hop QA problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' One type of ∗Work carried out as part of MSc thesis supervised by Huawei Noah’s Ark Lab, London †Work carried out while working at Huawei Noah’s Ark Lab, London Question: Where did the form of music played by Die Rhöner Säuwäntzt originate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Answer: United States Supports: Document 9 s1: Die Rhöner Säuwäntzt are a Skiffle- Bluesband from Eichenzell-Lütter in Hessen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Document 4 s1: Skiffle is a music genre with jazz, blues, folk and American folk influences [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='] s2: Originating as a term in the United States in the first half of the 20th century [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='] Figure 1: Example of a multi-hop answer and support prediction, as found in HotpotQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' these recent approaches extends well-performing single-hop machine reading comprehension mod- els to be multi-hop, such as DecompRC (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2019) and QFE (Nishida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The other avenue is to develop models specifi- cally aimed at multi-hop QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Among those, Graph Neural Networks (GNNs) have recently garnered a lot of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In GNN-based approaches, gaphs are employed to represent query and context con- tents (nodes) and the underlying relationships be- tween them (edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Information between nodes is simultaneously propagated via the edges with the help of a variety of GNNs, such as Graph Convolu- tional Network (GCN) (Kipf and Welling, 2017), Graph Attention Network (GAT) (Veliˇckovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2017), or Graph Recurrent Network (GRN) (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' With these GNNs, node representa- tions are obtained conditioned on the question and context documents, and used for the QA task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In this paper, we focus on one particular GNN ap- proach designed for the Hotpot QA benchmark, the Hierarchical Graph Network (HGN) introduced in Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' HGN constructs a hierarchical graph that integrates nodes from different granu- larity levels (question/paragraph/sentence/entity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The edges in the graph capture the interactions be- tween the information from heterogeneous levels of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This hierarchical graph structure arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='11792v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='CL] 27 Jan 2023 has been shown to be crucial to the model’s remark- able performance1 on both finding scattered pieces of supporting information across documents and the answer span prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The contribution of this work is three-fold: (i) we extend the edges of HGN with a new edge type between the query and sentences, completing its original structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (ii) we introduce a novel exten- sion of the Graph Attention Network – Graph At- tention with Hierarchies (GATH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' GATH allows for making use of the explicit hierarchical graph struc- ture, by propagating information through the graph in a sequential fashion based on the hierarchy’s levels, rather than updating all nodes simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (iii) We perform initial experiments on the HotpotQA benchmark, providing evidence of the effectiveness of our proposed extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Code related to graph completion and GATH will be made publicly available at redacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 2 Background To solve the multi-hop QA problem, two general research paths have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The first direc- tion focuses on extending the successful single- hop machine reading comprehension method to the multi-hop QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' DecompRC (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2019) de- composes the multi-hop reasoning problem into multiple single-hop sub-questions based on span predictions and applied traditional machine reading comprehension techniques on these sub-questions to obtain answers to the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Query-Focused Extractor (QFE) (Nishida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2019) reformulates the multi-hop QA task as a query-focused summa- rization task based on the extractive summarization model (Chen and Bansal, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The second research direction natively addresses the task as a multi-hop setting, and directly tries to gather the information from all context doc- uments in order to answer the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Many approaches based on the transformer architecture (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2017) address the multi-hop QA task as simply one of attention between all words in all available documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In such approaches, the problem quickly becomes intractable due to the long inputs involved, and they thus typically focus on alleviating the problems of using a full attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The Longformer (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2020), for example, introduces a windowed attention mechanism to localise the problem, allow- 1At the time of writing, HGN achieves SOTA results on HotpotQA, for GNN-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' ing for much longer input sequences to be handled than with standard BERT-based language models (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' However, recently more research effort has been put toward approaches that employ Graph Neural Networks, which allow for organising information from various sources into a graph structure before addressing the core task of Question Answering, mitigating the need for very-long-distance attention functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Coref-GRU (Dhingra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2018) integrates mul- tiple evidence associated with each entity mention by incorporating co-reference information using a collection of GRU layers of a gated-attention reader (Dhingra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' However, Coref- GRU only leverages co-references local to a sen- tence but ignores other useful global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' To address this problem, MHQA-GRN and MHQA- GCN (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2018a) integrate evidence in a more complex entity graph, with edges that also connect global evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Similarly, De Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (2019) also encode different relations between en- tity mentions in the documents and perform the graph reasoning via Graph Convolutional Network (GCN) (Kipf and Welling, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' All of the above methods which involve Graph Neural Networks only consider entity nodes and the relations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The HDE-Graph (Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2019) extends these works by creating a new type of graph with nodes corresponding to answer candi- dates, documents and entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Different edges are included into the graph to capture the interaction between these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' DFGN (Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2019) con- structs a dynamic entity graph and performs graph reasoning with a fusion block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This fusion block in- cludes iterative interactions between the graph and the documents (Doc2Graph and Graph2Doc flows) in the graph construction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Hierarchical Graph Network (Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2020) proposes a hier- archical graph that incorporates nodes on different levels of a hierarchy, including query, paragraph, sentence, and entity nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This hierarchical graph allows the model to aggregate query-related data from many sources at various granularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' One limitation that all of the above conventional QA graph neural networks share is that their in- formation propagation mechanisms do not directly utilise the (explicit or implicit) hierarchical prop- erty of the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In fields outside of Natural Language Processing, recent studies on hi- erarchical graph neural networks focus on passing information on each hierarchical level to the node at different attention weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In multi-agent reinforcement learning, HGAT (Ryu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2020) generates hierarchical state- embedding of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This HGAT model stacks inter-agent and inter-group graph attention net- works hierarchically to capture inter-group node interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' A two-level graph attention mechanism (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2020) was developed for propagating information in the close neighborhood of each node in the constructed hierarchical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' HATS (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2019) predicts stock trends using relational data on companies in the stock market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' HATS selectively aggregates information from different relation types with a hierarchically designed atten- tion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' By maintaining only important information at each level, HATS efficiently filters out relations (edges) not useful for trend prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' However, all previous studies on hierarchical graph neural networks only exploit the possible hierarchical structure on the graph node itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Dif- ferent from the above methods, our proposed hi- erarchical graph attention mechanism allows the graph node embeddings to be updated in the or- der of the hierarchical granularity level, instead of simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 3 Model As our proposed improvements are largely aimed at the established Hierarchical Graph Network (HGN) model (Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2020) for HotpotQA, we briefly describe the original architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' HGN builds a hierarchical graph with nodes from several granu- larity levels (question/paragraph/sentence/entity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This hierarchical graph structure is good at captur- ing the interaction between nodes from different granularity levels and has been shown beneficial to the model’s remarkable performance on both finding scattered pieces of supporting information across documents, and to answer span prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The full HGN model pipeline consists of four modules: (i) the Graph Construction Module se- lects query-related paragraphs and builds a hier- archical graph that contains edges between nodes from different granularity levels within the para- graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (ii) the Context Encoding Module gives an initial representation/embeddings for nodes in the graph via encoding layers that consist of a RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2019) encoder and a bi- attention layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (iii) the Graph Reasoning Mod- ule updates the initial representation of all nodes via reasoning over the hierarchical graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (iv) the Multi-task Prediction Module performs multiple sub-tasks including paragraph selection, support- ing facts prediction, entity prediction and answer span extraction, based on the representation of all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This process is summarized in Figure 2, as presented by the original authors of the HGN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We note that HGN still has limitations on its graph structure and the graph reasoning step, and in this work introduce according changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Our proposed extensions aim to further improve HGN through a more complete graph structure, and a novel hierarchical graph nodes update mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' As such, our method mainly targets the Graph Con- struction and Graph Reasoning Modules, described in more detail below, while we leave the Context Encoding and Multi-task Prediction Modules un- changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Graph Construction Module The Hierarchical Graph is built based on the given HotpotQA question-context pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This construc- tion process consists of two steps: (i) multi-hop reasoning paragraph retrieval from Wikipedia, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' selecting candidate paragraphs with potential multi- hop relationship to the question as paragraph nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (ii) adding edges between question, sentence and entity nodes within the retrieved paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In particular, the first step consists of retriev- ing “first-hop” paragraphs, that is, paragraphs of Wikipedia entries that belong to entities mentioned in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' After this, a number of “second- hop” paragraphs is selected, from Wikipedia arti- cles that are hyper-linked from these first hops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Our work keeps the original paragraph selection method, but introduces novel meaningful edges between graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Context Encoding Module With the hierarchical graph structure in place, rep- resentations of the nodes within the graph are ob- tained via the Context Encoding Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In this encoder, query and context are concatenated and fed into a pretrained RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The obtained representations are further passed into a bi-attention layer (Seo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2018) to enhance the cross interactions between the question and the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Through this encoding mechanism, the question node is finally represented as q ∈ Rd and the i-th paragraph/sentence/entity nodes are repre- sented by pi, si and ei ∈ Rd respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Figure 2: Model architecture of Hierarchical Graph Network (HGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This illustration was originally introduced in Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We include it here for completion, to provide an overview of HGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Graph Reasoning Module Intuitively, the initial representations of the graph nodes only carry the contextualized information contained within their local contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' To benefit from the hierarchy and information across differ- ent contexts, the Graph Reasoning Module further propagates information between the graph nodes using a single-layered Multi-head Graph Attention Network (GAT) (Veliˇckovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' How- ever, we believe the simultaneous node-update per- formed by standard GAT can be improved, in the presence of the explicitly given hierarchical prop- erty of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We therefore propose a novel hierarchical graph reasoning method that performs node updates sequentially, for different levels of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In this manner, nodes on certain granularity levels of the graph are allowed to first aggregate some information, before passing it on to their neighbours on other levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We speculate that this staggered information passing paradigm can be beneficial to the multi-hop Question An- swering task, by passing on more question-specific contextualized information to relevant nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Multi-task Prediction Module The final step of the HGN model is to jointly pre- dict answer and supporting facts for the question via multi-task learning based on the updated graph node representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This is decomposed into five sub-tasks: (i) paragraph selection determines if a paragraph contains the ground truth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (ii) sentence selection determines if a sentence from the selected paragraph is a supporting fact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (iii) answer span prediction finds the start and end indices of the ground-truth span;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (iv) answer type prediction pre- dicts the type of the question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (v) entity prediction determines if the answer can be found among the selected entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The above sub-tasks are jointly trained through multi-task learning with the final objective of the total loss from these sub-tasks: Ljoint =Lstart + Lend + λ1Lpara+ λ2Lsent + λ3Lentity + λ4Ltype (1) With HGN re-introduced for completeness, we describe our proposed extensions to the original architecture in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='1 Completion of the graph structure HGN constructs a hierarchical graph connecting the query node with the selected multi-hop para- graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Each selected paragraph contains sentences and entities which are also encoded as nodes in the hierarchical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The graph not only incorpo- rates the natural hierarchy existing in paragraphs, sentences and entities, but also includes helpful connections between them to faciliate the structual information propagation within the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Specif- ically, the graph consists of seven types of edges, which link the nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' These edges are (i) edges between the question node and first- hop paragraph nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (ii) edges between paragraph nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (iii) edges between sentences in the same paragraph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (iv) edges between paragraph nodes and the corresponding within-paragraph sentence nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (v) edges between second-hop paragraphs and the hyperlinked sentences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (vi) edges between ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Multi-task Prediction Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Graph Construction Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Paragraph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Supporting Facts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Entity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Answer Span ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Extraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='介 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Paragraph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='(P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Updated: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Gated Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='hyperlink ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='介 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Sentence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='S2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='S3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='S4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='S5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Graph Reasoning Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Initial Representations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Entity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='E3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='E4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='E1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Adriana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='个 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='New York ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Virginia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Greenwich ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Trigiani ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Village ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='City ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Context Encoding Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='P2Figure 3: Hierarchical Graph with (orange-colored) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='new question_sentence edges added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' the question node and its matching entity nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (vii) edges between sentence nodes and their corre- sponding within-sentence entity nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We note that the only type of edge that seems to be missing from the graph are question-sentence edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Hence, we first complete the hierarchi- cal graph by introducing novel question_sentence edges which connect the question node with all sentence nodes of selected paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Such new connections are introduced as edge (viii) in the hier- archical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The constructed hierarchical graph with novel edges added is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We reason that this more complete graph might help the model to learn more useful embedding because of the modification in the graph topology, which facilitates the information transmission be- tween the question and sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='2 Graph Attention with Hierarchies The Graph Reasoning Module updates the contex- tualized representations of graph nodes to capture the information aggregated from topological neigh- bours such that the local structures of these nodes can be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In HGN, this process is realized by the Graph Attention Network (GAT) (Veliˇckovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2017), a well-established GNN approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' However, we note that in the specific setting of Multi-hop QA with the presence of an explicit hierarchical graph structure, GAT might not be able to make full use of the information encoded in the graph, as it will not directly capture the crucial dependencies between “levels” of the hi- erarchical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' To address this problem, we propose a novel Graph Attention Network with Hierarchies (GATH) which updates nodes sequentially condi- tioned on an imposed order over the hierarchy lev- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This is expected to help the model more effec- tively processes the local observation of each node into an information-condensed and contextualized state representation for individual nodes on specific levels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' for paragraphs, before passing their in- formation on to their neighbours on other levels, such as to entity nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We expect this staggered flow of information might help the model aggregate information that is more useful and conditioned on the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The nodes in the graph are split into four cate- gories, and can be represented by q, P, S and E: P = {pi}np i=1 S = {si}ns i=1 E = {ei}ne i=1 with each node embedded with an embedding func- tion as described above, into a d-dimensional vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' These node representations are jointly repre- sent the graph nodes as H = {q, P, S, E} ∈ Rg×d, g = 1 + np + ns + ne GATH updates all initial node embedding H to H ′ through hierarchical graph updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Different from GAT, GATH updates the nodes representation sequentially, according to a pre-determined order of hierarchical levels, instead of simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' It takes the initial node representations H as input, but first only updates information of node features of the first hierarchical level while keeping other node embeddings unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' For example, if the first level to be updated is the paragraph level, we obtain the updated graph representation Hpara = {h1, h ′ 2, h ′ 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' h ′ 1+np, h2+np, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' , hg} Specifically, h ′ i = ∥K k=1LeakyRelu( � j∈Ni αk ijhjWk) (2) where ∥K k=1 represents concatenation of K heads, Wk is the weight matrix to be learned, Ni repre- sents the set of neighbouring nodes of node i and αk ij is the attention coefficient calculated by: αk ij = exp(LeakyRelu([hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' hj]wk eij)) � t∈Ni exp(LeakyRelu([hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' ht]wkeit)) (3) where [hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' hj] denotes the concatenation of hi and hj, and wk eij is the weight vector corresponding to the edge between node i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Based on the updated embeddings on the para- graph level Hpara, we might next consider updating Q edge (vi) edge (i) Paragraph edge (ii) Level P1 P2 édge (vili) edge (v) edge (iv) Sentence Level S1 S2 S3 S4 S5 edge (ii) edge (vii) Entity Level E1 E2 E3 E4the information on the sentence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' GATH propa- gates information to all nodes on the sentence level based on Hpara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This will output a further updated graph representation Hsent = {h1, h ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' h ′ 1+np+ns, h2+np+ns, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' , hg} with all nodes in P and S updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Continuing the process in this manner, we even- tually will have updated all node representations to obtain H ′ {h ′ 1, h ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', h ′ g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Algorithm 1 summa- rizes the above procedures in pseudo code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Ad- ditionally, these updating steps are combined and illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 4 Experiments In this section, we present experiments comparing our extended HGN models with GATH with the original one employing GAT, and provide a detailed analysis of the proposed improvements and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' For all experiments, we use RoBERTalarge as the base embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We train with a batch size of 16 and a learning rate of 1e−5 over 5 epochs, with λ1, λ3, λ4 = 1 and λ2 = 2, and we employ a dropout rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='2 on the transformer outputs, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='3 throughout the rest of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='1 Dataset The effects of the above proposed improvements are assessed based on HotpotQA (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' It is a dataset with 113k English Wikipedia-based question-answer pairs with two main features: (i) It requires reasoning over multiple documents with- out constraining itself to an existing knowledge base or knowledge schema;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (ii) Sentence-level sup- porting facts are given for the answer to each ques- tion, which explain the information sources that the answer comes from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The performance of models on HotpotQA is mainly assessed on two metrics, exact match (EM) and F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The model is ex- pected to not only provide an accurate answer to the question, but also to give supporting evidences for its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Thus, EM and F1 score are calculated for both answer spans and supporting facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' HotpotQA has two settings: Distractor and Full- wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In the distractor setting, context paragraphs consist of 2 gold truth paragraphs containing in- formation that is needed to solve the question, and 8 paragraphs retrieved from Wikipedia based on the question, serving as related yet uninformative distractors for the question-answer pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In the Fullwiki setting, all context paragraphs come from Wikipedia’s top search results, and they need to be pre-ranked and selected in a first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Compared with the distractor setting, this setting requires us to propose an additional paragraph selection model concerned with information retrieval, before we address multi-hop reasoning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' As all our pro- posed extensions aim at the graph construction and reasoning steps, we only perform these initial ex- periments to assess the impact of our approach in the distractor setting, where we are independent from the influence of such a retrieval system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='2 Experimental Results Using the HotpotQA dataset, the models with our extensions of graph completion and GATH are com- pared against the baseline model of HGN with stan- dard GAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Since it could reasonably be argued that GATH “simulates” a (partially) multi-layered GAT in the sense that some nodes are updated only after others have already been able to incorporate neighbouring information – which in standard GAT requires at least two full layers – we also include an HGN trained with a two-layer network rather than the single layer used in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Table 1 summarizes the results on the dev set of HotpotQA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 2Authors’ note: unfortunately, despite our best efforts we were not able to reproduce the numbers reported for HGN in Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (2020), even with their original, open-sourced code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We tried both the hyper-parameters as published in the paper, and the ones shipped with the code release however, the RoBERTalarge performance when training from scratch was consistently much lower than expected on dev (∼ 74 vs ∼ 70 joint F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We contacted the original authors, who were not able to help out with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In light of these discrepancies, we Algorithm 1 Graph Attention Network with Hierarchies (GATH) Input: H = hi, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content="., hg} Output: H' = {hi, h2, ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='.,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='] for t in 1 : total number of levels do create Ht = [ ] foriin 1 : g do if node i belongs to level t then ht ← IIK=LeakyRelu(ZjeN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Qtit) k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='(t)ht-1wk,(t) k,(t) EsE N, exp( LeakyRelu([ht-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='h-1]we;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content="(t)) else h ←ht-1 end if Append h, to Ht end for end for return H' - Htotal number of levelsFigure 4: Hierarchical node representation update process." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The grey-colored graph nodes are initial contextualized embedding given by the Context Encoding Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Through the paragraph level message passing layer, only the neighboring information of all paragraph nodes can be passed and renewed on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Similar steps repeat for sentence level and entity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' For convenience of labeling indices, we set np = 2, ns = 5, ne = 4 Completion of graph structure The HGN with new query-sentence edges improves over the base- line by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='7/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='4 on Joint EM and F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This supports our intuition that the the missing question- to-sentence edges can indeed bring advantages to the model’s abilities of both answer span extraction and supporting facts prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Graph Attention with Hierarchies GATH al- lows for pre-defining the order of level updates in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Given that the order in which the hierar- chy levels are updated is likely to affect the model’s performance, we perform experiments with dif- ferent orders (P/S/E3,4, E/S/P, S/E/P and S/P/E) and compare them to the baseline models with one and two-layer GAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' All the GATH-based ex- tended models outperform the baseline model on the answer-span extraction by an absolute gain of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='6 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='4 points on the answer extraction metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' On the other hand, the order of hierarchical lev- els does show an influence on the model’s evi- decided to focus only on dev set performance when assessing the impact of our extensions against re-trained vanilla HGN, as a fair comparison to the original model on test was not possible at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 3P/S/E abbreviates Paragraph/Sentence/Entity 4We exclude the query level update to make it more com- parable to the baseline model, which also excludes this update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' dence collection ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The “wrong” order leads to worse performance of the extended model, such as in the E/S/P and S/P/E cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' On most metrics, but specifically on joint F1 score, the extended GATH-based model with the or- der S/E/P outperforms not only the baseline model, but also the other GATH models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' It achieves a Joint EM/F1 score of 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='9/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='5, exceeding the baseline model’s performance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='2 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Interestingly, the 2-layer GAT version of HGN slightly under-performs when compared to the orig- inal HGN setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' While gaining 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='3 points in sup- port prediction F1, it loses the same amount of performance in answer prediction and joint scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' We assume this is why the original HGN calls for only one layer, when we could intuitively have ex- pected multi-layered networks to perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Combined query-sentence edges and GATH The above experimental results demonstrate the individual effectiveness of these two proposed im- provements of graph completion and GATH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Nat- urally, we are also interested in the performance resulting from combining both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The “HGN (Com- bined)” row in Table 1 represents the model com- bining graph completion and GATH-S/E/P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=" This combined model brings slight improvement over H Hsent Hpara h1 h1 edge (i) Paragraph edge (i) Paragraph Level P1 P2 Level P1 h2 h2 h2 edge (iv) edge (v) edge (iv) - dge (v) Sentence Sentence h's Level S2 63 h?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=" h3 Level S1 edge (ii) edge (vii)i h'4 Entity Level h4 h4 E1 EntityLevel hs Paragraph level nodes updating ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=". Sentence level nodes updating hg hg hg H' Hent edge(vi) hi edge (vi), h1 Paragraph Paragraph Level P1 P2 Level P1 h2 h2 Sentence hs Level S2 S3 $4 S5 Sentence S1 hs Level S3 SA edge (vii) : h4 h4 edge (vi) i Entity Level E2 E3 E4 EntityLevel hs E3 hs 1 ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='. Question level nodes updating hg hg Entity level nodes updatingAnswer Support Joint Model EM F1 P R EM F1 P R EM F1 P R HGN (baseline) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='4 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='9 HGN (2-layer GAT) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='1 78.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='3 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='1 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='5 HGN (Combined) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='9 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='8 Table 1: Performance of the proposed HGN with completed edges (HGN que_sent), GATH, and both extensions combined on the development set of HotpotQA in distractor setting, against the baseline model HGN with GAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' the other models on most metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' This final model sees further improvements, particularly in the an- swer span prediction task, and achieves the overall highest joint F1 score at 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='9, indicating that the contributions of graph completion and GATH are mutually benefitial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='3 Error Analysis In this section, we perform an error analysis on the concrete influence of the proposed HGN (com- bined) model based on question types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The major- ity of questions in HotpotQA fall under the bridge5 and comparison reasoning categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' As sug- gested by Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' (2020), we split comparison questions into comp-yn and comp-span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The former represents questions that should answer the compar- ison between two entities with “yes” or “no”, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' “Is Obama younger than Trump?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=', while the latter requires an answer span, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' “Who is younger, Obama or Trump?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Table 2 shows the performance of the original HGN model and the proposed model HGN-GATH (combined) on various types of reasoning questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Results indicate that comp-yn questions are easiest for both models, and the bridge type is the hardest to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' The analysis table shows that HGN (com- bined) is more effective than the original model on all of these reasoning kinds except support EM for comp-yn, though even here the much improved answer prediction leads to an overall improvement of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='42 on Joint EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 5requiring a bridging entity between support sentences, needed to arrive at the answer HGN-GAT Question Ans EM Sup EM Joint EM Pct(%) comp-yn 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='22 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='44 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='19 comp-span 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='50 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='04 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='49 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='90 bridge 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='08 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='01 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='73 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='91 HGN-GATH (Combined) Question Ans EM Sup EM Joint EM Pct(%) comp-yn 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='81 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='79 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='19 comp-span 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='42 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='53 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='34 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='90 bridge 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='95 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='08 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='71 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content='91 Table 2: Original HGN (top) and HGN-GATH com- bined (bottom) model results for various reasoning types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' ‘Pct’ signifies percentage of all questions per category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' 5 Conclusions and Future Work In this paper, we proposed two extensions to Hi- erarchical Graph Network (HGN) for the multi- hop Question Answering task on HotpotQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' First, we completed the hierarchical graph structure by adding new edges between the query and context sentence nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Second, we introduced GATH as the mechanism for neural node updates, a novel extension to GAT that can update node representa- tions sequentially, based on hierarchical levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' To the best of our knowledge, this is the first time the hierarchical graph structure is directly exploited in the update mechanism for information propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' Experimental results indicate the validity of our approaches individually, as well as when used jointly for the multi-hop QA problem, outperform- ing the currently best performing graph neural net- work based model, HGN, on HotpotQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} +page_content=' In the future, we would particularly like to in- tegrate hierarchical graph attention weights into GATH, as motivated by related research in Rein- forcement 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pages 9612–9619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFKT4oBgHgl3EQfWi5i/content/2301.11792v1.pdf'} diff --git a/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf b/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9f3a4d60c26944e75d1d62911e7bbda0457ee552 --- /dev/null +++ b/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a1df23e7f207db2f86bfcc845a2f0ccb429dffc4867b1d7e29cc5afb85bd964d +size 24150783 diff --git a/FNE1T4oBgHgl3EQf-gbR/content/tmp_files/2301.03571v1.pdf.txt b/FNE1T4oBgHgl3EQf-gbR/content/tmp_files/2301.03571v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..655cbc5c0407834b90eb1461260a66a5032652f8 --- /dev/null +++ b/FNE1T4oBgHgl3EQf-gbR/content/tmp_files/2301.03571v1.pdf.txt @@ -0,0 +1,1215 @@ +Dipolar Spin Liquid Ending with Quantum Critical Point in a Gd-based Triangular Magnet +Junsen Xiang,1, ∗ Cheng Su,2, ∗ Ning Xi,3, ∗ Zhendong Fu,4 Zhuo Chen,5 Hai Jin,6 Ziyu Chen,2 Zhao-Jun Mo,7 +Yang Qi,8, 9 Jun Shen,5, 10 Long Zhang,11, 12 Wentao Jin,2, † Wei Li,3, 12, 13, ‡ Peijie Sun,1, § and Gang Su11, 12, ¶ +1Beijing National Laboratory for Condensed Matter Physics, +Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China +2School of Physics, Beihang University, Beijing 100191, China +3CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China +4Neutron Platform, Songshan Lake Materials Laboratory, Dongguan 523808, China +5School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China +6Department of Astronomy, Tsinghua University, Beijing 100084, China +7Ganjiang Innovation Academy, Chinese Academy of Sciences, Ganzhou 341119, People’s Republic of China. +8State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China +9Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China +10Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China +11Kavli Institute for Theoretical Sciences, and School of Physical Sciences, +University of Chinese Academy of Sciences, Beijng 100049, China +12CAS Center for Excellence in Topological Quantum Computation, +University of Chinese Academy of Sciences, Beijng 100190, China +13Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China +(Dated: January 10, 2023) +By performing experimental and model studies of a triangular-lattice dipolar magnet KBaGd(BO3)2 (KBGB), +we find the highly frustrated magnet with a planar anisotropy hosts a strongly fluctuating dipolar spin liquid +(DSL) originating from the intriguing interplay between dipolar and Heisenberg interactions. The DSL con- +stitutes an extended regime in the field-temperature phase diagram, which gets narrowed in temperature range +as field increases and eventually ends with a quantum critical point (QCP) at Bc ≃ 0.75 T. Based on dipolar +Heisenberg model calculations, we identify the DSL as a Berezinskii-Kosterlitz-Thouless (BKT) phase. Due to +the tremendous entropy accumulation that can be related to the strong BKT and quantum fluctuations, unprece- +dented magnetic cooling effects are observed in the DSL regime and particularly near the QCP, making KBGB +a superior dipolar coolant over commercial Gd-based refrigerants. We establish a universal phase diagram for +triangular-lattice dipolar quantum magnets where emergent symmetry plays an essential role, and lay down +foundations for their applications in sub-Kelvin refrigeration. +Introduction.— Triangular-lattice quantum antiferromag- +nets have raised great research interest recently, due to the +unusual quantum spin states and transitions therein [1, 2]. +One prominent example is the quantum spin liquid (QSL) [3– +5] and its possible materialization in organic compounds [6– +8] and rare-earth triangular magnets [9–16]. Due to the in- +triguing spin frustration effects and two dimensionality (2D), +Berezinskii-Kosterlitz-Thouless (BKT) physics may appear in +the triangular quantum antiferromagnets. The Co-based quan- +tum antiferromagnet Na2BaCo(PO4)2 hosts persistent spin +fluctuations [17–20] till very low temperature, and is proposed +to posses spin supersolid state with BKT fluctuations of U(1) +phase [21]. Emergent symmetry, as a consequence of frustra- +tion, has also been disclosed on the triangular lattice, with a +recent example of rare-earth magnet TmMgGaO4 [22–27]. +Recently, it has been proposed that the dipolar interactions +can give rise to QSL in triangular-lattice quantum spin sys- +tems [29]. Lately such dipolar system has been realized in Yb- +based triangular compounds [30–34]. However, the dipolar +∗ These authors contributed equally to this work. +† wtjin@buaa.edu.cn +‡ w.li@itp.ac.cn +§ pjsun@iphy.ac.cn +¶ gsu@ucas.ac.cn +interactions are rather weak and it is very challenging for con- +ventional thermodynamic and spectroscopic measurements to +probe the exotic spin states due to dipolar interactions. On +the contrary, the rare-earth dipolar magnets with larger mo- +ments, e.g., Gd-based compounds with µeff ≈ 8µB and high +spin S = 7/2, are much less explored both in experiments +and theories. It is expected that the dipolar frustration ef- +fects are a priori more evident in these systems. Moreover, +in sub-Kelvin refrigeration for space applications [35, 36] and +quantum computations [37], high-spin frustrated magnets, es- +pecially those with spin-liquid like behaviors [38], can have +great entropy densities and cooling capacity, holding thus +strong promise as superior coolants. +In this work, we perform low-temperature thermodynam- +ics and magnetocalorics measurements on single-crystal sam- +ples of Gd-based triangular-lattice compound KBaGd(BO3)2 +(KBGB). The thermodynamic measurements suggest a dipo- +lar spin liquid state with no conventional ordering but strong +spin fluctuations, which are reflected in the algebraic specific +heat and imaginary dynamical susceptibility (χ +′′ +ac). We estab- +lish a dipolar Heisenberg model with both dipole-dipole and +Heisenberg interactions for KBGB. Monte Carlo (MC) sim- +ulations explain excellently the experimental measurements +and unveil the exotic spin states and transitions in the phase +diagram. In particular, the model simulations suggest a two- +step melting of the clock antiferromagnetic (AF) order via two +arXiv:2301.03571v1 [cond-mat.str-el] 9 Jan 2023 + +2 +-0.8 +0 +0.8 +-0.8 +0 +0.8 +Yy +Yx +-0.8 +0 +0.8 +-0.8 +0 +0.8 +Yy +Yx +-0.8 +0 +0.8 +-0.8 +0 +0.8 +Yy +Yx +6-clock AF +DSL +PM +Temperature +(b) +(c) +(d) +(e) +(a) +b +a +c +K/Ba +Gd +B +O +b +a +a∗ +Si +Sj +eij +FIG. 1. (a) shows the crystal structure of KBaGd(BO3)2, where (b) +triangular-lattice layers of GdO6 octahedra are separated by the Ba/K +layers with site mixing. The grey arrows refer to the spins on site i +and j, and the unit vector eij is also indicated. Dipole-dipole inter- +actions are bond-dependent and follow the ¯3m site symmetry. (c)-(e) +are histograms of the order parameter Ψxy ≡ Ψx + iΨy for the 6- +clock antiferromagnetic (AF) [28], dipolar spin liquid (DSL) with an +emergent U(1) symmetry, and the paramagnetic (PM) phases. +BKT transitions, between which a floating BKT phase emerge +with an emergent U(1) symmetry, well accounting for the ex- +perimentally observed spin liquid behaviors with enormous +low-temperature entropy. Consequently, giant magnetocaloric +effect (MCE) is observed in the quasi-adiabatic demagnetiza- +tion measurements, where we find a clear dip in temperature +which suggests the presence of quantum critical point (QCP) +near Bc ≃ 0.75 T. The lowest temperature of 70 mK clearly +surpasses that of commercial refrigerant Gd3Ga5O12 (GGG) +under similar conditions. Overall, the triangular-lattice rare- +earth dipolar magnets open an avenue for exploring exotic +spin states as well as finding superior sub-Kelvin coolants. +Crystal structure and effective model for KBaGd(BO3)2.— +Centimeter-sized single crystals of KBGB were synthesized +using the flux method as described in detail in Supplementary +Materials (SM) [28], and the X-ray diffraction measurements +suggest high quality of the single crystals. KBGB is found +to crystallize in a trigonal structure [40, 41] with space group +R-3m [c.f., Fig. 1(a)], and has a relatively high ionic density +of 6.4 nm−3. As shown in Fig. 1(b), magnetic Gd3+ ions with +4f 7 electron configuration (L = 0, S = 7/2) form perfect +triangular lattice. +The dipolar interaction between magnetic ions Gd3+ +has a characteristic energy Edp +∼ +2µ0µ2 +eff/4πa3 +≈ +0.05 meV (with µeff ≈ 8µB), which determines the low- +temperature spin states in KBGB. To simulate such Gd- +based dipolar magnet, we consider the following Hamil- +tonian, H += +JH +� +⟨i,j⟩NN Si · Sj + JD +� +i,j[Si · Sj − +3(Si · eij)(Sj · eij)]/r3 +ij, where eij(rij) refers to the unit vec- +tor(distance) between site i and j. JH and JD refer to the +nearest neighbor (NN) Heisenberg and dipole-dipole interac- +tions, respectively. As the dipolar interactions show rapid (cu- +bic) power-law decay and the longer range interactions can be +washed out, we keep only NN terms as +HDH = +� +⟨i,j⟩NN +J Si · Sj − D (Si · eij)(Sj · eij), +(1) +where J = JH + JD/a3 is the NN isotropic coupling and +D = 3JD/a3 refers to the dipolar anisotropic term. We find +the NN dipolar Heisenberg (DH) model with couplings J = +47 mK and D = 80 mK very well describe the compound +and accurately reproduce the experimental measurements on +KBGB. The MC simulations are performed on up to 60 × 60 +triangular lattice. Due to the high-spin state with S = 7/2, +classical MC simulations capture well the finite-temperature +properties of KBGB [28]. We guarantee the error bars to be +always smaller than the symbol size in the presented data. +Magnetic specific heat, susceptibility, and dipolar spin +liquid.— In Fig. 2(a) we show the zero-field specific heat +Cm measured down to 65 mK. There exists a round peak at +T ∗ ≃ 218 mK, below which the system exhibits Cm ∼ T 2 +with algebraic scaling, resembling that of 2D Heisenberg +or XY quantum spin model with U(1) symmetry [42, 43]. +The dipolar anisotropy in Eq. (1), like in spin-orbit magnets, +leads to a discretized C3 rotational symmetry, and it gener- +ically corresponds to a divergent Cm peak when transition- +ing to low-T symmetry breaking phase. +The presence of +round peak and T 2 scaling in Cm is very remarkable, which +suggests a liquid-like and strongly fluctuating spin state. In +Fig. 2(a), when compared to the renowned Gd-based refrig- +erant GGG [36, 39, 44, 45], KBGB has tremendous low- +temperature specific heat, far surpassing that of GGG. +In Fig. 2(b), we apply out-of-plane fields (B//c) to the com- +pound, and find also round peaks in Cm curves, which move +towards lower temperature with heights slightly reduced. This +suggests that the spin liquid states constitute an extended +phase that we dub as dipolar spin liquid (DSL). As field fur- +ther increases and exceeds about 0.75 T, the DSL behaviors +disappears [c.f., the contour plot of Cm/T in Fig. 3(b)], and +the Cm peak moves now to high-temperature side, with the +low-T peak and low-energy fluctuations quickly suppressed. +In Fig. 2(c), we perform magnetization measurements +on single-crystal sample of KBGB, and find a clear mag- +netic anisotropy between the out-of-plane (//c axis) and in- +plane (//a) directions. +This anisotropy can be clearly rec- +ognized in the different saturation magnetization moments +and transition field values, i.e., 1 T(0.5 T) along c(a) axis. +In Fig. 2(d), we perform low-temperature dc susceptibility +(χdc) measurements, and find χdc also exhibits a clear easy- +plane anisotropy. In addition, a small but sensible in-plane +anisotropy between a and a∗ [see inset of Fig. 2(c)] is also +observed, consistent with the intrinsic anisotropy in bond- +dependent dipolar interaction [c.f., Eq. (1)]. +To further explore the DSL, ac magnetic susceptibilities are +measured in Figs. 2(e,f), with χ′ +ac and χ′′ +ac for real and imag- +inary parts, respectively. The real χ′ +ac exhibits a frequency- +independent maximum and remains large even below the char- +acteristic temperature scale T ∗. +Therefore, although there +exist K/Ba site mixing in the compound, the spin-glass sce- +nario can be excluded in KBGB. Interestingly, the imaginary + +3 +1 +10 +100 +0 +2 +4 +6 +8 +10 +12 +Exp. +Model +χdc (emu·Oe-1·mol-1 +Gd) +T (K) +a +a* +c +0.1 T +-2 0 2 4 6 8 10 +0.0 +0.4 +0.8 +1.2 +χdc +-1 +θa -0.30 K +θa* -0.33 K + θc +-1.32 K +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +25 +50 +75 +T (K) +Cm/T (J·mol-1 +Gd·K-2) +KBGB +GGG +0 T +Cm/T ~ T +T* 218 mK +0.1 +1 +0.0 +0.5 +1.0 +1.5 +2.0 + 4943 Hz + 6253 Hz + 9984 Hz +χac'' (a.u.) +T (K) +T* +0 +1 +2 +3 +4 +0 +2 +4 +6 +8 +10 +B // a +B // c +Moment (µB/Gd) +B (T) + +0.4 K + 2.36 + 2.49 +Model +ga +gc +0.1 +1 +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +2.4 +91 Hz +955 Hz +2439 Hz +3087 Hz +3910 Hz +T (K) +χac' (a.u.) +T* +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +25 +50 +75 +0.25 T +0.5 T +0.75 T +Cm/T (J·mol-1 +Gd·K-2) +T (K) +1 T +2 T +3 T +4 T +(a) +(d) +(e) +(f) +(c) +(b) +a* +a +FIG. 2. Specific heat of KBGB under (a) zero and (b) finite fields along out-of-plane direction (B//c). An algebraic Cm ∼ T 2 scaling is +observed below the round peak temperature T ∗, and the Cm/T values far outweigh that of GGG [39] for T ≲ T ∗. In (b) we find the round +peak in Cm/T firstly moves towards lower temperature and later for B > Bc ≃ 0.75 T the low-temperature Cm quickly gets suppressed. (c) +shows the magnetization curves of the single-crystal KBGB sample for B//a and //c, and the results show excellent agreement with the DH +model calculations (solid lines). The saturation moments are µsat +a +≃ 8.26µB and µsat +c +≃ 8.72µB, from which we determine the Landé factors +ga ≃ 2.36 and gc ≃ 2.49, respectively. The as-grown KBGB single crystal is shown in the inset, with directions a and a∗ also indicated. (d) +shows the molar dc magnetic susceptibilities (χdc) measured along the a, a∗, and c axes, respectively, where the solid lines representing the +DH model calculations show excellent agreements. The inset shows the Curie-Weiss fittings in the paramagnetic regime 0.4 K ≤ T ≤ 10 K, +with the fitted Curie-Weiss temperatures θa,a∗,c also indicated. (e, f) present respectively the real and imaginary ac susceptibilities measured +with different frequencies. +ac susceptibility χ′′ +ac(T), although being featureless for low +frequencies ω ≲ 4 kHz, show a clear temperature-dependent +behavior for higher frequencies in Fig. 2(f). Considering that +χ′′(ω) can be directly related to the dynamical correlation +S(ω) through the fluctuation-dissipation theorem, χ′′(ω) ∝ +ω +T S(ω) (ω ≪ T), this clearly suggests the persistence of low- +energy spin fluctuations even below T ∗ and supports the spin- +liquid scenario. +Magnetocaloric effect and quantum critical point.— In +Fig. 3(a), we perform quasi-adiabatic demagnetization mea- +surements and obtain the isentropic curves. It is found that +KBGB clearly outperforms GGG in the minimal temperature, +i.e., Tm ≃ 70 mK (KBGB) vs. 322 mK (GGG), when starting +from the same initial condition of Ti = 2 K and Bi = 6 T. +In Fig. 3(b) we provide more of the isentropic lines from dif- +ferent initial conditions, and observe the highly asymmetric +isentropes, which “levels off” in the bright DSL regime as in- +dicated by large values of Cm/T. +For KBGB, the lowest temperature Tm is achieved at the +dip in isentropic lines and remains below 100 mK in the +small field side. This happens also for measurements starting +from rather low temperature Ti ≃ 95 mK, where the lowest +Tm ≃ 33 mK. Such unprecedented MCE response strongly +corroborates the existence of QCP at Bc ≃ 0.75 T. The mag- +netic Grüneisen ratio ΓB = +1 +T ( ∂T +∂B )S has been widely used +in the studies of heavy fermion [46–50] and low-dimensional +quantum spin systems [51–54]. In the inset of Fig. 3 an ev- +ident peak-dip structure with sign change is observed [55– +58], and the peak height exceeds 4 times that of GGG. Such a +prominent critical cooling effect provides valuable MCE evi- +dence for QCP in the compound KBGB. +Emergent symmetry in KBGB.— According to the magne- +tothermal and MCE measurements above, we arrive at the +phase diagram of KBGB in Fig. 3(b). +The two schematic +dashed lines, enclosing the DSL with large Cm/T, meet at a +QCP (Bc) where the demagnetization process reaches its low- +est temperature. Besides QCP, within the DSL regime we find +persistent spin fluctuations and cooling effects whose origin is +clarified by model calculations below. +We conduct MC calculations of the DH model [Eq. (1)] +for KBGB. As the model is highly frustrated in the out-of- +plane direction, the order parameter lies within the ab plane. +Note although the determined Landé factor gc ≃ 2.49 is +slightly larger than ga ≃ 2.36, the intrinsic planar anisotropy +of dipolar interaction leads to larger in-plane χdc (along a +and a∗ axes) than that along the c axis. The negative Curie- +Weiss temperatures fitted from the dc susceptibility reflect the +AF nature, and the slightly different θa ≃ −300 mK and +θa∗ ≃ −330 mK shows the in-plane anisotropy. In Fig. 2(d), +we find the anisotropic susceptibility and magnetization mea- + +4 +0 +1 +2 +3 +4 +5 +6 +0.1 +1 +T (K) +B (T) +70 mK +KBGB +GGG +B // c +322 mK +2 K +33 mK +0.0 +0.4 +0.8 +1.2 +0.0 +0.1 +0.2 +0.3 +B (T) +T (K) +0 +20 +40 +60 +Cm/T +DSL +QCP +Quasi-Adiabatic +PM +6-clock AF +(a) +(b) +ΓΒ (T-1) +0 1 2 3 4 +-1 +0 +1 +2 +3 +4×GGG +Bc +FIG. 3. (a) shows the quasi-adiabatic isentropes measured in KBGB +under out-of-plane field (see details in SM [28]). The KBGB curve +exhibits a clear dip at the lowest temperature Tm ≃ 70 mK, much +lower than that of GGG (Tm ≃ 322 mK). Starting from Ti ≃ 95 mK, +KBGB is observed to cool down to remarkably low temperature +Tm ≃ 33 mK in the dip (blue dotted line). The inset shows the mag- +netic Grüneisen ratio ΓB deduced from the curves in (a). (b) shows +the phase diagram of KBGB with the contour plot of Cm/T in the +background. The bright regime with large spin fluctuations represent +the DSL, with schematic dashed line boundaries, ending up with a +QCP at Bc ≃ 0.75 T. +sured along a and c axes can be well captured by the DH +model. Besides, the model calculations of specific heat also +obtain a round peak at about 270 mK, which again gets sup- +pressed as field increases (see SM [28]), very much resem- +bling the experimental data in Figs. 2(a,b). The comparisons +confirm that the compound KBGB can indeed be accurately +described by the DH model. +To characterize the spin states in the phase diagram, we in- +troduce the order parameter Ψxy ≡ meiθ = � +j eiQrj(mx +j + +imy +j ), +where j runs over the lattice sites and Q += +± 1 +2a∗, ± 1 +2b∗, ± 1 +2(a∗ − b∗) [28]. Histogram of the complex +order parameter Ψxy at various temperature are shown in +Figs. 1(c-e). At low temperature, the dipolar system exhibits +a 6-clock AF order corresponding to θ = 0, ±π/3, ±2π/3, +and π [28]. As temperature ramps up, the six points in the +histogram prolong and merge into a circle with emergent +U(1) symmetry, where the angle θ can choose arbitrary angle. +As temperature further enhances, eventually the amplitude m +vanishes and the system enters the conventional PM phase. +Recall that the 6-state clock model with an anisotropic term +∼ cos (6θ) undergoes two successive BKT transitions [59], +between which the anisotropic term becomes irrelevant per- +turbation. Based on this symmetry argument, we consider the +intermediate DSL in the system as BKT phase with emer- +gent U(1) symmetry and effectively described by 2D XY +model [60–63]. The emergent symmetry extends also to the +zero-temperature QCP as the clock term is dangerously irrele- +vant [60], and the transition directly between the 6-fold clock +symmetry broken and PM phases belong to the 3D XY univer- +sality class. Therefore, the emergent symmetry constitutes a +key for demystifying spin-liquid state and quantum criticality +in the compound KBGB. +0 +60 +120 +180 +0 +1 +2 +3 +4 +T (K) +Time (min) +4 T +6 T +0.1 +1 +0 +5 +10 +15 +20 +Sm (×10·J·Kg-1·K-1) +T (K) +∆Q +4 T +∆Sm +0 T +(a) +0.1 +1 +10 +0 +10 +20 +30 +40 +� � m (J·Kg-1·K-1) +T (K) +KBGB +GGG +4 T +2 T +1 T +(b) +FIG. 4. (a) The quasi-adiabatic demagnetization cooling curves of +KBGB, starting from two different initial conditions (Ti = 4 K, +Bi = 4 T) and (Ti = 2 K, Bi = 6 T), with reached lowest tem- +perature Tm ≃ 205 mK and 70 mK, respectively. Parasitic heat +loads are estimated to be 0.2 µW for Ti = 4 K environment and +0.05 µW for Ti = 2 K. The inset shows magnetic entropy under +zero and 4 T fields, with the shaded area representing the absorbed +heat ∆Q = 47.44 J·Kg−1 in the hold process. (b) plots the entropy +change ∆Sm vs. T, for fields decreasing from 1 T, 2 T, and 4 T to +zero, respectively. Comparisons to GGG are also presented [39, 44]. +Superior cooling performance.— Starting from 2 K en- +vironment, KBGB can reach as low as 70 mK as shown +in Fig. 4(a), such a low cooling temperature far surpasses +other Gd-based refrigerants, e.g., GGG (322 mK) and GdLiF4 +(480 mK) [64]. Besides, KBGB also exhibits long hold time +and large isothermal entropy change ∆Sm. In Fig. 4(a) we +show that KBGB remains in low temperature for a long period +after the field is exhausted. In the environment temperature of +2 K, 0.5 g KBGB remains below 140 mK for th ≈ 2 h under +0.05 µW heat leak, which can be ascribed to the large heat +absorption ∆Q depicted in the inset of Fig. 4(a). +The isothermal entropy change ∆Sm characterizes the +cooling capacity of refrigerants. In Fig. 4(b), we compare +∆Sm of KBGB with that of GGG, and find that in the whole +temperature range concerned KBGB has significantly larger +∆Sm for 1 T field. Moreover, the maximal ∆Sm of KBGB lo- +cates below 1 K [shaded regime in Fig. 4(b)], and the entropy +change in KBGB exceeds that of GGG in this sub-Kelvin +regime of central interest. Overall, the low cooling temper- +ature Tm, long hold time th, and enormous entropy change +∆Sm in the sub-Kelvin regime lead to the conclusion that +KBGB serves a superior quantum magnet coolant. +Discussions and outlook.— The pursue for high entropy +density and low ordering temperature constitutes two oppos- +ing factors hard to fulfill simultaneously in optimizing sub- +Kelvin refrigerants. Here the spin frustration and quantum +criticality in the dipolar system come to the rescue. We show +that the compound KBaGd(BO3)2 with high Gd3+ ion density +yet form a disordered and strongly fluctuating spin liquid till +extremely low temperature, giving rise to the superior cooling +capacity due to the entropy accumulation near QCP. We use +the DH model within NN interactions to describe KBGB and +find it well reproduces the experimental results. Inclusion of +further neighboring dipolar couplings will not change the con- + +5 +clusion here, as it has been shown to maintain the universality +class of BKT transitions in planar dipolar models [63, 65]. +The scenario of DSL ending up with emergent U(1) QCP +may also be applicable to other dipolar quantum magnets. Re- +cent progress in experimental studies reveal several families of +rare-earth triangular quantum dipolar antiferromagnets, e.g., +Ba3REB3O9/Ba3REB9O18 (with RE a rare-earth ion) [32, 33] +and ABaRE(BO3)2 (with A an alkali ion) [66, 67]. It has been +observed that in Ba3YbB3O9 that 80% entropy remain below +56 mK [31], despite a dipolar energy scale of about 160 mK, +suggesting that the DSL may also play a role in the Yb-based +dipolar compounds. Therefore, this work opens a venue for +hunting exotic spin states as well as superior quantum coolants +in triangular dipolar magnets. +Note added.— Upon finishing the present work, we are +aware of a recent work [68] also conducting MCE study of +KBGB with however polycrystalline samples, where they find +strong cooling effect down to 121 mK. +Acknowledgements.— W.L. is indebted to Yuan Wan and +Tao Shi for helpful discussions. W.J. and C.S. acknowledge +the support from the beamline 1W1A of the Beijing Syn- +chrotron Radiation Facility. +This work was supported by +the National Natural Science Foundation of China (Grant +Nos. 12222412, 11834014, 11974036, 12047503, 12074023, +12074024, 12174387, and 12141002), National Key R +& D Program of China (Grant No. 2018YFA0305800), +Strategic +Priority +Research +Program +of +CAS +(Grant +No. XDB28000000), and CAS Project for Young Scien- +tists in Basic Research (Grant No. YSBR-057). 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D 36, 515 (1987). + +8 +Intensity (arb. units) +Intensity (arb. units) +(003) +(006) +(009) +(0012) +FIG. S1. (a) shows the powder XRD pattern of KBGB measured at room temperature and corresponding Rietveld refinement. The open circle +and red solid line represent the observed and calculated intensities, respectively, while the blue solid line shows their difference. The olive +vertical bars mark the expected reflections for KBGB. (b) Single-crystal XRD scan along the (0,0,L) direction for one representative crystal, +revealing only peaks that are well indexed by (0,0,3n). The insets show the image of the as-grown KBGB crystals and the rocking-curve +scan of the (0,0,12) reflection fitted by a Gaussian profile. The very narrow peak width of FWHM = 0.041◦ indicates excellent quality of the +crystals. +Supplementary Materials +Dipolar Spin Liquid Ending with Quantum Critical Point in a Gd-based Triangular Magnet +Xiang et al. +Section 1. +SAMPLE PREPARATION AND STRUCTURE CHARACTERIZATION +Polycrystalline samples of KBGB were firstly prepared by standard solid-state reaction method as reported in Ref. 40. Sto- +ichiometric mixtures of K2CO3 (99.99%), BaCO3 (99.95%), H3BO3 (99.99%) and Gd2O3 (99.99%) (with 6% excess H3BO3 +and 5% excess of K2CO3 and BaCO3) were thoroughly ground and pelletized. Then the pellet was placed into an aluminum +crucible and sintered at 900◦C in air for 10 h. This sintering process was repeated for several times to minimize possible +impurities. +Single-crystal samples of KBGB were grown using the flux method as reported in Ref. 41. The pre-obtained polycrystalline +KBGB with high purity was mixed with the H3BO3 (99.99%) and KF (99.9%) fluxes in a molar ratio of 2:3:[2-3], and thoroughly +ground. The mixture was transferred into a Pt crucible, heated up to 980◦C in air for 24 h, and then slowly cooled to 790◦C with +a rate of 2◦C/h. After the furnace cooling, centimeter-sized crystals were obtained on top of the fluxes. +The phase purity of the polycrystalline KBGB sample was confirmed by powder XRD at room temperature, performed +on a Bruker D8 ADVANCE diffractometer in Bragg-Brentano geometry with Cu-Kα radiation (λ = 1.5406 Å). As shown in +Fig. S1(a), the powder XRD pattern can be well fitted with the previously reported trigonal phase of KBGB [40] (a = b = +5.4676(1) Å, c = 17.9514(3) Å), without any visible impurity peaks, indicating high purity of the synthesized KBGB powders. +The quality of the single-crystal KBGB sample was checked by high-resolution synchrotron XRD (λ = 1.54564 Å) measure- +ments at room temperature, performed on the 1W1A beamline at the Beijing Synchrotron Radiation Facility (BSRF), China. As +shown in Fig. S1(b), a long L scan, equivalent to a θ-2θ scan with respect to the normal direction of the plate-like KBGB crystal, +only shows Bragg reflections well indexed by (0, 0, 3n) as expected for the R-3m space group. The peak width (full width at half +maximum, FWHM) observed in the rocking-curve scan of the (0, 0, 12) peak is very small, 0.041(2)◦, as shown in the inset of +Fig. S1(b), which suggests excellent crystal quality. KBGB is relatively easy to synthesize and has excellent chemical stability, +paving its viable way for applications in advanced cryogenics. + +9 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.1 +430 mK +280 mK +195 mK +150 mK +95 mK +T (K) +B (T) +0.4 +0.03 +0.06 - 0.09 T·min-1 +Bi = 3 T +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +KBGB +GGG +T (K) +B (T) +0.15 T·min-1 +(a) +(c) +(b) +0.0 +0.6 +1.2 +0.1 +0.2 +0.3 +0.4 +0.5 +3.0 +2.5 +2.0 +0 +1 +2 +3 +-1 +0 +1 +2 +ΓB (T-1) +Bc +FIG. S2. (a) Illustration of the two-stage quasi-adiabatic demagnetization cooling device for the measurements of 0.5 g KBGB single crystals. +(b) shows the measured isentropic curves of KBGB starting from various initial conditions (Ti = 2 K, Bi = 4 T), (2 K, 6 T), and (4 K, 4 T), +respectively, where the lowest temperature are found to be significantly lower than those of GGG. The inset zooms in the small-field range +(B ≤ 1.2 T). (c) The DR-based measurements with an initial temperature Ti ≤ 430 mK and field Bi = 3 T, where the lowest achieved +temperature is Tm ≃ 33 mK. The inset shows the magnetic Grüneisen ratio ΓB deduced from the low-temperature isentropic T-B lines in the +main plot, where the sign change is evident and the peak becomes more and more pronounced as the initial temperature Ti lowers. +Section 2. +MAGNETOTHERMAL MEASUREMENTS +Comprehensive magnetothermal measurements were performed on single-crystal samples of KBGB. The low-temperature +specific heat (Cp, T ≥ 50 mK) and ac susceptibility (χac, T ≥ 50 mK) measurements were conducted using the Quantum +Design Physical Property Measurement System (PPMS) equipped with a 3He–4He dilution refrigerator (DR) insert. The specific +heat data were measured under various out-of-plane fields (B//c) with the semi-adiabatic heat pulse method. The phonon +contributions are negligible below 2 K as estimated via a Debye T 3 analysis of high-temperature Cp data. The ac susceptibility +(χac), as a function of temperature, was measured in zero dc field under different ac frequencies, with the amplitude of the ac +excitation field set as 3 Oe. The dc magnetic susceptibility χdc, as a function of temperature down to 0.4 K, was measured +using a Quantum Design Magnetic Property Measurement System (MPMS) equipped with a 3He insert. The isothermal dc +magnetization curves in the field up to 7 T applied along the a and c axes were measured at 0.4 K with the same setup. +Section 3. +MAGNETOCALORIC MEASUREMENTS +Magnetocaloric effect (MCE) of the frustrated dipolar magnet KBGB was characterized using a homemade setup integrated +into the PPMS, for initial temperature 2 K ≤ Ti ≤ 4 K. A DR-based setup is also exploited for MCE measurements with low +initial temperature Ti ≤ 500 mK. +A. +PPMS-based setup for quasi-adiabatic demagnetization measurements +As shown in Fig. S2, a homemade PPMS-based construction for quasi-adiabatic demagnetization process is set up, inspired +by the Hagmann-Richards design for space applications [69]. An additional guard stage consisting of copper cylinders and +Gd3Ga5O12 (GGG) crystals (20 g), a conventional coolant, offer thermal intercepts between the sample stage and the PPMS +chamber. In experiments, plate-like KBGB single crystals (with a total mass of 0.5 g) are stacked along the c-axis and fixed +on a silver foil by cryogenic glue. To improve the thermal insulation, a Vespel straw is used to support the sample pillar inside +the copper cylinder. The guard stage is suppported by PEEK tubes to reduce the thermal exchange with the environment. The +electrical connection of the thermometer (a field-calibrated RuO2 chip) on top of the pillar is made by two pairs of twisted + +bnck +COWWGLCISI +bbW2 +bEEK +Ccc (To a) +Ws12 +N6abel +blgid2 +CCC (To a) +KBCB 2luajG +p 2.0-10 +manganese wires (25 µm in diameter and approximately 60 cm in length) to reduce the heat leak. A thermal shield protects +the sample from radiant heating and reduce other parasitic heat loads from the PPMS chamber. Demagnetization cooling +measurements are performed by gradually decreasing the fields from the initial field Bi at a rate of ˙B = 0.15 T·min−1. +The parasitic heat load can be estimated from the temperature change rate of sample after the magnet field being exhausted, +i.e., in the hold process with B = 0. To be specific, the parasitic heat load is estimated by ˙Q = C0 ˙T, where C0 is heat capacity +of the sample and ˙T is the temperature change rate. For example, when starting from an initial condition of 2 K, it is found +that ˙T ≈ 5 × 10−6 K/s. Considering C0 ≃ 0.01 J/K for 0.5 g KBGB samples, we thus figure out the parasitic heat load as +˙Q ≈ 0.05 µW. +In Fig. S2 we show the isentropic lines of KBGB obtained through the quasi-adiabatic demagnetization measurements, and +make a comparison with the widely used refrigerant GGG. The results with different initial conditions lead to the same conclusion +that KBGB clearly outperforms GGG in the lowest cooling temperature. +B. +The DR-based quasi-adiabatic demagnetization measurements +To perform MCE measurements from a lower initial temperature below 500 mK, a standard DR heat capacity sample mount +is used, which provides a quasi-adiabatic condition with high vacuum in the 3He–4He dilution insert of PPMS. The thermometer +used is a RuO2 semiconductor. It has been carefully calibrated as functions of temperature (50 mK-4 K) and magnetic field +(0-5 T), and also extrapolated to 30 mK according to the scaling behavior ln(R − R0) ∼ T −1/4 [67]. +The polymer strips are used to support the sample platform. A KBGB single crystal with a much smaller mass of 2.3 mg is +used here, to avoid large magnetic torque that may break the suspended lines in the sample mount. To decrease the irreversible +heating effect on the DR mount, the field sweep rate ˙B has been reduced to 0.06 - 0.09 T·min−1. Due to the small mass of +the sample, the parasitic heat loads have a stronger influence in the MCE measurements. However, a prominent dip can still be +observed in the quasi-adiabatic cooling curve that clearly signals the existence of a QCP in Fig. S2(c). +(a) θ = 0 +(b) θ = Τ +π 3 +(c) θ = 2 Τ +π 3 +(d) θ = π +(e) θ = 4 Τ +π 3 +(f) θ = 5 Τ +π 3 +Q1 = ±b*/2 +Q3 = ±b*/2 +Q2 = ±(b*-a*)/2 +b* +a* +Q1 +Q3 +Q2 +(g) First Brillouin Zone +FIG. S3. (a)-(f) show the magnetic configurations of the stripe order with 6-fold degeneracy, i.e., 6-clock AF, which can be labeled with angle +θ (in complex order parameter Ψxy), and also by ordering vector Q1 (blue dots), Q2 (yellow dots), and Q3 (red dots) shown in (g). +Section 4. +MONTE CARLO SIMULATIONS +As the spin quantum number S = 7/2 is large in KBGB, here we use the classical Monte Carlo simulations with standard +Metropolis algorithm and single spin update and [70, 71]. The largest system size is 60 × 60, and we calculate the snapshots of + +11 +the ground-state spin configurations in Figs. S3(a-f). The corresponding ordering wave vectors Q = ± 1 +2a∗, ± 1 +2b∗, ± 1 +2(a∗ −b∗), +with a∗, b∗ are the three ordering wave vectors for the 6-clock AF order shown in Fig. S3(g). The phase angle θ of the complex +order parameter Ψxy can only take 6 discretized values that correspond to the 6-fold degenerate ground states (corresponding to +Q1, Q2, Q3, see below). +In Figs. 1(c-e) of the main text, we show histograms of the complex order parameter Ψxy ≡ meiθ under magnetic field +B = 0.68 T and at different temperature, i.e., (c) T = 0.05 K (6-clock AF), (d) T = 0.14 K (DSL), and (e) T = 0.25 K (PM), +respectively. To count the histograms, we collect 5 × 106 MC samples on a L = 12 × 12 lattice for statistics. +The MC simulation results of specific heat are shown in Fig. S4, where the contour plot in Fig. S4(a) resembles the experi- +mental data in Fig. 3(b) of the main text. The round peak in Cm is located at T ∗ ≃ 270 mK, and the peak heights are converged +with system sizes, as indicated in the inset of Fig. S4(b). As magnetic fields are applied along the out-of-plane direction, similar +to the experiments, we also observe that the Cm peaks move towards low temperature side, with heights lowered, in Fig. S4(c). +0.1 +0.2 +0.3 +0.4 +0.5 +1 +2 + B= 0 T + B= 0.39 T + B= 0.68 T +Cm +T (K) +L = 60 +0.2 +0.4 +1 +2 + L=36 + L=24 + L=60 + L=54 + L=48 +Cm +T (K) +0.0 +0.5 +1.0 +0.0 +0.2 +0.4 +0.6 +B (T) +T(K) +0.4 +0.8 +1.2 +1.6 +2 +(a) +(b) +(c) +Cm +0.25 +0.29 +1.5 +2.5 +Cm +T (K) +FIG. S4. The calculated results of specific heat Cm. (a) shows the contour plot of Cm data under out-of-plane field B. The (b) zero-field Cm +curves for different system sizes and (c) Cm curves for different fields are also presented. The inset in (b) compares the Cm data near crossover +temperature T ∗ ≃ 270 mK. The MC simulations are performed on the HD model [Eq. (1) in the main text] with couplings J = 47 mK and +D = 80 mK. (c) compares the specific heat curves under zero and finite magnetic fields. +In the simulations, we use the natural unit (J = 1) in the MC calculations and thus the following process is required for +comparing the model calculations to experimental data in SI units: (1) We replace the Si operators in Eq. (1) of the main text +by classical vectors, Si → Sni ≡ 7/2 ni, where ni is a unit vector; (2) The value of temperature T in natural unit should +be multiplied by a factor of J = 0.047 K; (3) Multiply the magnetic field B in natural unit (i.e., B/JS = 1) by a factor of +JkB/(gcµB) ≃ 0.028 T. + diff --git a/FNE1T4oBgHgl3EQf-gbR/content/tmp_files/load_file.txt b/FNE1T4oBgHgl3EQf-gbR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4e86f213dae27b0d7bc714926ccff44004919b0 --- /dev/null +++ b/FNE1T4oBgHgl3EQf-gbR/content/tmp_files/load_file.txt @@ -0,0 +1,1375 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf,len=1374 +page_content='Dipolar Spin Liquid Ending with Quantum Critical Point in a Gd-based Triangular Magnet Junsen Xiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' ∗ Cheng Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' ∗ Ning Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' ∗ Zhendong Fu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 Zhuo Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 Hai Jin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='6 Ziyu Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 Zhao-Jun Mo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='7 Yang Qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 9 Jun Shen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 10 Long Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 12 Wentao Jin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' † Wei Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' ‡ Peijie Sun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' § and Gang Su11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' ¶ 1Beijing National Laboratory for Condensed Matter Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 2School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 3CAS Key Laboratory of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Institute of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 4Neutron Platform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Songshan Lake Materials Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Dongguan 523808,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 5School of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijing Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijing 100081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 6Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijing 100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 7Ganjiang Innovation Academy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Ganzhou 341119,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' People’s Republic of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 8State Key Laboratory of Surface Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Shanghai 200433,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 9Center for Field Theory and Particle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Shanghai 200433,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 10Technical Institute of Physics and Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 11Kavli Institute for Theoretical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' and School of Physical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijng 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 12CAS Center for Excellence in Topological Quantum Computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijng 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China 13Peng Huanwu Collaborative Center for Research and Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' China (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2023) By performing experimental and model studies of a triangular-lattice dipolar magnet KBaGd(BO3)2 (KBGB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' we find the highly frustrated magnet with a planar anisotropy hosts a strongly fluctuating dipolar spin liquid (DSL) originating from the intriguing interplay between dipolar and Heisenberg interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The DSL con- stitutes an extended regime in the field-temperature phase diagram, which gets narrowed in temperature range as field increases and eventually ends with a quantum critical point (QCP) at Bc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='75 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Based on dipolar Heisenberg model calculations, we identify the DSL as a Berezinskii-Kosterlitz-Thouless (BKT) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Due to the tremendous entropy accumulation that can be related to the strong BKT and quantum fluctuations, unprece- dented magnetic cooling effects are observed in the DSL regime and particularly near the QCP, making KBGB a superior dipolar coolant over commercial Gd-based refrigerants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' We establish a universal phase diagram for triangular-lattice dipolar quantum magnets where emergent symmetry plays an essential role, and lay down foundations for their applications in sub-Kelvin refrigeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='— Triangular-lattice quantum antiferromag- nets have raised great research interest recently, due to the unusual quantum spin states and transitions therein [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' One prominent example is the quantum spin liquid (QSL) [3– 5] and its possible materialization in organic compounds [6– 8] and rare-earth triangular magnets [9–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Due to the in- triguing spin frustration effects and two dimensionality (2D), Berezinskii-Kosterlitz-Thouless (BKT) physics may appear in the triangular quantum antiferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The Co-based quan- tum antiferromagnet Na2BaCo(PO4)2 hosts persistent spin fluctuations [17–20] till very low temperature, and is proposed to posses spin supersolid state with BKT fluctuations of U(1) phase [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Emergent symmetry, as a consequence of frustra- tion, has also been disclosed on the triangular lattice, with a recent example of rare-earth magnet TmMgGaO4 [22–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Recently, it has been proposed that the dipolar interactions can give rise to QSL in triangular-lattice quantum spin sys- tems [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Lately such dipolar system has been realized in Yb- based triangular compounds [30–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' However, the dipolar ∗ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' † wtjin@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='cn ‡ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='li@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='cn § pjsun@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='cn ¶ gsu@ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='cn interactions are rather weak and it is very challenging for con- ventional thermodynamic and spectroscopic measurements to probe the exotic spin states due to dipolar interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' On the contrary, the rare-earth dipolar magnets with larger mo- ments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', Gd-based compounds with µeff ≈ 8µB and high spin S = 7/2, are much less explored both in experiments and theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' It is expected that the dipolar frustration ef- fects are a priori more evident in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Moreover, in sub-Kelvin refrigeration for space applications [35, 36] and quantum computations [37], high-spin frustrated magnets, es- pecially those with spin-liquid like behaviors [38], can have great entropy densities and cooling capacity, holding thus strong promise as superior coolants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In this work, we perform low-temperature thermodynam- ics and magnetocalorics measurements on single-crystal sam- ples of Gd-based triangular-lattice compound KBaGd(BO3)2 (KBGB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The thermodynamic measurements suggest a dipo- lar spin liquid state with no conventional ordering but strong spin fluctuations, which are reflected in the algebraic specific heat and imaginary dynamical susceptibility (χ ′′ ac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' We estab- lish a dipolar Heisenberg model with both dipole-dipole and Heisenberg interactions for KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Monte Carlo (MC) sim- ulations explain excellently the experimental measurements and unveil the exotic spin states and transitions in the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In particular, the model simulations suggest a two- step melting of the clock antiferromagnetic (AF) order via two arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='03571v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='str-el] 9 Jan 2023 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 Yy Yx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 Yy Yx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 Yy Yx 6-clock AF DSL PM Temperature (b) (c) (d) (e) (a) b a c K/Ba Gd B O b a a∗ Si Sj eij FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (a) shows the crystal structure of KBaGd(BO3)2, where (b) triangular-lattice layers of GdO6 octahedra are separated by the Ba/K layers with site mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The grey arrows refer to the spins on site i and j, and the unit vector eij is also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Dipole-dipole inter- actions are bond-dependent and follow the ¯3m site symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (c)-(e) are histograms of the order parameter Ψxy ≡ Ψx + iΨy for the 6- clock antiferromagnetic (AF) [28], dipolar spin liquid (DSL) with an emergent U(1) symmetry, and the paramagnetic (PM) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' BKT transitions, between which a floating BKT phase emerge with an emergent U(1) symmetry, well accounting for the ex- perimentally observed spin liquid behaviors with enormous low-temperature entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Consequently, giant magnetocaloric effect (MCE) is observed in the quasi-adiabatic demagnetiza- tion measurements, where we find a clear dip in temperature which suggests the presence of quantum critical point (QCP) near Bc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='75 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The lowest temperature of 70 mK clearly surpasses that of commercial refrigerant Gd3Ga5O12 (GGG) under similar conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Overall, the triangular-lattice rare- earth dipolar magnets open an avenue for exploring exotic spin states as well as finding superior sub-Kelvin coolants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Crystal structure and effective model for KBaGd(BO3)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='— Centimeter-sized single crystals of KBGB were synthesized using the flux method as described in detail in Supplementary Materials (SM) [28], and the X-ray diffraction measurements suggest high quality of the single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' KBGB is found to crystallize in a trigonal structure [40, 41] with space group R-3m [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 1(a)], and has a relatively high ionic density of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 nm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 1(b), magnetic Gd3+ ions with 4f 7 electron configuration (L = 0, S = 7/2) form perfect triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The dipolar interaction between magnetic ions Gd3+ has a characteristic energy Edp ∼ 2µ0µ2 eff/4πa3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='05 meV (with µeff ≈ 8µB), which determines the low- temperature spin states in KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' To simulate such Gd- based dipolar magnet, we consider the following Hamil- tonian, H = JH � ⟨i,j⟩NN Si · Sj + JD � i,j[Si · Sj − 3(Si · eij)(Sj · eij)]/r3 ij, where eij(rij) refers to the unit vec- tor(distance) between site i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' JH and JD refer to the nearest neighbor (NN) Heisenberg and dipole-dipole interac- tions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' As the dipolar interactions show rapid (cu- bic) power-law decay and the longer range interactions can be washed out, we keep only NN terms as HDH = � ⟨i,j⟩NN J Si · Sj − D (Si · eij)(Sj · eij), (1) where J = JH + JD/a3 is the NN isotropic coupling and D = 3JD/a3 refers to the dipolar anisotropic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' We find the NN dipolar Heisenberg (DH) model with couplings J = 47 mK and D = 80 mK very well describe the compound and accurately reproduce the experimental measurements on KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The MC simulations are performed on up to 60 × 60 triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Due to the high-spin state with S = 7/2, classical MC simulations capture well the finite-temperature properties of KBGB [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' We guarantee the error bars to be always smaller than the symbol size in the presented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Magnetic specific heat, susceptibility, and dipolar spin liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='— In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(a) we show the zero-field specific heat Cm measured down to 65 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' There exists a round peak at T ∗ ≃ 218 mK, below which the system exhibits Cm ∼ T 2 with algebraic scaling, resembling that of 2D Heisenberg or XY quantum spin model with U(1) symmetry [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The dipolar anisotropy in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (1), like in spin-orbit magnets, leads to a discretized C3 rotational symmetry, and it gener- ically corresponds to a divergent Cm peak when transition- ing to low-T symmetry breaking phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The presence of round peak and T 2 scaling in Cm is very remarkable, which suggests a liquid-like and strongly fluctuating spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(a), when compared to the renowned Gd-based refrig- erant GGG [36, 39, 44, 45], KBGB has tremendous low- temperature specific heat, far surpassing that of GGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(b), we apply out-of-plane fields (B//c) to the com- pound, and find also round peaks in Cm curves, which move towards lower temperature with heights slightly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' This suggests that the spin liquid states constitute an extended phase that we dub as dipolar spin liquid (DSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' As field fur- ther increases and exceeds about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='75 T, the DSL behaviors disappears [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', the contour plot of Cm/T in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 3(b)], and the Cm peak moves now to high-temperature side, with the low-T peak and low-energy fluctuations quickly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(c), we perform magnetization measurements on single-crystal sample of KBGB, and find a clear mag- netic anisotropy between the out-of-plane (//c axis) and in- plane (//a) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' This anisotropy can be clearly rec- ognized in the different saturation magnetization moments and transition field values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', 1 T(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 T) along c(a) axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(d), we perform low-temperature dc susceptibility (χdc) measurements, and find χdc also exhibits a clear easy- plane anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In addition, a small but sensible in-plane anisotropy between a and a∗ [see inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(c)] is also observed, consistent with the intrinsic anisotropy in bond- dependent dipolar interaction [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' To further explore the DSL, ac magnetic susceptibilities are measured in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(e,f), with χ′ ac and χ′′ ac for real and imag- inary parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The real χ′ ac exhibits a frequency- independent maximum and remains large even below the char- acteristic temperature scale T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Therefore, although there exist K/Ba site mixing in the compound, the spin-glass sce- nario can be excluded in KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Interestingly, the imaginary 3 1 10 100 0 2 4 6 8 10 12 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Model χdc (emu·Oe-1·mol-1 Gd) T (K) a a* c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 T 2 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 χdc 1 θa -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='30 K θa* -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='33 K θc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='32 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 0 25 50 75 T (K) Cm/T (J·mol-1 Gd·K-2) KBGB GGG 0 T Cm/T ~ T T* 218 mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content="0 4943 Hz 6253 Hz 9984 Hz χac'' (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=') T (K) T* 0 1 2 3 4 0 2 4 6 8 10 B // a B // c Moment (µB/Gd) B (T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='49 Model ga gc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content="4 91 Hz 955 Hz 2439 Hz 3087 Hz 3910 Hz T (K) χac' (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=') T* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 0 25 50 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='25 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='75 T Cm/T (J·mol-1 Gd·K-2) T (K) 1 T 2 T 3 T 4 T (a) (d) (e) (f) (c) (b) a* a FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Specific heat of KBGB under (a) zero and (b) finite fields along out-of-plane direction (B//c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' An algebraic Cm ∼ T 2 scaling is observed below the round peak temperature T ∗, and the Cm/T values far outweigh that of GGG [39] for T ≲ T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In (b) we find the round peak in Cm/T firstly moves towards lower temperature and later for B > Bc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='75 T the low-temperature Cm quickly gets suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (c) shows the magnetization curves of the single-crystal KBGB sample for B//a and //c, and the results show excellent agreement with the DH model calculations (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The saturation moments are µsat a ≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='26µB and µsat c ≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='72µB, from which we determine the Landé factors ga ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='36 and gc ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='49, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The as-grown KBGB single crystal is shown in the inset, with directions a and a∗ also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (d) shows the molar dc magnetic susceptibilities (χdc) measured along the a, a∗, and c axes, respectively, where the solid lines representing the DH model calculations show excellent agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The inset shows the Curie-Weiss fittings in the paramagnetic regime 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 K ≤ T ≤ 10 K, with the fitted Curie-Weiss temperatures θa,a∗,c also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (e, f) present respectively the real and imaginary ac susceptibilities measured with different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' ac susceptibility χ′′ ac(T), although being featureless for low frequencies ω ≲ 4 kHz, show a clear temperature-dependent behavior for higher frequencies in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Considering that χ′′(ω) can be directly related to the dynamical correlation S(ω) through the fluctuation-dissipation theorem, χ′′(ω) ∝ ω T S(ω) (ω ≪ T), this clearly suggests the persistence of low- energy spin fluctuations even below T ∗ and supports the spin- liquid scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Magnetocaloric effect and quantum critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='— In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 3(a), we perform quasi-adiabatic demagnetization mea- surements and obtain the isentropic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' It is found that KBGB clearly outperforms GGG in the minimal temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', Tm ≃ 70 mK (KBGB) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 322 mK (GGG), when starting from the same initial condition of Ti = 2 K and Bi = 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 3(b) we provide more of the isentropic lines from dif- ferent initial conditions, and observe the highly asymmetric isentropes, which “levels off” in the bright DSL regime as in- dicated by large values of Cm/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' For KBGB, the lowest temperature Tm is achieved at the dip in isentropic lines and remains below 100 mK in the small field side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' This happens also for measurements starting from rather low temperature Ti ≃ 95 mK, where the lowest Tm ≃ 33 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Such unprecedented MCE response strongly corroborates the existence of QCP at Bc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='75 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The mag- netic Grüneisen ratio ΓB = 1 T ( ∂T ∂B )S has been widely used in the studies of heavy fermion [46–50] and low-dimensional quantum spin systems [51–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 3 an ev- ident peak-dip structure with sign change is observed [55– 58], and the peak height exceeds 4 times that of GGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Such a prominent critical cooling effect provides valuable MCE evi- dence for QCP in the compound KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Emergent symmetry in KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='— According to the magne- tothermal and MCE measurements above, we arrive at the phase diagram of KBGB in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The two schematic dashed lines, enclosing the DSL with large Cm/T, meet at a QCP (Bc) where the demagnetization process reaches its low- est temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Besides QCP, within the DSL regime we find persistent spin fluctuations and cooling effects whose origin is clarified by model calculations below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' We conduct MC calculations of the DH model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (1)] for KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' As the model is highly frustrated in the out-of- plane direction, the order parameter lies within the ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Note although the determined Landé factor gc ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='49 is slightly larger than ga ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='36, the intrinsic planar anisotropy of dipolar interaction leads to larger in-plane χdc (along a and a∗ axes) than that along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The negative Curie- Weiss temperatures fitted from the dc susceptibility reflect the AF nature, and the slightly different θa ≃ −300 mK and θa∗ ≃ −330 mK shows the in-plane anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(d), we find the anisotropic susceptibility and magnetization mea- 4 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 1 T (K) B (T) 70 mK KBGB GGG B // c 322 mK 2 K 33 mK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='3 B (T) T (K) 0 20 40 60 Cm/T DSL QCP Quasi-Adiabatic PM 6-clock AF (a) (b) ΓΒ (T-1) 0 1 2 3 4 1 0 1 2 3 4×GGG Bc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (a) shows the quasi-adiabatic isentropes measured in KBGB under out-of-plane field (see details in SM [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The KBGB curve exhibits a clear dip at the lowest temperature Tm ≃ 70 mK, much lower than that of GGG (Tm ≃ 322 mK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Starting from Ti ≃ 95 mK, KBGB is observed to cool down to remarkably low temperature Tm ≃ 33 mK in the dip (blue dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The inset shows the mag- netic Grüneisen ratio ΓB deduced from the curves in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (b) shows the phase diagram of KBGB with the contour plot of Cm/T in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The bright regime with large spin fluctuations represent the DSL, with schematic dashed line boundaries, ending up with a QCP at Bc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='75 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' sured along a and c axes can be well captured by the DH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Besides, the model calculations of specific heat also obtain a round peak at about 270 mK, which again gets sup- pressed as field increases (see SM [28]), very much resem- bling the experimental data in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The comparisons confirm that the compound KBGB can indeed be accurately described by the DH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' To characterize the spin states in the phase diagram, we in- troduce the order parameter Ψxy ≡ meiθ = � j eiQrj(mx j + imy j ), where j runs over the lattice sites and Q = ± 1 2a∗, ± 1 2b∗, ± 1 2(a∗ − b∗) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Histogram of the complex order parameter Ψxy at various temperature are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 1(c-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' At low temperature, the dipolar system exhibits a 6-clock AF order corresponding to θ = 0, ±π/3, ±2π/3, and π [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' As temperature ramps up, the six points in the histogram prolong and merge into a circle with emergent U(1) symmetry, where the angle θ can choose arbitrary angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' As temperature further enhances, eventually the amplitude m vanishes and the system enters the conventional PM phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Recall that the 6-state clock model with an anisotropic term ∼ cos (6θ) undergoes two successive BKT transitions [59], between which the anisotropic term becomes irrelevant per- turbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Based on this symmetry argument, we consider the intermediate DSL in the system as BKT phase with emer- gent U(1) symmetry and effectively described by 2D XY model [60–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The emergent symmetry extends also to the zero-temperature QCP as the clock term is dangerously irrele- vant [60], and the transition directly between the 6-fold clock symmetry broken and PM phases belong to the 3D XY univer- sality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Therefore, the emergent symmetry constitutes a key for demystifying spin-liquid state and quantum criticality in the compound KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 0 60 120 180 0 1 2 3 4 T (K) Time (min) 4 T 6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 1 0 5 10 15 20 Sm (×10·J·Kg-1·K-1) T (K) ∆Q 4 T ∆Sm 0 T (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 1 10 0 10 20 30 40 � � m (J·Kg-1·K-1) T (K) KBGB GGG 4 T 2 T 1 T (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (a) The quasi-adiabatic demagnetization cooling curves of KBGB, starting from two different initial conditions (Ti = 4 K, Bi = 4 T) and (Ti = 2 K, Bi = 6 T), with reached lowest tem- perature Tm ≃ 205 mK and 70 mK, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Parasitic heat loads are estimated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 µW for Ti = 4 K environment and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='05 µW for Ti = 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The inset shows magnetic entropy under zero and 4 T fields, with the shaded area representing the absorbed heat ∆Q = 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='44 J·Kg−1 in the hold process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (b) plots the entropy change ∆Sm vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' T, for fields decreasing from 1 T, 2 T, and 4 T to zero, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Comparisons to GGG are also presented [39, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Superior cooling performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='— Starting from 2 K en- vironment, KBGB can reach as low as 70 mK as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 4(a), such a low cooling temperature far surpasses other Gd-based refrigerants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', GGG (322 mK) and GdLiF4 (480 mK) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Besides, KBGB also exhibits long hold time and large isothermal entropy change ∆Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 4(a) we show that KBGB remains in low temperature for a long period after the field is exhausted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In the environment temperature of 2 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 g KBGB remains below 140 mK for th ≈ 2 h under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='05 µW heat leak, which can be ascribed to the large heat absorption ∆Q depicted in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The isothermal entropy change ∆Sm characterizes the cooling capacity of refrigerants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 4(b), we compare ∆Sm of KBGB with that of GGG, and find that in the whole temperature range concerned KBGB has significantly larger ∆Sm for 1 T field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Moreover, the maximal ∆Sm of KBGB lo- cates below 1 K [shaded regime in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 4(b)], and the entropy change in KBGB exceeds that of GGG in this sub-Kelvin regime of central interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Overall, the low cooling temper- ature Tm, long hold time th, and enormous entropy change ∆Sm in the sub-Kelvin regime lead to the conclusion that KBGB serves a superior quantum magnet coolant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Discussions and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='— The pursue for high entropy density and low ordering temperature constitutes two oppos- ing factors hard to fulfill simultaneously in optimizing sub- Kelvin refrigerants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Here the spin frustration and quantum criticality in the dipolar system come to the rescue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' We show that the compound KBaGd(BO3)2 with high Gd3+ ion density yet form a disordered and strongly fluctuating spin liquid till extremely low temperature, giving rise to the superior cooling capacity due to the entropy accumulation near QCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' We use the DH model within NN interactions to describe KBGB and find it well reproduces the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Inclusion of further neighboring dipolar couplings will not change the con- 5 clusion here, as it has been shown to maintain the universality class of BKT transitions in planar dipolar models [63, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The scenario of DSL ending up with emergent U(1) QCP may also be applicable to other dipolar quantum magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Re- cent progress in experimental studies reveal several families of rare-earth triangular quantum dipolar antiferromagnets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', Ba3REB3O9/Ba3REB9O18 (with RE a rare-earth ion) [32, 33] and ABaRE(BO3)2 (with A an alkali ion) [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' It has been observed that in Ba3YbB3O9 that 80% entropy remain below 56 mK [31], despite a dipolar energy scale of about 160 mK, suggesting that the DSL may also play a role in the Yb-based dipolar compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Therefore, this work opens a venue for hunting exotic spin states as well as superior quantum coolants in triangular dipolar magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Note added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='— Upon finishing the present work, we are aware of a recent work [68] also conducting MCE study of KBGB with however polycrystalline samples, where they find strong cooling effect down to 121 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='— W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' is indebted to Yuan Wan and Tao Shi for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' acknowledge the support from the beamline 1W1A of the Beijing Syn- chrotron Radiation Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' This work was supported by the National Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 12222412, 11834014, 11974036, 12047503, 12074023, 12074024, 12174387, and 12141002), National Key R & D Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2018YFA0305800), Strategic Priority Research Program of CAS (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' XDB28000000), and CAS Project for Young Scien- tists in Basic Research (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' YSBR-057).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' We thank the HPC-ITP for the technical support and generous allocation of CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' This work was supported by the Synergetic Extreme Condition User Facility (SECUF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} 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Neutron scatter- ing investigation of proposed Kosterlitz-Thouless transitions in the triangular-lattice ising antiferromagnet TmMgGaO4, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' B 103, 064424 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' [28] Supplementary Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 1 describes the KBGB sample prepara- tion and their XRD charactherization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The magnetothermal and magnetocaloric measurements are 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 2, 42 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' [68] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Jesche, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Winterhalter-Stocker, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Hirschberger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Bel- lon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Bachus, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Tokiwa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Tsirlin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Gegen- wart, Adiabatic demagnetization cooling well below the mag- netic ordering temperature in the triangular antiferromagnet KBaGd(BO3)2, arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='12483 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' [69] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Hagmann and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Richards, Two-stage magnetic refrigerator for astronomical applications with reservoir temperatures above 4 K, Cryogenics 34, 221 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' [70] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Sandvik, Computational studies of quantum spin sys- tems, AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 1297, 135 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' [71] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Creutz, Overrelaxation and monte carlo simulation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' D 36, 515 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 8 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' units) Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' units) (003) (006) (009) (0012) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (a) shows the powder XRD pattern of KBGB measured at room temperature and corresponding Rietveld refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The open circle and red solid line represent the observed and calculated intensities, respectively, while the blue solid line shows their difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The olive vertical bars mark the expected reflections for KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (b) Single-crystal XRD scan along the (0,0,L) direction for one representative crystal, revealing only peaks that are well indexed by (0,0,3n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The insets show the image of the as-grown KBGB crystals and the rocking-curve scan of the (0,0,12) reflection fitted by a Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The very narrow peak width of FWHM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='041◦ indicates excellent quality of the crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Supplementary Materials Dipolar Spin Liquid Ending with Quantum Critical Point in a Gd-based Triangular Magnet Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' SAMPLE PREPARATION AND STRUCTURE CHARACTERIZATION Polycrystalline samples of KBGB were firstly prepared by standard solid-state reaction method as reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Sto- ichiometric mixtures of K2CO3 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='99%), BaCO3 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='95%), H3BO3 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='99%) and Gd2O3 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='99%) (with 6% excess H3BO3 and 5% excess of K2CO3 and BaCO3) were thoroughly ground and pelletized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Then the pellet was placed into an aluminum crucible and sintered at 900◦C in air for 10 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' This sintering process was repeated for several times to minimize possible impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Single-crystal samples of KBGB were grown using the flux method as reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The pre-obtained polycrystalline KBGB with high purity was mixed with the H3BO3 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='99%) and KF (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='9%) fluxes in a molar ratio of 2:3:[2-3], and thoroughly ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The mixture was transferred into a Pt crucible, heated up to 980◦C in air for 24 h, and then slowly cooled to 790◦C with a rate of 2◦C/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' After the furnace cooling, centimeter-sized crystals were obtained on top of the fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The phase purity of the polycrystalline KBGB sample was confirmed by powder XRD at room temperature, performed on a Bruker D8 ADVANCE diffractometer in Bragg-Brentano geometry with Cu-Kα radiation (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5406 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S1(a), the powder XRD pattern can be well fitted with the previously reported trigonal phase of KBGB [40] (a = b = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4676(1) Å, c = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='9514(3) Å), without any visible impurity peaks, indicating high purity of the synthesized KBGB powders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The quality of the single-crystal KBGB sample was checked by high-resolution synchrotron XRD (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='54564 Å) measure- ments at room temperature, performed on the 1W1A beamline at the Beijing Synchrotron Radiation Facility (BSRF), China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S1(b), a long L scan, equivalent to a θ-2θ scan with respect to the normal direction of the plate-like KBGB crystal, only shows Bragg reflections well indexed by (0, 0, 3n) as expected for the R-3m space group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The peak width (full width at half maximum, FWHM) observed in the rocking-curve scan of the (0, 0, 12) peak is very small, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='041(2)◦, as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S1(b), which suggests excellent crystal quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' KBGB is relatively easy to synthesize and has excellent chemical stability, paving its viable way for applications in advanced cryogenics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 430 mK 280 mK 195 mK 150 mK 95 mK T (K) B (T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='06 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='09 T·min-1 Bi = 3 T 0 1 2 3 4 5 6 0 1 2 3 4 KBGB GGG T (K) B (T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='15 T·min-1 (a) (c) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0 1 2 3 1 0 1 2 ΓB (T-1) Bc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (a) Illustration of the two-stage quasi-adiabatic demagnetization cooling device for the measurements of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 g KBGB single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (b) shows the measured isentropic curves of KBGB starting from various initial conditions (Ti = 2 K, Bi = 4 T), (2 K, 6 T), and (4 K, 4 T), respectively, where the lowest temperature are found to be significantly lower than those of GGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The inset zooms in the small-field range (B ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (c) The DR-based measurements with an initial temperature Ti ≤ 430 mK and field Bi = 3 T, where the lowest achieved temperature is Tm ≃ 33 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The inset shows the magnetic Grüneisen ratio ΓB deduced from the low-temperature isentropic T-B lines in the main plot, where the sign change is evident and the peak becomes more and more pronounced as the initial temperature Ti lowers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' MAGNETOTHERMAL MEASUREMENTS Comprehensive magnetothermal measurements were performed on single-crystal samples of KBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The low-temperature specific heat (Cp, T ≥ 50 mK) and ac susceptibility (χac, T ≥ 50 mK) measurements were conducted using the Quantum Design Physical Property Measurement System (PPMS) equipped with a 3He–4He dilution refrigerator (DR) insert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The specific heat data were measured under various out-of-plane fields (B//c) with the semi-adiabatic heat pulse method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The phonon contributions are negligible below 2 K as estimated via a Debye T 3 analysis of high-temperature Cp data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The ac susceptibility (χac), as a function of temperature, was measured in zero dc field under different ac frequencies, with the amplitude of the ac excitation field set as 3 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The dc magnetic susceptibility χdc, as a function of temperature down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 K, was measured using a Quantum Design Magnetic Property Measurement System (MPMS) equipped with a 3He insert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The isothermal dc magnetization curves in the field up to 7 T applied along the a and c axes were measured at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 K with the same setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' MAGNETOCALORIC MEASUREMENTS Magnetocaloric effect (MCE) of the frustrated dipolar magnet KBGB was characterized using a homemade setup integrated into the PPMS, for initial temperature 2 K ≤ Ti ≤ 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' A DR-based setup is also exploited for MCE measurements with low initial temperature Ti ≤ 500 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' PPMS-based setup for quasi-adiabatic demagnetization measurements As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S2, a homemade PPMS-based construction for quasi-adiabatic demagnetization process is set up, inspired by the Hagmann-Richards design for space applications [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' An additional guard stage consisting of copper cylinders and Gd3Ga5O12 (GGG) crystals (20 g), a conventional coolant, offer thermal intercepts between the sample stage and the PPMS chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In experiments, plate-like KBGB single crystals (with a total mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 g) are stacked along the c-axis and fixed on a silver foil by cryogenic glue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' To improve the thermal insulation, a Vespel straw is used to support the sample pillar inside the copper cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The guard stage is suppported by PEEK tubes to reduce the thermal exchange with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The electrical connection of the thermometer (a field-calibrated RuO2 chip) on top of the pillar is made by two pairs of twisted bnck COWWGLCISI bbW2 bEEK Ccc (To a) Ws12 N6abel blgid2 CCC (To a) KBCB 2luajG p 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0-10 manganese wires (25 µm in diameter and approximately 60 cm in length) to reduce the heat leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' A thermal shield protects the sample from radiant heating and reduce other parasitic heat loads from the PPMS chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Demagnetization cooling measurements are performed by gradually decreasing the fields from the initial field Bi at a rate of ˙B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='15 T·min−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The parasitic heat load can be estimated from the temperature change rate of sample after the magnet field being exhausted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', in the hold process with B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' To be specific, the parasitic heat load is estimated by ˙Q = C0 ˙T, where C0 is heat capacity of the sample and ˙T is the temperature change rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' For example, when starting from an initial condition of 2 K, it is found that ˙T ≈ 5 × 10−6 K/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Considering C0 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='01 J/K for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 g KBGB samples, we thus figure out the parasitic heat load as ˙Q ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='05 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S2 we show the isentropic lines of KBGB obtained through the quasi-adiabatic demagnetization measurements, and make a comparison with the widely used refrigerant GGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The results with different initial conditions lead to the same conclusion that KBGB clearly outperforms GGG in the lowest cooling temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The DR-based quasi-adiabatic demagnetization measurements To perform MCE measurements from a lower initial temperature below 500 mK, a standard DR heat capacity sample mount is used, which provides a quasi-adiabatic condition with high vacuum in the 3He–4He dilution insert of PPMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The thermometer used is a RuO2 semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' It has been carefully calibrated as functions of temperature (50 mK-4 K) and magnetic field (0-5 T), and also extrapolated to 30 mK according to the scaling behavior ln(R − R0) ∼ T −1/4 [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The polymer strips are used to support the sample platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' A KBGB single crystal with a much smaller mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='3 mg is used here, to avoid large magnetic torque that may break the suspended lines in the sample mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' To decrease the irreversible heating effect on the DR mount, the field sweep rate ˙B has been reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='06 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='09 T·min−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Due to the small mass of the sample, the parasitic heat loads have a stronger influence in the MCE measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' However, a prominent dip can still be observed in the quasi-adiabatic cooling curve that clearly signals the existence of a QCP in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (a) θ = 0 (b) θ = Τ π 3 (c) θ = 2 Τ π 3 (d) θ = π (e) θ = 4 Τ π 3 (f) θ = 5 Τ π 3 Q1 = ±b*/2 Q3 = ±b*/2 Q2 = ±(b*-a*)/2 b* a* Q1 Q3 Q2 (g) First Brillouin Zone FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (a)-(f) show the magnetic configurations of the stripe order with 6-fold degeneracy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', 6-clock AF, which can be labeled with angle θ (in complex order parameter Ψxy), and also by ordering vector Q1 (blue dots), Q2 (yellow dots), and Q3 (red dots) shown in (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' MONTE CARLO SIMULATIONS As the spin quantum number S = 7/2 is large in KBGB, here we use the classical Monte Carlo simulations with standard Metropolis algorithm and single spin update and [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The largest system size is 60 × 60, and we calculate the snapshots of 11 the ground-state spin configurations in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S3(a-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The corresponding ordering wave vectors Q = ± 1 2a∗, ± 1 2b∗, ± 1 2(a∗ −b∗), with a∗, b∗ are the three ordering wave vectors for the 6-clock AF order shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S3(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The phase angle θ of the complex order parameter Ψxy can only take 6 discretized values that correspond to the 6-fold degenerate ground states (corresponding to Q1, Q2, Q3, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 1(c-e) of the main text, we show histograms of the complex order parameter Ψxy ≡ meiθ under magnetic field B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='68 T and at different temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', (c) T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='05 K (6-clock AF), (d) T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='14 K (DSL), and (e) T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='25 K (PM), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' To count the histograms, we collect 5 × 106 MC samples on a L = 12 × 12 lattice for statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The MC simulation results of specific heat are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S4, where the contour plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S4(a) resembles the experi- mental data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 3(b) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The round peak in Cm is located at T ∗ ≃ 270 mK, and the peak heights are converged with system sizes, as indicated in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' As magnetic fields are applied along the out-of-plane direction, similar to the experiments, we also observe that the Cm peaks move towards low temperature side, with heights lowered, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 1 2 B= 0 T B= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='39 T B= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='68 T Cm T (K) L = 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 1 2 L=36 L=24 L=60 L=54 L=48 Cm T (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='6 B (T) T(K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='6 2 (a) (b) (c) Cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='5 Cm T (K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The calculated results of specific heat Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (a) shows the contour plot of Cm data under out-of-plane field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The (b) zero-field Cm curves for different system sizes and (c) Cm curves for different fields are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The inset in (b) compares the Cm data near crossover temperature T ∗ ≃ 270 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' The MC simulations are performed on the HD model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (1) in the main text] with couplings J = 47 mK and D = 80 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (c) compares the specific heat curves under zero and finite magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' In the simulations, we use the natural unit (J = 1) in the MC calculations and thus the following process is required for comparing the model calculations to experimental data in SI units: (1) We replace the Si operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (1) of the main text by classical vectors, Si → Sni ≡ 7/2 ni, where ni is a unit vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (2) The value of temperature T in natural unit should be multiplied by a factor of J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='047 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=' (3) Multiply the magnetic field B in natural unit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content=', B/JS = 1) by a factor of JkB/(gcµB) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} +page_content='028 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQf-gbR/content/2301.03571v1.pdf'} diff --git a/FdAzT4oBgHgl3EQfUfww/content/tmp_files/2301.01266v1.pdf.txt b/FdAzT4oBgHgl3EQfUfww/content/tmp_files/2301.01266v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f09ba435aeb1f5945f6a2902d52539860dea01a --- /dev/null +++ b/FdAzT4oBgHgl3EQfUfww/content/tmp_files/2301.01266v1.pdf.txt @@ -0,0 +1,5838 @@ +HIGGS-COULOMB CORRESPONDENCE AND WALL-CROSSING IN ABELIAN GLSMS +KONSTANTIN ALESHKIN AND CHIU-CHU MELISSA LIU +Dedicated to the memory of Professor Bumsig Kim +Abstract. We compute I-functions and central charges for abelian GLSMs using virtual matrix factorizations of +Favero and Kim. In the Calabi-Yau case we provide analytic continuation for the central charges by explicit integral +formulas. The integrals in question are called hemisphere partition functions and we call the integral representation +Higgs-Coulomb correspondence. We then use it to prove GIT stability wall-crossing for central charges. +Contents +1. +Introduction +1 +2. +Geometry of gauged linear sigma models and A-model state spaces +2 +3. +Categories of B-branes and K-theories +10 +4. +The Higgs Branch +10 +5. +The Coulomb Branch +32 +Appendix A. +Convergence of multivariate hypergeometric functions +41 +References +46 +1. Introduction +2d gauged linear sigma models (GLSMs) were introduced by Witten in 1993. Following [FJR], the input data +of a GLSM is a 5-tuple (V, G, C∗ +R, W, ζ), where V is a finite dimensional complex vector space, G ⊂ GL(V ) is +a reductive linear group known as the gauged group, C∗ +R ∼= C∗ acts linearly on V and the action commutes +with the G-action, W : V → C is a G-invariant polynomial which is quasi-homogeneous with respect to the C∗ +R- +action, and ζ is a G-character with the property V ss +G (ζ) = V s +G(ζ), i.e., every ζ-semistable point is ζ-stable, so +that the GIT quotient stack Xζ = [V//ζG] = [V ss +G (ζ)/G] is an orbifold (i.e. smooth DM stack with trivial generic +stabilizer). The space of stability conditions is ˆG ⊗Z R ∼= RdimC Z(G), where ˆG = Hom(G, C∗) is the group of G +characters and Z(G) is the center of G; it is decomposed into chambers called phases. The G-invariant polynomial +W descends to wζ : Xζ → C; the pair (Xζ, wζ) is a Landau-Ginzburg (LG) model, where wζ is known as the +superpotential. GLSM invariants of (V, G, C∗ +R, W, ζ) are, roughly speaking, virtual counts of curves in the critical +locus Zζ := Crit(wζ) = [(Crit(W) ∩ V ss +G (ζ)) /G] which is often assumed to be compact/proper but can be singular. +Different phases of a GLSM (that is GLSMs which differ only by the choice of a stability parameter) are closely +related to each other. For example, let G = C∗ act on V = C6 by weights (1, 1, 1, 1, 1, −5) with W = pW5 where +W5 = x5 +1 + · · · + x5 +5 is the Fermat quintic polynomial in 5 variables. In the CY/geometric phase ζ > 0, Xζ = KP4, +Zζ = X5 := {W5 = 0} ⊂ P4 is the Fermat quintic threefold, and GLSM invariants are (up to sign) Gromov-Witten +(GW) invariants of X5. +In the LG phase ζ < 0, Xζ ∼= [C5/µ5], where µ5 is the group of 5-th roots of unity +acts diagonally on C5, Zζ is supported at the origin, and GLSM invariants are Fan-Jarvis-Ruan-Witten (FJRW) +invariants of the affine LG model ([C5/µ5], W5). Chiodo-Ruan [CR] proved genus-zero LG/CY correspondence +for quintic threefolds relating GW invariants of X5 and FJRW invariants of ([C5/µ5], W5). Their proof can be +summarized into two steps. +(1) (ϵ-wall-crossing) In the CY (resp. LG) phase, the Givental-style mirror theorem says the J-function which +governs the genus-zero GW (resp. FJRW) is related to the I-function, which can be expressed in terms of +explicit hypergeometric series, by explicit change of variables known as the mirror map. +(2) (ζ-wall-crossing) The I-function admits a Mellin-Barnes integral representation. I-functions in the two +phases are related by analytic continuation given by deforming the contour of integration in C. +The interpretation of Step (1) as ϵ-wall-crossing appeared in later work. For each ϵ ∈ Q>0, Ciocan-Fontanine–Kim– +Maulik [CKM] introduced ϵ-stable quasimaps to certain GIT quotient W//G. Ciocan-Fontanine and Kim [CK] +1 +arXiv:2301.01266v1 [math.AG] 3 Jan 2023 + +introduced Jϵ which is a generating function of invariants defined by genus-zero ϵ-stable quasimaps; Jϵ specializes +to the I-function and the J-function as ϵ → 0+ and ϵ → +∞, respectively. In the presence of a good torus action, +they proved ϵ-wall-crossing which relates Jϵ to the J-function J = J∞ for any ϵ ∈ Q>0, by change of variables. +They computed the I-function I = J0+ explicitly. In particular, they recover the mirror theorem in the geometric +phase first proved by Givental [Gi96] and Lian-Liu-Yau [LLY]. The mirror theorem in the LG phase was first proved +by Chiodo-Ruan [CR] and later reproved by Ross-Ruan [RR] via ϵ-wall-crossing. +For a general GLSM, GLSM invariants are defined by integrating against virtual cycles on moduli of ϵ-stable LG +quasimaps. The virtual cycle is constructed for narrow sectors by Fan-Jarvis-Ruan [FJR] via cosection localization, +and for both narrow and broad sectors by Favero-Kim [FK] via matrix factorization; Favero-Kim’s construction +generalizes previous constructions for affine LG models [PV16] and for convex hybrid models [CFGKS]. When +ϵ > 0, the definition relies on a good lift ˜ζ which is a character of the group Γ ⊂ GL(V ) generated by G and C∗ +R, +such that V ss +Γ (˜ζ) = V ss +G (ζ). Such a good lift does not always exist. At ϵ = 0+, a good lift is not needed. In this +paper we focus on the ϵ → 0+ stability condition and study genus-zero GLSM invariants and ζ-wall-crossing for +abelian GLSMs where G = (C∗)κ. In this case, Xζ is a smooth toric DM stack. Let �T-be the diagonal subgroup +of GL(V ). Using the work of Favero-Kim [FK], we define and compute K-theoretic GLSM I-function IK +w which +takes values in the K-theory of category of matrix factorizations on the inertial stack IXζ, and the (cohomological) +GLSM I-function Iw which takes values in the GLSM state space Hw. We also define and compute K-theorectic +�T-equivariant I-function IK +� +T of (V, G, C∗ +R, 0, ζ) which takes values in K � +T (IXζ), and �T-equivariant I-function I � +T of +(V, G, C∗ +R, 0, ζ) which takes values in H � +T (IXζ). +In Gromov-Witten theory I-functions fail to capture integral structure on cohomology [Ir09]. Hosono defined an +object called a central charge that sees the Gamma integral structure. Integral structures are crucial for integral +representations. Central charges are power series which are constructed from both J-function and objects of the +derived category of coherent sheaves of the target manifold. Hori and Romo [HR] constructed explicit analytic +functions called hemisphere partition functions and conjectured that their power series expansions are equal to +the central charges in appropriate cases. +We define the GLSM central charge of a matrix factorization B of +(Xζ, wζ) as Zw(B) = ⟨Iw, Γwchw([B])⟩ where Γw is an appropriate version of Iritani’s Γ-class and [B] is the K- +theory class of B. We also define the �T-equivariant central charge of a �T-equivariant perfect complex B on Xζ as +Z � +T (B) = ⟨I � +T , Γ � +T ch � +T ([B])⟩. We show that our central charges indeed have integral representations of the hemisphere +partition function form (Theorem 5.6). +We call this representation Higgs-Coulomb correspondence because in +physics GLSM central charges can be obtained by a version of the Higgs branch localization and hemisphere +partition functions are computed by the Coulomb branch localization (c.f. [BC]). +The integral representations +we obtain depend continuously on the complexified stability parameter θ = ζ + 2π√−1B and do not have any +restrictions on ζ. +In this philosophy ζ-wall-crossing follows immediately by analytic continuation in ζ (Theorem 5.21). Remarkably, +matrix factorizations of the central charges in question are related by the Fourier-Mukai transform [BP10]. Let ζ± +represent stability conditions in two adjacent chambers and B+ be a matrix factorizations in the phase corresponding +to ζ+. Then, the analytic continuation of the central charge of B+ is a central charge of B− = FM(B+). Particular +Fourier-Mukai kernel is choosen by B = Im(θ)/2π and convergence of the integral representation is equivalent to +the so-called Grade Restriction Rule [HL,BFK,CIJS]. +(1.1) +B +B+ +B− +π+ +π− +F M +Acknowledgements. We wish to thank Daniel Halpern-Leistner, Kentaro Hori, Hiroshi Iritani, Andrei Okounkov, +Tudor P˘adurariu, Renata Picciotto, Alexander Polishchuk, Che Shen, Yefeng Shen, Mark Shoemaker, and Yang +Zhou for helpful communications. We thank the hospitality and support of the Simons Center for Geometry and +Physics (SCGP) during the program Integrability, Enumerative Geometry and Quantization (August 22-September +23, 2022) where part of the paper was completed. The authors are partially supported by NSF grant DMS-1564497. +2. Geometry of gauged linear sigma models and A-model state spaces +In this paper, all the schemes and algebraic stacks are defined over Spec C, where C is the field of complex +numbers. +2 + +2.1. Gauged linear sigma models. We start with the setup of a general gauged linear sigma model (GLSM) +following [FJR,FK]. Part of our formulation in Section 4 is closer to that in the more general setting in [CJR]. +The input data of a GLSM is a 5-tuple (V, G, C∗ +R, W, ζ), where +(1) (linear space) V = SpecC[x1, . . . , xn+κ] ≃ Cn+κ is a finite dimensional complex vector space, where κ = +dim G. +(2) (gauge group) G ⊂ GL(V ) is a reductive algebraic group. +(3) (vector R-symmetry) C∗ +R ∼= C∗ acts linearly and faithfully on V , so we may view C∗ +R as a subgroup of +GL(V ). We assume that +(a) the intersection G ∩ C∗ +R is finite, and +(b) the C∗ +R-action commutes with the G-action. +The finite group G ∩ C∗ +R must be cyclic, generated by an element J of finite order r ∈ Z>0, given explicitly +in Equation (2.2) below. The surjective group homomorphism C∗ +R → C∗ +ω := C∗/⟨J⟩ is a degree r covering +map. Let Γ := GC∗ +R ⊂ GL(V ) be the subgroup generated by G and C∗ +R. Then we have a short exact +sequence of groups: +(2.1) +1 → G → Γ +χ→ C∗ +ω → 1. +By (b), the Γ-action on V induces a C∗ +ω-action on the smooth Artin stack [V/G] in the sense of [Ro]. The +R-charges are +(q1, . . . , qn+κ) = +�2c1 +r , . . . , 2cn+κ +r +� +, +where c1, . . . , cn+κ ∈ Z are the weights of the C∗ +R-action on V ≃ Cn+κ. (Note that gcd(c1, . . . , cn+κ) = 1 +since C∗ +R acts faithfully on V .) Let +(2.2) +J = (e2π√−1c1/r, . . . , e2π√−1cn+κ/r). +(4) (superpotential) W : V → C is a G-invariant regular function which is a quasi-homogeneous polynomial of +degree r with respect to the C∗ +R-action on V ; in other words, W ∈ C[x1, . . . , xn+κ]G and +W(tc1x1, . . . , tcn+κxn+κ) = trW(x1, . . . , xn+κ), +t ∈ C∗ +R, +(x1, . . . , xn+κ) ∈ V. +It descends to a function w : [V/G] → C of degree 1 with respect to the C∗ +ω-action on [V/G]. +(5) (stability condition) ζ ∈ Hom(G, C∗) = Hom(Gab, C∗), where Gab = G/[G, G] is the abelianization of G. +We view Hom(G, C∗) as an additive group and let χζ : G → C∗ denote the associated G-character. Let +V ss +G (ζ) (respectively V s +G(ζ)) be the set of semistable (respectively stable) points in V determined by the +G-linearization on the trivial line bundle V × C → V given by g · (v, t) = (g · p, χζ(g)t). We assume that +V s +G(ζ) = V ss +G (ζ). Then the quotient stack Xζ := [V ss +G (ζ)/G] is an orbifold (i.e. a smooth Deligne-Mumford +stack with trivial generic stabilizer) of dimension n. The GIT quotient Xζ := V �ζ G = V ss +G (ζ)/G is the +coarse moduli space of Xζ. +Let Zζ := [(Crit(W) ∩ V ss +G (ζ))/G] be the critical locus of wζ := w|Xζ : Xζ → C. We say ζ is in a geometric phase +if Crit(W) ∩ V ss(ζ) is non-singular, which implies Zζ is an orbifold. +The central charge of the GLSM is (cf. [FJR, Definition 3.2.3]) +(2.3) +ˆc := dim V − dim G − 2ˆq = n − 2ˆq, +where ˆq = 1 +2 +n+κ +� +j=1 +qj = 1 +r +n+κ +� +j=1 +cj. +Remark 2.1. By the condition (4) above, the superpotential W is semi-invariant with respect to Γ in the sense +of [PV11, Section 2], i.e., +(2.4) +W(γ · x) = χ(γ)W(x) +for any γ ∈ Γ and x ∈ V. +Remark 2.2. In this paper the stability condition ζ is an element in Hom(G, C∗) and corresponds to the symbol θ +in [FJR,FK]. The 1-dimensional torus C∗ +ω in this paper corresponds to C∗ +ω in [CJR]. +Remark 2.3. At this point, we do not assume the critical locus Zζ is proper. This allows us to include the case +without superpotential as a special case where the superpotential and the R-charges are zero, i.e. W = 0 and qj = 0 +for j = 1, . . . , n + κ; note that in this special case the C∗ +R-action on V is trivial, and in particular, not faithful. +3 + +2.2. Abelian GLSMs. In the rest of this paper, we consider abelian GLSMs where the gauge group G = (C∗)κ is +a complex algebraic torus. In this case [G, G] = {1} and Gab = G. +Up to an inner automorphism of GL(V ), we may assume the image of ρV : G → GL(V ) is contained in the +diagonal torus �T ≃ (C∗)n+κ ⊂ GL(V ) ∼= GLn+κ(C). Then Xζ is an n-dimensional toric orbifold, and its coarse +moduli Xζ is a semi-projective simplicial toric variety which contains T := �T/G ∼= (C∗)n as a Zariski dense open +subset. We have a short exact sequence of abelian groups +(2.5) +1 → G +ρV +−→ �T −→ T → 1. +Remark 2.4. The notation in this subsection is similar to but slightly different from that in [CIJ]: G, �T, and T +in this paper correspond to K, T, and Q in [CIJ, Section 4.3], respectively; κ and n in this paper correspond to r +and m − r in [CIJ, Section 4.1], respectively. +• Applying Hom(C∗, −) to (4.38), we obtain the following short exact sequence of cocharater lattices +(2.6) +0 → L := Hom(C∗, G) −→ � +N := Hom(C∗, �T) −→ N := Hom(C∗, T) → 0, +where L ∼= Zκ, � +N ∼= Zn+κ, and N ∼= Zn. +• Applying Hom(−, C∗) to (4.38), or equivalently dualizing (2.6), we obtain the following short exact sequence +of character lattices: +(2.7) +0 → M := Hom(T, C∗) −→ � +M := Hom( �T, C∗) −→ L∨ := Hom(G, C∗) → 0 +The map L → � +N ∼= Zn+κ is given by (D1, . . . , Dn+κ) where Di ∈ L∨. The stability condition ζ is an element +in Hom(G, C∗) = L∨. If we choose a Z-basis {ξ1, . . . , ξκ} of L (which is equivalent to a choice of an isomorphism +G ≃ (C∗)κ) and let {ξ∗ +1, . . . , ξ∗ +κ} be the dual Z-basis of L∨, then +Di = +κ +� +a=1 +Qa +i ξ∗ +a +for some Qa +i ∈ Z. Given any t = �κ +a=1 taξ∗ +a ∈ L∨, where t1, . . . , tκ ∈ Z, let χt : G → C∗ be the corresponding G +character given by +(2.8) +χt(s1, . . . , sκ) = +κ +� +a=1 +sta +a . +Then the map G ≃ (C∗)κ → �T ∼= (C∗)n+κ is given by +(2.9) +s = (s1, . . . , sκ) ∈ G �→ +� +χD1(s), . . . , χDn+κ(s) +� +∈ �T, +where χDi(s) = +κ +� +a=1 +sQa +i +a . +Given a lattice Λ ∼= Zr and a field F, we define ΛF := Λ ⊗Z F ∼= Fr; in this paper, F = Q, R, or C. +Remark 2.5. The map G → �T is injective iff D1, . . . , Dn+κ generate the lattice L∨ over Z. In Section 5 (The +Coulomb Branch), we work with the weaker assumption that D1, . . . , Dn+κ span the vector space L∨ +Q over Q, or +equivalently, the kernel K of the group homomorphism G → �T is finite. Then Xζ is a smooth toric DM stack with +a generic stabilizer K; it is a toric orbifold iff K is trivial. It is also possible to work in this generality in Section +4 (The Higgs branch). We assume K is trivial in Section 4 mainly because [FJR] and [FK] assume so, but K can +be non-trivial in orbifold quasimap theory [CCK] which can be viewed as a mathematical theory of GLSM without +superpotential. +Let GR ≃ U(1)κ be the maximal compact subgroup of G ≃ (C∗)κ. Then LR is canonically isomorphic to the Lie +algebra gR of GR. The G-action on V = SpecC[x1, . . . , xn+κ] restricts to a Hamiltonian GR-action on the K¨ahler +manifold (V, +√−1 +2 +�n+κ +i=1 dxi ∧ d¯xi) with a moment map +µ : V −→ g∨ +R = L∨ +R, +(x1, . . . , xn+κ) = 1 +2 +κ +� +a=1 +Qa +i |xi|2ξ∗ +a. +Then +Xζ = [V ss +G (ζ)/G] = [µ−1(ζ)/GR]. +From this perspective, the stability condition ζ is a regular value of the moment map µ, and can be an element in +L∨ +R ∼= Rκ. +4 + +2.3. Anticones and the extended stacky fan. The triple (V, G, ζ), which is part of the input data +(V, G, C∗ +R, W, ζ) of the given abelian GLSM, determines a set Aζ of anticones and an extended stacky fan +Σζ = (N, Σζ, β, Sζ). We describe them in this subsection, and describe V ss +G (ζ) ⊂ V in terms of anticones. +We fix an isomorphism V = SpecC[x1, . . . , xn+κ] ∼= Cn+κ, which determines an ordered Z-basis (�e1, . . . , �en+κ) +of � +N. In particular, +� +N = +n+κ +� +i=1 +Z�ei. +Let vi ∈ N be the image of ei under � +N → N. Define β = (v1, . . . , vn+κ). +Given a subset I of {1, . . . , n + κ}, let I′ = {1, . . . , n + κ} \ I be its complement, and define +∠I = { +� +i∈I +aiDi : ai ∈ R, ai > 0} ⊂ L∨ +R, +σI = { +� +i∈I +aivi : ai ∈ R, ai ≥ 0} ⊂ NR. +If I = ∅ is the empty set, define σ∅ = {0}. +For a fixed G-action on V , a stability condition ζ ∈ L∨ +R determines the following three sets. +Aζ += +{I ⊂ {1, . . . , n + κ} : ζ ∈ ∠I}, +Σζ += +{σI : I′ ∈ Aζ}, +Sζ += +{i ∈ {1, . . . , n + κ} : σ{i} /∈ Σζ} = {i ∈ {1, . . . , n + κ} : {i}′ /∈ Aζ}. +Note that Sζ ⊂ I for any I ∈ Aζ. +Assumption 2.6. We choose the stability condition ζ ∈ L∨ +R such that the following three equivalent conditions are +satisfied. +(i) For any I ∈ Aζ, {Di : i ∈ I} spans L∨ +Q as a vector space over Q. +(ii) For any I ∈ Aζ, {vi : i ∈ I′} is a set of linearly independent vectors in NQ, or equivalently, σI′ is a +simplicial cone in NR. +(iii) Σζ is a simplicial fan in NR. +Elements in Σζ are called cones, while elements in Aζ are called anticones. By (i), the cardinality |I| of any +anticone I ∈ Aζ is greater or equal to κ. Let Σζ(d) be the set of d-dimensional cones in Σζ. Then σ ∈ Σζ(d) iff +σ = σI where |I| = d and I′ ∈ Aζ. Let +Amin +ζ += {I ∈ Aζ : |I| = κ} = {I ∈ Aζ : σI′ ∈ Σζ(n)} +be the set of minimal anti-cones. +The irrelevant ideal of ζ is the ideal Bζ in C[x] := C[x1, . . . , xn+κ] generated by {xI := � +i∈I xi : I ∈ Aζ}. Let +Zζ = Z(Bζ) be the closed subvariety of V = SpecC[x] defined by the irrelevant ideal Bζ ⊂ C[x], and let Uζ = V \Zζ. +If ζ ∈ L∨ is a G character, then Uζ = V ss +G (ζ), and Zζ = V un +G (ζ) is the set of unstable points defined by ζ. +For any I ∈ Aζ, define +(2.10) +VI = V \ Z(xI) = {(x1, . . . , xn+κ) ∈ V : xi ̸= 0 if i ∈ I} = (C∗)I × CI′. +Then +Uζ = +� +I∈Aζ +VI. +Note that if I, J ∈ Aζ and I ⊂ J then VJ ⊂ VI. Therefore, +Uζ = +� +I∈Amin +ζ +VI. +We now give an alternative description of Zζ. If I ⊂ {1, . . . , n + κ}, and I /∈ Aζ, or equivalently σI′ /∈ Σζ, define +ZI = CI × {0}I′ = {(x1, . . . , xn+κ) ∈ V : xi = 0 if i ∈ I′}. +Then +Zζ = +� +I /∈Aζ +ZI. +Define +Cζ := +� +I∈Aζ +∠I = +� +I∈Amin +ζ +∠I ⊂ L∨ +R. +The open cone Cζ is called the extended ample cone in [CIJ]. It is a chamber in the space of stability conditions: if +ζ′ ∈ Cζ then Σζ′ = Σζ and Uζ′ = Uζ. We recall the following facts. +5 + +(1) The quotient stack +Xζ = [Uζ/G] +is the smooth toric Deligne-Mumford (DM) stack defined by the stacky fan (N, Σζ, β). See Borisov-Chen- +Smith [BCS] for definition of toric Deligne-Mumford stacks in terms of stacky fans. +(2) The coarse moduli space of Xζ is the categorical (and geometric) quotient +Xζ = Uζ/G +which is the toric variety defined by the simplicial fan Σζ ⊂ NR. See [Fu93, CLS] for an introduction of +toric varieties, and in particular the definition of general normal toric varieties in terms of fans. +(3) If ζ ∈ L∨ is a G-character then +Xζ = [V ss +G (ζ)/G] +is the GIT quotient stack, and +Xζ = V ss +G (ζ)/G = V �ζ G +is the GIT quotient. +(4) The triple (V, G, ζ) determines a particular presentation of Xζ as a quotient stack [Uζ/G] and an extended +stacky fan +Σζ = (N, Σζ, β, Sζ), +a notion introduced by Jiang [Ji08]. +2.4. Closed toric substacks and their generic stabilizers. The �T-divisor �Dj = {xj = 0} ⊂ V = SpecC[x] +restricts to a �T-divisor �Dj∩Uζ ⊂ Uζ which descends to a T-divisor Dj = [( �Dj∩Uζ)/G] in the toric stack Xζ = [Uζ/G] +and a T-divisor Dj in the toric variety Xζ. Note that Dj and Dj are empty if j ∈ Sζ. +Given any σ ∈ Σζ(d), where 0 ≤ d ≤ n, we have σ = σI′ for some I ∈ Aζ with |I| = κ + n − d. Let +V(σ) = +� +i∈I′ +Di ⊂ Xζ, +V (σ) = +� +i∈I′ +Di ⊂ Xζ. +Then V(σ) (resp. V (σ)) is an (n − d)-dimensional closed toric substack (resp. subvariety) of Xζ (resp. Xζ). The +generic stabilizer of the toric stack V(σ) is the finite group +Gσ = +� +i∈I +Ker(χDi) ⊂ G +where χDi : G → C∗ is defined as in (2.9). If τ, σ ∈ Σζ and τ ⊂ σ then V(τ) ⊃ V(σ), so Gτ ⊂ Gσ. In particular, +V({0}) = Xζ and G{0} is trivial. If I ∈ Amin +ζ +, then σI′ ∈ Σζ(n) and pI := V(σI′) ≃ [•/GσI′ ] = BGσI′ is the unique +T-fixed point in +XI := [VI/G] ≃ +�� +(C∗)I × CI′� +/G +� +≃ [CI′/GσI′ ] ≃ [Cn/GσI′ ]. +Here • = SpecC is a point, BGσI′ is the classifying space of GσI′ , and VI is defined by Equation (2.10). +2.5. Line bundles. Let +U T +j := OXζ(−Dj) ∈ PicT (Xζ), +uT +j := −(c1)T (U T +j ) ∈ H2 +T (Xζ; Q). +Then uT +j is the T-equivariant Poincar´e dual of Dj. Note that uT +j = 0 if j ∈ Sζ. The T-equivariant Chern character +of U T +j is chT (U T +j ) = e−uT +j . +The T-equivariant line bundles U T +j generate KT (Xζ) as an algebra over Z. Any group homomorphism A → T +induces a map [Xζ/A] → [Xζ/T] = [V ss +G (ζ)/ �T] and ring homomorphisms +KT (Xζ) → KA(Xζ), +φ∗ : H∗ +T (Xζ; Q) → H∗ +A(Xζ; Q). +Let Uj ∈ Pic(Xζ) (resp. uj ∈ H2(Xζ; Q)) be the image of U T +j +∈ PicT (Xζ) (resp. uT +j ∈ H2 +T (Xζ; Q)) under +the surjective group homomorphism PicT (Xζ) → Pic(Xζ) (resp. H2 +T (Xζ; Q) → H2(Xζ; Q)) induced by the group +homomorphism {1} → T. Then c1(Uj) = −uj and ch(Uj) = e−uj. +Let U � +T +j ∈ Pic � +T (Xζ) (resp. u � +T +j ∈ H2 +� +T (Xζ; Q)) be the image of U T +j ∈ PicT (Xζ) (resp. uT +j ∈ H2 +T (Xζ; Q)) under the +group homomorphism PicT (Xζ) → Pic � +T (Xζ) (resp. H2 +T (Xζ; Q) → H2 +� +T (Xζ; Q)) induced by the group homomorphism +�T → T = �T/G. Then (c1) � +T (U � +T +j ) = −u � +T +j and ch � +T (U � +T +j ) = e−u +� +T +j . +For i = 1, . . . , n + κ, let χDi : G → C∗ be defined as in (2.9). For a = 1, . . . , κ, let χξ∗ +a : G → C∗ be the character +associated to ξ∗ +a ∈ L∨, i.e., χξ∗ +a(s1, . . . , sκ) = sa. Let G act on Uζ × C by +s · (x1, . . . , xn+κ, y) = (χD1(s)x1, . . . , χDn+κxκ, χξ∗ +a(s−1)y), +6 + +This defines a G-equivariant line bundle on Uζ, or equivalently a line bundle Pa on Xζ = [Uζ/G]. Let pa = −c1(Pa) ∈ +H2(Xζ; Q). Then ch(Pa) = e−pa. For j = 1, . . . , n + κ, we have +(2.11) +Uj = +κ +� +a=1 +P +Qa +j +a +∈ Pic(Xζ), +uj = +κ +� +a=1 +Qa +j pa ∈ H2(Xζ; Q). +Let Λj ∈ Pic � +T (•) = Pic(B �T) be the �T-equivariant line bundle over a point • defined by the �T-character �tj, and +let λj = −(c1) � +T (Λj) ∈ H2 +� +T (•; Z) = H2(B �T; Z). Then +K � +T (•) = Z[Λ±1 +1 , . . . , Λ±1 +n+κ], +H∗(B �T; Z) = Z[λ1, . . . , λn+κ]. +For later convenience, we introduce the following definition. +Definition 2.7. Let I ∈ Amin +ζ +be a minimal anticone. Then {Di : i ∈ I} is a Q-basis of L∨ +Q. Let {D∗,I +i +: i ∈ I} be +the dual Q-basis of LQ, i.e., for any i, j ∈ I, +⟨Di, D∗,I +j ⟩ = δij +where ⟨ , ⟩ : L∨ +Q × LQ → Q is the natural pairing between dual vector spaces over Q. +Given any I ∈ Amin +ζ +, the inclusion ιI : pI �→ Xζ of the torus fixed point pI induces a ring homomorphism +ι∗ +I : H∗ +� +T (Xζ; Q) −→ H∗ +� +T (pI; Q) ≃ H∗(B �T; Q) = Q[λ1, . . . , λn+κ]. +For i = 1, . . . , n + κ, +(2.12) +ι∗ +Iu +� +T +i = +� +j∈I +⟨Di, D∗,I +j ⟩λj − λi +∀I ∈ Amin +ζ +. +In particular, if i ∈ I then ι∗ +Iu � +T +i = 0, which is consistent with the fact that the T-fixed point pI is not contained in +the divisor Di. +Let �T × G act on Uζ × C by +(�t, s) · (x1, . . . , xn+κ, y) = (�t1χD1(s)x1, . . . , �tn+κχDn+κxκ, χξ∗ +a(s−1)y), +where �t = (�t1, . . . , �tn+κ) ∈ �T and s ∈ G. This defines a �T × G-equivariant line bundle on Uζ, or equivalently a +�T-equivariant line bundle P � +T +a on Xζ = [Uζ/G]. Let p � +T +a = −c1(P � +T +a ) ∈ H2 +� +T (Xζ; Q). Then ch � +T (P � +T +a ) = e−p +� +T +a . For +a = 1 . . . , κ, +(2.13) +ι∗ +Ip +� +T +a = +� +j∈I +⟨ξ∗ +a, D∗,I +j ⟩λj +∀I ∈ Amin +ζ +. +We have +(2.14) +U +� +T +j = +κ +� +a=1 +(P +� +T +a )Qa +j · Λ−1 +j +∈ Pic � +T (Xζ), +u +� +T +j = +κ +� +a=1 +Qa +j p +� +T +a − λj ∈ H2 +� +T (Xζ; Q). +Under K � +T (Xζ) −→ K(Xζ), U � +T +j , P � +T +a , and Λj are mapped to Uj, Pa, and 1, respectively; under H2 +� +T (Xζ; Q) → +H2(Xζ; Q), u � +T +j , p � +T +a , and λj are mapped to uj, pa, and 0, respectively. Our definitions of u � +T +j , U � +T +j , p � +T +a , P � +T +a , λj, Λj +are consistent with the convention in Givental’s papers [GiV,GiVI] on permutation-equivariant quantum K-theory +of toric manifolds. +2.6. The inertia stack. +2.6.1. The inertia stack of a general algebraic stack. Given a general algebraic stack X, the inertia stack IX of X +is the fiber product +IX +� +� +X +∆ +� +X +∆ � +∆ � X × X +where ∆ : X → X ×X is the diagonal morphism. IX is an algebraic stack, and in particular a groupoid. An object +in the groupoid IX is a pair (x, g) where x is an object in the groupoid X and k is an element in the automorphism +group AutX (x) = HomX (x, x) of x. Morphisms between two objects in IX are +HomIX ((x1, g1), (x2, g2)) = {h ∈ HomX (x1, x2) : h ◦ g1 = g2 ◦ h}. +7 + +The map (x, g) �→ (x, g−1), where (x, g) is an object in IX, defines an involution +inv : IX → IX. +2.6.2. The inertia stack of a quotient stack. If X = [U/G] is a quotient stack, where U is a scheme and G is an +algebraic group, then the inertia stack is also a quotient stack: +IX = [IU/G] +where IU := {(x, g) ∈ U × G | g · x = x} is a closed subscheme of U × G, and the G-action on IU is given by +h · (x, g) = (h · x, hgh−1), +where h ∈ G and (x, g) ∈ IU. +In particular, if G is abelian then the action is given by h · (x, g) = (h · x, g). +The involution inv : IX → IX is induced by the G-equivariant involution IU → IU given by (x, g) �→ (x, g−1). +2.6.3. The inertial stack of the toric orbifold Xζ. Let Xζ = [Uζ/G] be as in Section 2.3. To describe its inertia stack +IXζ more explicitly, we introduce some definitions. Given σ = σI ∈ Σζ, where I ⊂ {1, . . . , n + κ}, define +(2.15) +Box(σ) := +� +v ∈ N : v = +� +i∈I +civi, 0 ≤ ci < 1 +� +and +(2.16) +Box′(σ) := +� +v ∈ N : v = +� +i∈I +civi, 0 < ci < 1 +� +. +Define +(2.17) +Box(Σζ) := +� +σ∈Σζ +Box(σ) = +� +σ∈Σζ(n) +Box(σ). +which is a finite set. For any v ∈ Box(Σζ) there exists a unique σ ∈ Σζ such that v ∈ Box′(σ). Therefore, +(2.18) +Box(Σζ) = +� +σ∈Σζ +Box′(σ), +where the right hand side of (2.18) is a disjoint union. Given any σ = σI ∈ Σζ, where I′ ∈ Aζ, define +(2.19) +age(v) = +� +i∈I +ci ∈ Q +if v = +� +i∈I +civi ∈ Box′(σ). +Suppose that σ = σI ∈ Σζ where I′ ∈ Aζ. There is a bijection Box(σ) −→ Gσ given by +(2.20) +v = +� +i∈I +civi �→ g(v) = (a1, · · · , an+κ) ∈ Gσ ⊂ G ⊂ �T ≃ (C∗)n+κ where ai = +� +e2π√−1ci, +i ∈ I, +1, +i ∈ I′. +The map v �→ g(v) defines a bijection +Box(Σζ) = +� +σ∈Σζ +Box(σ) −→ +� +σ∈Σζ +Gσ = {g ∈ G : (Uζ)g is non-empty.}, +where (Uζ)g = {x ∈ Uζ : g · x = x}. We have +(2.21) +IUζ = {(x, g) ∈ Uζ × G : g · x = x} = +� +v∈Box(Σζ) +(Uζ)g(v) × {g(v)}. +The above union is a disjoint union of connected components. +IXζ = [IUζ/G] = +� +v∈Box(Σζ) +Xζ,v +where Xζ,v ≃ [(Uζ)g(v)/G]. +In particular, g(0) = 1 is the identity element of G and Xζ,0 ≃ Xζ. There is an involution inv : Box(Σζ) → Box(Σζ) +characterized by g(inv(v)) = g(v)−1 ∈ G. The involution inv : IXζ −→ IXζ maps Xζ,v isomorphically to Xζ,inv(v). +Observe that +age(v) + age(inv(v)) + dim Xζ,v = n = dim Xζ. +Let +(2.22) +η = (eπ√−1q1, . . . , eπ√−1qn+κ) ∈ C∗ +R. +8 + +Then η2 = J = (e2π√−1q1, . . . , e2π√−1qn+κ) and W(η · x) = −W(x). Let invR : IXζ → IXζ be the map induced by +the map IUζ → IUζ given by (x, g) �→ (η · x, g−1). Then invR maps Xζ,v isomorphically to Xζ,inv(v). Note that in +general invR ◦ invR is not the identity map, i.e., invR is not an involution. +2.7. A-model GLSM state spaces. Let M be a large positive number such that the real part of any critical +values of W is less than M, and define w∞ +ζ := (ReW)−1(M, ∞) ⊂ Xζ. As a graded vector space over Q, the rational +GLSM state space of (V, G, C∗ +R, W, ζ) is +(2.23) +Hw,Q = +� +v∈Box(Σζ) +H∗(Xζ,v, w∞ +ζ,v; Q)[2 (age(v) − ˆq)]. +where wζ,v = wζ +�� +Xζ,v. In particular, Hw,Q ∼= H∗(IXζ; Q) as a vector space over Q. Note that invR maps (Xζ,v, wζ,v) +diffeomorphically to (Xζ,inv(v), −wζ,inv(v)) and induces an isomorphism +(2.24) +inv∗ +R : H∗(Xζ,inv(v), −w∞ +ζ,inv(v); Q) +∼ += +−→ H∗(Xζ,v, w∞ +ζ,v; Q). +By [FK, Corollary 4.3], Crit(wζ,v) ⊂ Zζ = Crit(wζ). If Zζ is proper then there is a nondegenerate pairing +(2.25) +H∗(Xζ,v, w∞ +ζ,v; Q) × H∗(Xζ,v, −w∞ +ζ,v; Q) → Q. +for all v ∈ Box(Σζ). Combining (2.24) and (2.25), we obtain a nondegenerate pairing +H∗(Xζ,v, w∞ +ζ,v; Q) × H∗(Xζ,inv(v), w∞ +ζ,inv(v); Q) → Q +for all v ∈ Box(Σζ), thus a non-degenerate pairing ( , )w : Hw,Q × Hw,Q → Q. +As a graded vector space over C, the GLSM state space of (V, G, C∗ +R, W, ζ) is +Hw = +� +v∈Box(Σζ) +H∗ � +Xζ,v, (Ω• +Xζ,v, dwζ,v) +� +[2(age(v) − ˆq)] +where +H∗ � +Xζ,v, (Ω• +Xζ,v, dwζ,v) +� +≃ H∗(Xζ,v, w∞ +ζ,v; C). +Therefore, Hw ≃ Hw,Q ⊗Q C as a graded vector space over C. +We say v ∈ Box(Σζ) is narrow if Xζ,v is compact. If v is narrow then wζ,v is constant and w∞ +ζ,v is empty, so +H∗(Xζ,v, w∞ +ζ,v; C) = H∗(Xζ,v; C). +In this case, we call the above vector space a narrow sector. We say v is broad if v is not narrow, and call +H∗(Xζ,v, w∞ +ζ,v; C) +a broad sector. +We also consider a closely related GLSM (V, G, C∗ +R, 0, ζ) obtained by replacing the superpotental W by zero. As +a graded vector space over C, the GLSM state space of (V, G, C∗ +R, 0, ζ) is +(2.26) +H = +� +v∈Box(Σζ) +H∗(Xζ,v; C)[2 (age(v) − ˆq)]. +Note that inv and invR are homotopic as maps from Xζ,v to Xζ,inv(v), so inv∗ +R = inv∗ : H∗(Xζ,inv(v); C) → +H∗(Xζ,v; C). +The action of �T on V commutes with the action of its subgroup G and C∗ +R, and preserves the zero superpotential +0 and the semistable locus V ss +G (ζ). We define the �T-equivariant state space H � +T of (V, G, C∗ +R, 0, ζ) as follows. As a +graded vector space over C, +(2.27) +H � +T = +� +v∈Box(Σζ) +H∗ +� +T (Xζ,v; C)[2 (age(v) − ˆq)] +where each H∗ +� +T (Xζ,v; C) is a module over H∗ +� +T (•; C) = H∗(B �T; C) = C[λ1, . . . , λn+κ]. Let C(λ) := C(λ1, . . . , λ) be +the fractional field of C[λ] := C[λ1, . . . , λn+κ]. There is a non-degenerate pairing +H � +T ⊗C[λ] C(λ) × H � +T ⊗C[λ] C(λ) → C(λ). +9 + +3. Categories of B-branes and K-theories +Given an abelian GLSM, we consider several versions of the category of B-branes and the A-model state space. +dg category +category of B-branes +K-theory +A-model state space +MF(Xζ, wζ) +DMF(Xζ, wζ) ≃ DSg(Xζ,0) +K(DMF(Xζ, wζ)) +Hw = +� +v∈Box(Σζ) +H∗(Xζ,v, w∞ +ζ,v) +Perf � +T (Xζ) +Db +� +T (Xζ) +K � +T (Xζ) +H � +T = +� +v∈Box(Σζ) +H∗ +� +T (Xζ,v) +Perf(Xζ) +Db(Xζ) +K(Xζ) +H = +� +v∈Box(Σζ) +H∗(Xζ,v) +Db +c(Xζ) +Kc(Xζ) +Hc = +� +v∈Box(Σζ) +H∗ +c (Xζ,v) +In each version, the Chern character sends the K-theory class of a B-brane to an element in the A-model state +space. +Kw = K(DMF(Xζ, wζ)) +chw +−→ +Hw := +� +v∈Box(Σζ) +Hw,v, +Hw,v = H∗(Xζ,v, wζ,v; C). +K � +T = K � +T (Xζ) +ch � +T +−→ +H � +T = +� +v∈Box(Σζ) +H � +T ,v, +H � +T ,v = H∗ +� +T (Xζ,v; C). +K = K(Xζ) +ch +−→ +H = +� +v∈Box(Σζ) +Hv, +Hv := H∗(Xζ,v; C). +Kc = Kc(Xζ) +chc +−→ +Hc = +� +v∈Box(Σζ) +Hc,v, +Hc,v = H∗ +c (Xζ,v; C). +K � +T is a commutative ring with unity and an algebra over K � +T (•) = K(B �T) = Z[Λ± +1 , . . . , Λ± +n+κ], and K is a +commutative ring with unity and an algebra over K(•) = Z. There is a surjective ring homomorphism K � +T → K. +Kc and Kw are modules over the ring K. There is a map Kc → K which is a morphism of K-modules; the image +Kct is an ideal in the ring K. Taking Euler characteristic defines non-degenerate pairings: +K � +T × K � +T → Q(Λ1, . . . , Λn+κ), +K × Kc → Z, +Kw × Kw → Z. +Fix v ∈ Box(Σζ). +H � +T ,v is a commutative ring with unity and an algebra over H∗ +� +T (•) = H∗(B �T) = +C[λ1, . . . , λn+κ], and Hv is a commutative ring with unity and an algebra over H∗(•) = C. There is a surjec- +tive ring homomorphism H � +T ,v → Hv. Hc,v and Hw,v are modules over the ring Hv. There is a map Hc,v → Hv +which is a morphism of Hv-modules; the image Hct,v is an ideal in the ring Hv. We have. non-degenerate pairings: +H � +T ,v × H � +T ,inv(v) → C(λ1, . . . , λn+κ), +Hv × Hc,inv(v) → C, +Hw,v × Hw,inv(v) → C. +4. The Higgs Branch +4.1. A mathematical theory of GLSM: an overview. Let (V, G, C∗ +R, W, ζ) be the input date of a GLSM. The +first four components (V, G, C∗ +R, W) give rise to the following diagram: +V +W +� +� +[V/G] +w +� +� +□ +C +� +[V/Γ] +ˆw +� +� +[C/C∗ +ω] +� +BΓ = [•/Γ] +Bχ � BC∗ +ω = [•/C∗ +ω] +where the middle square is Cartesian, the upper triangle and the lower square are commutative, and the bottom +right arrow Bχ : BΓ → BC∗ +ω is induced by the group homomorphism χ : Γ → C∗ +ω. A Landau-Ginzburg (LG) +quasimap to (V, G, C∗ +R, W, ζ) is a birational map from an orbicurve C to [V ss +G (ζ)/Γ] which extends to a representable +10 + +morphism f : C → [V/Γ] of smooth Artin stacks and satisfies certain stability conditions such that the following +diagram commutes: +[V/Γ] +π +� +C +f +� +P +� +ωlog +C +� +BΓ +Bχ +� +BC∗ +ω +Recall that BΓ is the classifying space of principal Γ-bundles, and [V/Γ] is the classifying space of a principal +Γ-bundle P → C together with a section u : C → P ×Γ V , where P ×Γ V → C is the rank n + κ vector bundle +associated to the representation Γ → GL(V ). A section u : C → P ×Γ V is equivalent to a Γ-equivariant map +˜f : P → V . More explicitly, we have the following cartesian diagram +P +� +� +• +� +C = P/Γ +π◦f � BΓ = [•/Γ] +Let pr1 : P × V −→ P and pr2 : P × V → V be the projection to the first and second factors, respectively. We +have a commutative diagram +P × V +pr1 +� +pr2 +� V +� +P +� +(idP , ˜ +f) +� +˜ +f +� +• +where all the arrows are Γ-equivariant. Taking the quotient of the above diagram by the Γ-action, we obtain the +following commutative diagram +P ×Γ V +� +� [V/Γ] +π +� +C +π◦f � +u +� +f +� +BΓ = [•/Γ] +4.2. Twisted curves and their moduli. We follow the presentation of [AV02,AGV] on twisted curves. A genus- +g, ℓ-pointed twisted prestable curve is a connected proper one-dimensional DM stack C together with ℓ disjoint +zero-dimensional integral closed substacks z1, . . . , zℓ ⊂ C, such that +(i) C is ´etale locally a nodal curve; +(ii) formally locally near a node, C is isomorphic to the quotient stack +[Spec(C[x, y]/(xy))/µr], +where η · (x, y) = (ηx, η−1y), η ∈ µr; +(iii) each marking zi ⊂ C is contained in the smooth locus of C; +(iv) C is a scheme away from the markings and the singular points of C; the coarse moduli space C of C is a +nodal curve of arithmetic genus g. +Let π : C → C be the coarse moduli morphism; let zi = π(zi). The resulting (C, z1, . . . , zℓ) is a genus-g, ℓ-pointed +prestable curve. We say (C, z1, . . . , zℓ) is stable if (C, z1, . . . , zℓ) is stable. The moduli Mtw +g,ℓ of genus-g, ℓ-pointed +twisted prestable curves is a smooth Artin stack of dimension 3g − 3 + ℓ. For i = 1, . . . , ℓ, let Li be the line bundle +on Mtw +g,ℓ whose fiber over (C, z1, . . . , zℓ) is the T ∗ +zjC, the cotangent line to the coarse curve C at zi. +The space of infinitesimal automorphisms of (C, z1, . . . , zℓ) is +Ext0 +OC(ΩC(z1 + · · · + zℓ), OC), +while the space of infinitesimal deformations of (C, z1, . . . , zℓ) is +Ext1 +OC(ΩC(z1 + · · · + zℓ), OC). +(C, z1, . . . zℓ) is stable if and only if Ext0 +OC(ΩC(z1 + · · · + zℓ), OC) = 0. +11 + +4.3. Line bundles over a twisted curve. Let (C, z1, . . . , zℓ) be a genus-g, ℓ-pointed twisted prestable curve, +where zi = Bµri. A line bundle L on C defines a morphism C −→ BC∗ such that we have the following Cartesian +diagram +L +� +� +□ +[C/C∗] +� +C +� [•/C] = BC∗ +where [C/C∗] → BC∗ is the universal line bundle over the classifying space BC∗ of principal C∗-bundles. +Given any line bundle L over C, there exists a positive integer m such that L⊗m = π∗M for some line bundle M +over the coarse moduli space C. Define +deg L = 1 +m deg M ∈ Q +• If zi is a scheme point, we define agezi(L) = 0. +• If zi = Bµri, where ri > 1, then the restriction Lzi of L to zi is an element in Pic(Bµri) = Hom(µri, C∗) ∼= +Z/riZ. There is a unique ai ∈ {0, 1, . . . , ri − 1} such that +Lzi ∼= (TziC)⊗ai. +Define agezi(L) = ai +ri +∈ (0, 1) ∩ Q. +There is a unique line bundle L on the coarse moduli C such that +L ≃ π∗L ⊗ OC( +ℓ +� +i=1 +aizi). +where +deg +� +OC +� +ℓ +� +i=1 +aizi +�� += +ℓ +� +i=1 +agezi(L), +deg(π∗L) = deg L ∈ Z. +So +deg L − +ℓ +� +i=1 +agezi(L) ∈ Z. +For i = 0, 1, hi(C, L) = hi(C, L), so +(4.1) +χ(C, L) = χ(C, L) = deg L + 1 − g − +ℓ +� +j=1 +agezj(L) +which is a special case of Kawasaki’s orbifold version of Riemann-Roch theorem [Ka79]. +The space of infinitesimal automorphisms of L on a fixed twisted prestable curve C is +Ext0 +OC(L, L) ≃ H0(C, OC). +The space of infinitesimal deformations of L on a fixed twisted prestable curved C is +Ext1 +OC(L, L) ≃ H1(C, OC). +4.4. Universal moduli of principal Γ-bundles. Let Mg,ℓ(BΓ) denote the moduli of pairs ((C, z1, . . . , zℓ), P), +where +• (C, z1, . . . , zℓ) is a genus-g, ℓ-pointed twisted prestable curve; +• P is a principal Γ-bundle over C which corresponds to a representable morphism C −→ BΓ. +Then Mg,ℓ(BΓ) is a smooth Artin stack. Note that Mg,ℓ(BΓ) can be identified with the Hom-stack +HomM(CM, BΓ × M) +where M = Mtw +g,ℓ. Forgetting the principal bundle P defines a (non-representable) morphism of smooth Artin stacks +πD/M : D := Mg,ℓ(BΓ) −→ M +which is smooth of relative dimension dim Γ(g − 1) = (κ + 1)(g − 1). So Mg,ℓ(BΓ) is a smooth Artin stack of +dimension +3g − 3 + ℓ + (κ + 1)(g − 1) = (4 + κ)(g − 1) + ℓ = (dim BΓ − 3)(1 − g) + ℓ +12 + +where dim BΓ = − dim Γ = −κ − 1. Let πD : CD → D := Mg,ℓ(BΓ) be the universal curve, let fD : CD → BΓ +be the universal map, and let PD → CD be the universal principal Γ-bundle. We have the following cartesian +diagrams: +CD +� +πD +� +□ +CM +πM +� +D +πD/M � M +PD +� +� +□ +• +� +CD +fD +� BΓ +Let �L := Hom(C∗, Γ) ∼= Zκ+1 be the cocharacter lattice of Γ. Its dual lattice �L∨ = Hom(Γ, C∗) is the character +lattice of Γ. A principal Γ-bundle P −→ C determines a map C = P/Γ → BΓ = [•/Γ] whose degree is an element +βΓ ∈ H2(BΓ; Q) = �LQ +characterized by the following property. For any Γ-character λ ∈ Hom(Γ, C∗) = �L∨ = H2(BΓ; Z), let P ×λ C → C +be the line bundle associated to the representation λ : Γ → C∗ = GL(1). Then +� +βΓ +λ = deg(P ×λ C) = +� +[C] +c1(P ×λ C) ∈ Q, +where +• +� +βΓ +λ denotes the natural pairing between βΓ ∈ �LQ = H2(BΓ; Q) and λ ∈ �L∨ ⊂ �L∨ +Q = H2(BΓ; Q), and +• +� +[C] +c1(P ×λ C) denotes the natural pairing between [C] ∈ H2(C; Q) and c1(P ×λ C) ∈ H2(C; Q). +In other words, βΓ ∈ H2(BΓ; Q) is the image of [C] ∈ H2(C; Q) under P∗ : H2(C; Q) −→ H2(BΓ; Q). +The monodromy of P at zj = Bµrj is an element γj ∈ Γ of order rj. The subset +�LQ/�L ∼= (Q/Z)κ+1 ⊂ �LC/�L ∼= (C/Z)κ+1 = (C∗)κ+1 +can be identified with +{γ ∈ Γ : ord(γ) is finite.}. +The monodromies (γ1, . . . , γℓ) ∈ (�LQ/�L)ℓ and the degree βΓ ∈ �LQ satisfy the following compatibility condition: +ℓ +� +j=1 +γj = βΓ + �L ∈ �LQ/�L. +Let Mg,⃗γ(BΓ, βΓ) ⊂ Mg,ℓ(BΓ) be the open and closed substack of pairs ((C, z1, . . . , zℓ), P) with degree βΓ ∈ �LQ +and monodromies ⃗γ = (γ1, . . . , γℓ) ∈ (�LQ/�L)ℓ. Then +Mg,ℓ(BΓ) = +� +⃗γ=(γ1,...,γℓ)∈(�LQ/�L)ℓ +βΓ∈�LQ, �ℓ +i=1 γi=βΓ+�L +Mg,⃗γ(BΓ, βΓ). +4.5. Universal moduli of Γ-structures. Polishchuk-Vaintrob introduced Γ-structures [PV16] which is an alter- +native formulation for W-structures in [FJR11]. +Given an object (C, z1, . . . , zℓ) in Mg,ℓ(•), a Γ-structure on (C, z1, . . . , zℓ) is a pair (P, ρ) where ((C, z1, . . . , zℓ), P) +is an object in Mg,ℓ(BΓ) and +ρ : P ×χ C +∼ += +−→ ωlog +C +is an isomorphism of line bundles on C. +Let Bg,ℓ be the moduli space of triples ((C, z1, . . . , zℓ), P, ρ), where (C, z1, . . . , zℓ) is an object in Mtw +g,ℓ(•) and +(P, ρ) is a Γ-structure on (C, z1, . . . , zℓ). We have a commutative diagram +B = Bg,ℓ +πB/D� +πB/M +� +D := Mg,ℓ(BΓ) +πD/M +� +M = Mtw +g,ℓ +where πB/D : B −→ D is given by forgetting ρ. +The map πB/M : B → M is smooth of relative dimension +dim G(g − 1) = κ(g − 1), so Bg,ℓ is a smooth Artin stack of dimension +3g − 3 + ℓ + κ(g − 1) = (3 + κ)(g − 1) + ℓ = (dim BG − 3)(1 − g) + ℓ. +13 + +The map +(ρV , ρR) : G × C∗ +R ∼= (C∗)κ+1 −→ Γ ∼= (C∗)κ+1 +(h, t) �→ ht +is a surjective group homomorphism and a covering map of degree r = ord(J). It induces +(ρV , ρR)∗ : H2(BG; Z) × H2(BC∗ +R; Z) = L × +� +Z[P1] +� +−→ H2(BΓ; Z) = �L +which is an inclusion of lattices of finite index r. (We recall that BC∗ +R = P∞, and we let [P1] ∈ H2(P∞; Z) be the +class of P1 ⊂ P∞.) Therefore, we obtain an isomorphism +(ρV , ρR)∗ : H2(BG; Q) × H2(BC∗ +R; Q) = LQ × +� +Q[P1] +� +∼ += +−→ H2(BΓ; Q) = �LQ. +Let (βG, βR) = (ρV , ρR)−1 +∗ (βΓ). Then βR = 2g − 2 + ℓ +r +[P1] ∈ Q[P1] = H2(BC∗ +R; Q) = H2(P∞; Q). +Let Bg,ℓ(βG) ⊂ Bg,ℓ be the open and closed substack of triples ((C, z1, . . . , zℓ), P, ρ) with degree +βΓ = +� +βG, 2g − 2 + ℓ +r +[P1] +� +∈ LQ × +� +Q[P1] +� ∼= H2(BΓ; Q). +Then +Bg,ℓ = +� +βG∈LQ +Bg,ℓ(βG). +For each βG, let Bg,⃗γ(βG) ⊂ Bg,ℓ(βG) be the open and closed substack of triples ((C, z1, . . . , zℓ), P, ρ) with mon- +odromies ⃗γ = (γ1, . . . , γℓ) ∈ (LQ/L)ℓ. Then +Bg,ℓ(βG) = +� +⃗γ=(γ1,...,γℓ)∈(LQ/L)ℓ +�ℓ +i=1 γi=βG+L +Bg,⃗γ(βG). +4.6. Moduli of sections. In this subsection we fix g, ℓ, βG and ⃗γ = (γ1, . . . , γℓ) and let B = Bg,⃗γ(βG). +Following [BF97, Section 1], we introduce the following definition. Given a coherent sheaf F of OX-modules on +an algebraic stack X, let C(F) := SpecX(SymF∨) be the abelian cone associated to F. In particular, when F is +locally free, C(F) = tot(F) is the vector bundle associated to F. +Let PB → CB be the universal principal Γ-bundle over the universal curve πB : CB → B. Recall that Γ is a +subgroup of the diagonal torus �T ⊂ GL(V ), so +VB := PB ×Γ V = +n+κ +� +i=1 +Li,B +where each Li,B is a line bundle over CB. Consider the moduli of sections +Bg,⃗γ(V, βG) := C (πB∗VB) +which parametrizes 4-tuples ((C, z1, . . . , zℓ), P, ρ, u) where the triple ((C, z1, . . . , zℓ), P, ρ) is an object in B(•) and +u = (u1, . . . , un+κ) ∈ H0(C, P ×Γ V ) = H0(C, +n+κ +� +j=1 +Lj) +where uj ∈ H0(C, Lj). Let +Bg,ℓ(V, βG) := C +� +πBg,ℓ(βG)∗VBg,ℓ(βG) +� +. +Then +Bg,ℓ(V, βG) = +� +⃗γ=(γ1,...,γℓ)∈(LQ/L)ℓ +�ℓ +i=1 γi=βG+L +Bg,⃗γ(V, βG). +Given a fixed triple ((C, z1, . . . , zℓ), P, ρ), the space of infinitesimal deformations of the section uj of the line +bundle Lj is H0(C, Lj) and the space of obstructions to deforming uj is H1(C, Lj). We now compute +χ(C, Lj) = h0(C, Lj) − h1(C, Lj). +The line bundle Lj is of degree +⟨Dj, βG⟩ + qj +2 (2g − 2 + ℓ) +14 + +and has monodromy e2π√−1ageγi(Dj) around zi, where ageγi(Dj) ∈ Q ∩ [0, 1) is the unique representative in [0, 1) of +the pairing ⟨Dj, βG⟩ ∈ Q/Z. Note that ageγi(Dj) = ageziLj. We have +⟨Dj, βG⟩ + qj +2 (2g − 2 + ℓ) − +ℓ +� +i=1 +ageγi(Dj) ∈ Z. +By Kawasaki’s orbifold version of the Riemann-Roch theorem [Ka79], +χ(C, Lj) = ⟨Dj, βG⟩ + qj +2 (2g − 2 + ℓ) + (1 − g) − +ℓ +� +i=1 +ageγi(Dj). +Let +(4.2) +(c1)G(V ) = +n+κ +� +j=1 +Dj ∈ L∨, +ageγ(V ) = +n+κ +� +j=1 +ageγi(Dj) ∈ Q, +and recall that ˆq = 1 +2 +�n+κ +j=1 qj. Then +χ(C, P ×Γ V ) = +n+κ +� +j=1 +χ(C, Lj) = ⟨(c1)G(V ), βG⟩ + (n + κ − 2ˆq)(1 − g) − +ℓ +� +i=1 +� +ageγi(V ) − ˆq +� +. +There is a map πS/B : S = Bg,⃗γ(V, βG) −→ B = Bg,⃗γ(βG) given by ((C, z1, . . . , zℓ), P, ρ, u) �→ ((C, z1, . . . , zℓ), P, ρ), +i.e, forgetting the section u. The map πS/B is virtually smooth: there is a relative perfect obstruction theory ES/B, +where +E∨ +S/B = π∗ +S/BRπB∗VB. +The relative virtual dimension of πS/B is +(4.3) +dvir +S/B = ⟨(c1)G(V ), βG⟩ + (n + κ − 2ˆq)(1 − g) − +ℓ +� +i=1 +� +ageγi(V ) − ˆq +� +. +Therefore, the moduli of sections S is a possibly singular, but virtually smooth Artin stack; it is equipped with a +perfect obstruction theory ES of virtual dimension +dvir +S = dvir +S/B + dim B = ⟨(c1)G(V ), βG⟩ + (n + κ − 2ˆq)(1 − g) − +ℓ +� +i=1 +� +ageγi(V ) − ˆq +� ++ (κ + 3)(g − 1) + ℓ +which can be rewritten as +(4.4) +dvir +S = ⟨(c1)G(V ), βG⟩ + (ˆc − 3)(1 − g) + ℓ − +ℓ +� +i=1 +� +ageγi(V ) − ˆq +� +where ˆc = n − 2ˆq is the central charge of the GLSM. Note that if γ ∈ LQ/L corresponds to v ∈ Box(Σζ) then +ageγ(V ) = age(v). +The definition of the central charge ˆc and the degree shift 2 (age(v) − ˆq) in the definition of the A-model state spaces +are motivated by the formula (4.4) of the virtual dimension, which is consistent with [FJR, Lemma 6.1.7]. +Let PS → CS be the universal principal Γ-bundle over the universal curve πS : CS → S, and let uS : CS → +VS := PS ×Γ V be the universal section. We have the following commutative diagram. +VS +� +� +□ +VB +� +CS +� +πS +� +uS +� +□ +CB +� +πB +� +□ +CM +πM +� +S +πS/B � B +πB/M � M +15 + +4.7. Landau-Ginzburg quasimaps and their moduli. +Definition 4.1 (prestable LG quasimaps). A prestable genus-g, ℓ-pointed, degree βG Landau-Ginzburg (LG) +quasimap to the 5-tuple X = (V, G, C∗ +R, W, ζ) is a 4-tuple Q = ((C, z1, . . . , zℓ), P, ρ, u), which is an object in +Bg,ℓ(V, βG)(•) such that the base locus +B(Q) := u−1(P ×Γ V us +G (ζ)) ⊂ C +of Q is purely zero-dimensional and is disjoint from the marked points and the nodes in C. +Remark 4.2. Note that the �T-action preserves V ss +G (ζ) and V us +G (ζ). In particular, Γ acts on V us +G (ζ). +Let LGpre +g,ℓ (X, βG) be the moduli of prestable genus-g, ℓ-pointed, degree βG LG quasimaps to X. It is an open +substack of Bg,ℓ(V, βG). There are evaluation maps +evi : LGpre +g,ℓ (X, βG) → IXζ = +� +v∈Box(Σ) +Xζ,v, +i = 1, . . . , ℓ. +Given ⃗v = (v1, . . . , vℓ) ∈ Box(Σζ)ℓ, let +LGg,⃗v(X, βG) := +ℓ� +i=1 +ev−1 +i +(Xζ,vi) . +Then LGg,⃗v(X, βG) is an open substack of Bg,⃗γ=(γ1,...,γℓ)(V, βG), where γi ∈ LQ/L corresponds to vi ∈ Box(Σ). +Therefore, the virtual dimension of LGg,⃗v(X, βG) is +(4.5) +⟨(c1)G(V ), βG⟩ + (ˆc − 3)(1 − g) + ℓ − +ℓ +� +i=1 +(age(v) − ˆq) . +Definition 4.3 (good lift). ˜ζ ∈ �L∨ is a lift of ζ ∈ L∨ if ζ is the image of ˜ζ under +�L∨ = Hom(Γ, C∗) → L∨ = Hom(G, C∗). +˜ζ is a good lift of ζ if V ss +Γ (˜ζ) = V ss +G (ζ). +For any ˜ζ ∈ Hom(Γ, C∗) = �L∨, let χ˜ζ : Γ → C∗ denote the corresponding Γ-character, let L˜ζ ∈ PicΓ(V ) denote +the Γ-equivariant line bundle on V determined by χ˜ζ +χ +˜ζ1+˜ζ2 = χ +˜ζ1χ +˜ζ2, +L˜ζ1+˜ζ2 = L˜ζ1 ⊗ L˜ζ2. +A section s ∈ H0(V, L˜ζ)Γ defines a Γ-equivariant map V → C which induces a morphism +s : P ×Γ V → P ×χ˜ +ζ C. +Given any u ∈ H0(C, P ×Γ V ), let u∗s := s◦u ∈ H0(C, P ×χζ C). The following definition is [FJR, Definition 4.2.10] +(which is essentially [CKM, Definition 7.1.1]) in slightly different notation. +Definition 4.4 (length). Let Q = ((C, z1, . . . , zℓ), P, ρ, u) be a prestable LG quasimap to X = (V, G, C∗ +R, W, ζ), let +˜ζ ∈ �L∨ be a good lift of ζ ∈ L∨. The length of a point y in C with respect to Q and ˜ζ is defined to be +(4.6) +ℓy(Q, ˜ζ) := min +�ordy(u∗s) +m +��� s ∈ H0(V, Lm˜ζ = L⊗m +˜ζ +)Γ, m ∈ Z>0 +� +where ordy(u∗s) is the order of vanishing of the section u∗s ∈ H0(C, P ×χm˜ +ζ C) at y. +Definition 4.5 (ϵ-stable LG quasimaps). Let ˜ζ ∈ Hom(Γ, C∗) be a good lift of ζ ∈ Hom(G, C∗), and let ϵ be a +positive rational number. A prestable LG quasimap Q = ((C, z1, . . . , zℓ), P, ρ, u) is ϵ-stable with respect to ˜ζ if +(1) ωlog +C +⊗ (P טζ C)ϵ ∈ Pic(C) ⊗Z Q is an ample Q line bundle on C, and +(2) ϵℓy(Q, ˜ζ) ≤ 1 for every y ∈ C. +Remark 4.6. Let Cv (respectively Cv) be the connected component of the normalization of C (respectively C) +associated to a vertex v in the dual graph of the coarse moduli C of C. Let gv be the genus of Cv and let ℓv be +the number of points on Cv mapped to a marked point or a node under the normalization map, and let βΓ(v) ∈ +H2(BΓ; Q) = �LQ be the degree of Cv → C +P→ BΓ. Condition (1) in Definition 4.5 is equivalent to the following +condition: +2gv − 2 + ℓv + ϵ +� +βΓ(v) +˜ζ > 0 +for all vertex v in the dual graph of C. +16 + +Remark 4.7. Let m ∈ Z>0, ϵ ∈ Q>0, and ˜ζ ∈ �L∨. Then Q is ϵ-stable with respect to ˜ζ iff it is (ϵ/m)-stable with +respect to m˜ζ; see [CKM, Remark 7.1.4] for the analogous statement in quasimap theory. Given ν ∈ �L∨ +Q, choose +m ∈ Z>0 such that mν ∈ �L∨ and define Q to be ν-stable if it is (1/m)-stable with respect to mν; the definition is +independent of the choice of m and agrees with [FK, Definition 2.6]. +Let LGpre +g,ℓ (X, βG) (respectively LGϵ,˜ζ +g,ℓ(X, βG)) be the moduli of genus-g, ℓ-pointed, degree βG prestable (respec- +tively ϵ-stable with respect to ˜ζ) quasimaps to X := (V, G, C∗ +R, W, ζ). More generally, let Y be a Γ-invariant closed +subscheme of V such that Y ∩ V s +G(ζ) = Y ∩ V ss +G (ζ) is non-empty (e.g. Y = Crit(W)), and let Y = (Y, G, C∗ +R, W, ζ). +Let LGpre +g,ℓ (Y, βG) be the closed substack of LGpre +g,ℓ (X, βG) parametrizing Q = ((C, z1, . . . , zℓ), P, ρ, u) such that +u : C → [Y/Γ] ⊂ [V/Γ], and define LGϵ,˜ζ +g,ℓ(Y, βG) similarly. Fan-Jarvis-Ruan proved the following result: +Theorem 4.8 ( [FJR]). LGϵ,˜ζ +g,ℓ(Y, βG) is a separated Deligne-Mumford stack of finite type. It is proper over SpecC +if [(Y ∩ V ss +G (ζ))/G] is. +Given ⃗v = (v1, . . . , vℓ) ∈ Box(Σζ)ℓ, let +LGϵ,˜ζ +g,⃗v(X, βG) := LGϵ,˜ζ +g,ℓ(X, βG) ∩ LGg,⃗v(X, βG). +Then +LGϵ,˜ζ +g,ℓ(X, βG) = +� +⃗v∈Box(Σζ)ℓ +LGϵ,˜ζ +g,⃗v(X, βG). +In Section 4.8 and Section 4.9 below, we fix g,⃗v, ϵ, βG, and let +X = LGϵ,˜ζ +g,⃗v(X, βG), +Z = LGϵ,˜ζ +g,⃗v(Z, βG) +where Z = (Crit(W), G, C∗ +R, W, ζ). By Theorem 4.8, if +Zζ = [(Crit(W) ∩ V ss +G (ζ)) /G] = Crit(wζ) +is proper, then Z is proper. +4.8. Cosection localized virtual cycle and cosection localized virtual structure sheaf. Recall that v ∈ +Box(Σζ) is narrow if Xζ,v is compact. We assume v1, . . . , vℓ are narrow in this subsection. Under this assumption, +Fan-Jarvis-Ruan [FJR] constructed a cosection δ : ObX → OX whose zero locus is Z. Applying [KL13], they obtain +a cosection localized virtual cycle +[X]vir +loc ∈ A∗(Z; Q) +such that +ι∗[X]vir +loc = [X]vir ∈ A∗(X; Q) +where ι : Z → X is the inclusion, and [X]vir is the Behrend-Fantechi virtual fundamental class [BF97] defined by +the perfect obstruction theory described in previous subsections. When Z is proper, [X]vir +loc can be used to define +(cohomological) GLSM invariants in the narrow sector1. The construction of cosection localized virtual cycle in the +narrow sector in [FJR] can be viewed as generalization of Chang-Li-Li’s construction of Witten’s top Chern class +via cosection localization [CLL]. +We now consider the following particularly nice case, as in [BF97, Proposition 5.6]. +Situation 4.9. X is smooth of dimension r0 and ObX is locally free of rank r1. +In Situation 4.9, ObX := tot(ObX) is a vector bundle over X of rank r1, called the obstruction bundle, and the +virtual dimension is r = r0 − r1. Let δ∨ : OX → Ob∨ +X be the dual of the cosection, which is a section of Ob∨ +X. +[X]vir += +cr1(ObX) ∩ [X] = (−1)r1cr1(Ob∨ +X) ∩ [X] ∈ Ar(X; Q), +(4.7) +[X]vir +loc += +(−1)r1cr1(Ob∨ +X, δ∨) ∩ [X] ∈ Ar(Z; Q). +(4.8) +where +• [X] ∈ Ar0(X; Q) is the fundamental class of the smooth DM stack X, +• cr1(Ob∨ +X) ∩ − : Ak(X; Q) → Ak−r1(X; Q) is the top Chern class, and +• cr1(Ob∨ +X, δ∨) ∩ − : Ak(X; Q) → Ak−r1(Z; Q) is the localized top Chern class [Fu98, Chapter 14]. +1In [FJR, Section 6], GLSM correlators are defined for compact type insertions [FJR, Definition 4.1.4] which are more general than +narrow insertions. See [Sh] for subtleties of defining GLSM invariants involving compact type insertions which are not narrow, as well +as an alternative construction of genus-zero compact type GLSM invariants under additional assumptions. +17 + +Let K0(X) (resp. K0(X)) denote the Grothendieck group generated by coherent sheaves (resp. locally free +sheaves) on X with relations [F] = [F ′] + [F ′′] whenever there is a short exact sequence 0 → F ′ → F → F ′′ → 0. +Applying [KL18], one obtains a cosection localized virtual structure sheaf +Ovir +X,loc ∈ K0(Z) +such that +ι∗Ovir +X,loc = Ovir +X ∈ K0(X) +where Ovir +X is the virtual structure sheaf defined in [BF97] (see [BF97, Remark 5.4]) and [Lee]. When Z is proper, +Ovir +loc an be used to define K-theoretic GLSM invariants in the narrow sector. +In Situation 4.9, +Ovir +X = +r1 +� +i=0 +(−1)i ∧i Ob∨ +X = (−1)r1 det(Ob∨ +X) +r1 +� +i=0 +(−1)i ∧i ObX ∈ K0(X). +Note that +td(ObX)ch(Ovir +X ) = cr1(ObX), +so +[X]vir = td(ObX)ch(Ovir +X ) ∩ [X]. +The section δ∨ : OX → Ob∨ defines a Koszul complex +(4.9) +K(δ∨) := Symr1 � +ObX +iδ∨ +→ OX +� += +� +0 → ∧r1ObX +iδ∨ +→ ∧r1−1ObX → · · · → ∧1ObX +iδ∨ +→ OX → 0 +� +which is exact on X − Z. If e1, . . . , ek are local sections of ObX then +iδ∨(e1 ∧ · · · ∧ ek) = +k +� +i=1 +(−1)i−1⟨δ∨, ei⟩e1 ∧ · · · ∧ ei−1 ∧ ei+1 ∧ · · · ∧ ek. +Note that +td(Ob∨ +X)chX +Z (K(δ∨)) = cr1(Ob∨ +X, δ∨), +where chX +Z (K(δ∨))∩ : A∗(X; Q) → A∗(Z; Q) is the localized Chern character [Fu98, Chapter 18]. +4.9. Virtual factorization. Favero-Kim [FK] construct GLSM invariants for general choice of stability in both +narrow and broad sectors via matrix factorization, generalizing constructions in [PV16,CFGKS]. In this subsection +we briefly describe the construction (in slightly different notation). +4.9.1. The Artin stack C. Let ⃗γ = (γ1, . . . , γℓ), where γi ∈ LQ/L corresponds to vi ∈ Box(Σζ). Let B := Bg,⃗γ(βG) +be the universal moduli of Γ-structures defined in Section 4.5. Let PB → CB be the universal principal Γ-bundle +over the universal curve πB : CB → B, let VB = PB ×Γ V , and let C(πB∗VB) be the moduli of sections, which is +an abelian cone over B, as in Section 4.6. Then X is an open substack of C(πB∗VB), and the image of X under the +projection C(πB∗VB) → B (forgetting the section) is contained in a finite type open substack B◦ ⊂ B. Therefore, +X is an open substack of +C := C (πB◦∗VB◦) = B◦ ×B C (πB∗VB) . +For i = 1, . . . , ℓ, there are evaluations maps +(4.10) +evC +i : C −→ Xvi := [V g(vi)/G] +which restricts to +(4.11) +evi : X −→ Xζ,vi = +�� +V g(vi) ∩ V ss +G (ζ) +� +/G +� +⊂ Xvi. +18 + +4.9.2. The smooth Artin stack A and the smooth DM stack U. Over the finite type smooth Artin stack B◦, +RπB◦∗VB◦ admits a global resolution +RπB◦∗VB◦ = [A +dA +−→ B] +where A, B are locally free sheaves of OB◦-modules. +Let πA/B◦ : A := tot(A) → B◦ be the projection, let +tA ∈ Γ(A, π∗ +A/B◦A) be the tautological section, and let βA := +� +π∗ +A/B◦dA +� +◦ tA ∈ Γ(A, π∗ +A/B◦B). The zero locus of +tA is the zero section in A = tot(A), and the zero locus of βA is C. There exists an open substack U ⊂ A such +that U is a DM stack of finite type and the following diagram is a Cartesian square. +X = Z(βU) +ιX +� +jX +� +U +jU +� +C = Z(βA) +ιC +� A +In the above Cartesian diagram, the two vertical arrows are open embeddings, the two horizontal arrows are closed +embeddings, and βU = j∗ +UβA ∈ Γ(U, BU), where BU := j∗ +Uπ∗ +A/B◦B. The virtual dimension r of X is +r = dim B + rankA − rankB = dim U − rankB. +We have +[X]vir = crankB (BU, βU) ∩ [U], +where +• [U] ∈ Ar+rankB(U; Q) is the fundamental class of the smooth DM stack U, +• crankB (BU, βU) ∩ − : Ar+rankB(U) → Ar(X) is the localized top Chern class, and +• [X]vir ∈ Ar(X; Q) is the Behrend-Fantechi virtual fundamental class of X. +4.9.3. Evaluation maps. For i = 1, . . . , ℓ, the evaluation map evC +i +: C → Xvi extends to evA +i +: A → Xvi which +restricts to evU +i : U → Xζ,vi, so that we have the following commutative diagram +X +ιX � +� +U +evU +i � +evU +i � +� +Xζ,vi +� +C +ιC +� A +evA +i +� Xvi +where evU +i ◦ ιX = evi and evA +i ◦ ιC = evC +i , and all the vertical arrows are open embeddings. Let +⃗v = (v1, . . . , vℓ), +X⃗v := +ℓ +� +i=1 +Xvi, +Xζ,⃗v := +ℓ +� +i=1 +Xζ,vi. +A and evA +i are chosen such that +(4.12) +evA := +ℓ +� +i=1 +evA +i : A → X⃗v +is a surjective smooth map between smooth Artin stacks. More explicitly, given any object ξ = ((C, z1, . . . , zℓ), P, ρ) +in B◦(•), which corresponds to a morphism SpecC → B◦, let Aξ := SpecC ×B◦ A be the fiber of A = tot(A) over +ξ. We have the following linear maps between complex vector spaces: +(4.13) +H0(C, P ×Γ V ) +ιC,ξ +−→ Aξ +evU +ξ +−→ +ℓ +� +i=1 +H0(zi, (P ×Γ V )|zi) = +ℓ +� +i=1 +V g(vi) +where ιC,ξ is injective and evU +ξ is surjective. As a consequence, +(4.14) +evU := +ℓ +� +i=1 +evU +i : U → Xζ,⃗v +is a smooth map between smooth DM stacks. +19 + +4.9.4. Superpotentials and matrix factorizations. Let wvi : Xvi = [V g(vi)/G] → C denote the restriction of w : +[V/G] → C. Define a superpotential wA on A: +(4.15) +wA := +ℓ +� +i=1 +(evA +i )∗wvi ∈ Γ(A, OA) +which restricts to a superpotential on U: +(4.16) +wU := j∗ +UwA = +ℓ +� +i=1 +(evU +i )∗wζ,vi ∈ Γ(U, OU). +The sum of residues of a meromorphic 1-form on a curve is zero, so +ι∗ +CwA = 0, +ι∗ +XwU = 0. +When the GLSM X is a convex hypbrid model, it is shown in [CFGKS] that +(a) U can be chosen to be separated over SpecC. +(b) There exists a cosection α∨ +A : π∗ +A/B◦B → OA, or equivalently a section αA : OA → π∗ +A/B◦B∨, such that +⟨αA, βA⟩ = −wA. +(c) Let αU := j∗ +UαA ∈ Γ(U, B∨ +U). Then +Z = Z(αU) ∩ Z(βU) ⊂ X = Z(βU) ⊂ U. +In Section 4.14, we will provide explicit construction of U and α∨ +A satisfying (a)-(c) for the genus-zero one-pointed +moduli spaces used to define the GLSM I-functions for all abelian GLSMs. +When αA exists, one obtains a Koszul matrix factorization {αA, βA} of (A, −wA) defined by +{αA, βA} = +� � +i Λ2iπ∗ +A/B◦B∨ +∂ +� +� +i Λ2i+1π∗ +A/B◦B∨ +∂ +� +� +, +where ∂ = iβA + αA ∧ . +Then +KU := {αU, βU} = j∗ +U{αA, βA} +is a Koszul matrix factorization of (U, −wU). It is called the fundamental factorization in [PV16, CFGKS] and +called the virtual factorization in [FK]. +4.9.5. Localized Chern character and the virtual fundamental class. Let U be a smooth DM stack over C and let +w : U → C be a regular function. In [FK, Section B.4], Favero-Kim define the Atiyah class and the localized Chern +character of a matrix factorization for (U, w), following the construction of Kim-Polishchuk [KP22] when U is a +smooth scheme. Applying the definition to KU, one obtains a localized Chern character +chU +Z KU ∈ Heven +Z +(U, (Ω• +U, −dwU)) . +The virtual fundamental class of X is defined to be +[U]vir +w := +� +ℓ +� +i=1 +ri +� +tdchU +Z KU ∈ Heven +Z +(U, (Ω• +U, −dwU)) . +where ri = ord(g(vi)), or equivalently zi = Bµri. +4.10. Effective classes. Given X = (V, G, C∗ +R, W, ζ), and g, ℓ ∈ Z≥0, define +Keff(X)g,ℓ := {βG ∈ LQ : LGpre +g,ℓ (X, βG) is nonempty}. +We now give a more explicit description of Keff(X)g,ℓ when ℓ = 1. Let ((C, z), P, ρ, u) be an object in LGpre +g,1(X, βG). +Then +u(z) ∈ Xζ = +� +I∈Amin +ζ +XI. +If u(z) ∈ XI then ui(x) ̸= 0 for i ∈ I. We observe that +ui(z) ̸= 0 ⇒ deg Li ≥ 0 and agez(L) = 0 ⇔ deg Li ∈ Z≥0 +20 + +since deg Li − agez(L) ∈ Z. Therefore, +Keff(X)g,1 ⊂ +� +I∈Amin +ζ +Keff +I (X)g,1, +where +(4.17) +Keff +I (X)g,1 = {βG ∈ LQ : ⟨Di, βG⟩ + qi +2 (2g − 1) ∈ Z≥0 for all i ∈ I}. +Indeed, it is not hard to see that +(4.18) +Keff(X)g,1 = +� +I∈Amin +ζ +Keff +I (X)g,1. +Let {D∗I +i +: i ∈ I} be the Q-basis of LQ ∼= Qκ dual to the Q-basis {Di : i ∈ I} of L∨ +Q: for any i, j ∈ I, +⟨Di, D∗I +j ⟩ = δij +Then +Keff +I (Xζ, wζ)g,1 = +� � +i∈I +(mi − qi +2 (2g − 1))D∗I +i +: mi ∈ Z≥0 +� +4.11. Stacky loop spaces. In orbifold quasimap theory [CKK], the Jϵ-function is defined via C∗ localization on +genus-zero ϵ-stable quasimap graph spaces. When the gauge group G = (C∗)κ is an algebraic torus, the small +I-function I = J0+|t=0 can be computed completely explicitly by C∗ localization on stacky loop spaces. Similarly, +we may define the small Jϵ-function of a GLSM via C∗ localization on genus-zero ϵ-stable LG quasimap graph +spaces. When the gauge group G = (C∗)κ is an algebraic torus, the small I-function I = J0+|t=0 of the GLSM +can be computed by C∗ localization on the LG version of stacky loop spaces. In this paper, we define and compute +K-theoretic and cohomological I-functions via C∗ localization on stacky loop spaces defined in this subsection. +4.11.1. Classical version. Given any subset I of {1, . . . , n + κ}, define +VI = {(x1, . . . , xκ+n) ∈ V = Cn+κ | xi ̸= 0 if i ∈ I} = CI′ × (C∗)I +where I′ = {1, . . . , n + κ} \ I. Then +V ss +G (ζ) = +� +I∈Amin +ζ +VI, +and +Xζ = [V ss +G (ζ)/G] = +� +I∈Amin +ζ +XI +where XI = [VI/G] ⊂ Xζ is an open toric substack which is an affine toric orbifold defined by the n-dimensional +cone σI′ spanned by {vi : i ∈ I′}. XI contains a unique torus fixed (possibly stacky) point +pI = +�� +{0}I′ × (C∗)I� � +G +� += +�� +{0} +¯I × {1}I� � +GσI′ +� +∼= BGσI′ +where GσI′ ⊂ G is the stabilizer of the point {0}I′ × {1}I ∈ V (see Section 2.4). Let +NσI′ = +� +i∈I +Zvi. +Then GσI′ is a finite abelian group, and +GσI′ ∼= N/NσI′ , +Hom(GσI′ , C∗) ∼= L∨�� � +i∈I +ZDi +� +. +21 + +4.11.2. Quantum version. We introduce the following convention. Given any rational number m/a, where m ∈ Z, +a ∈ Z>0, and m, a are coprime, we define +Hi(P1, OP1(m +a )) := Hi(P[a, 1], OP[a,1](m)), +i = 0, 1. +Recall that the total space of OP1[a,1](m) is [ +� +(C2 − {0}) × C +� +/C∗], where C∗ acts by weights (a, 1, m). +Given any β ∈ Keff(Xζ, wζ)0,1, we let +(4.19) +Vβ = +n+κ +� +i=1 +Vβ,i, +where Vβ,i = H0 � +P1, OP1(⟨Di, β⟩ − qi +2 ) +� +, +and let +(4.20) +Wβ = +n+κ +� +i=1 +Wβ,i, +where Wβ,i = H1 � +P1, OP1(⟨Di, β⟩ − qi +2 ) +� +. +Let χDi : G → C∗, where 1 ≤ i ≤ n + κ, be defined as in Section 2. Let G act on Vβ,i and Wβ,i by g · u = χDi(g)u +where g ∈ G and u ∈ Vβ,i or Wβ,i. +Definition 4.10 (stacky loop space). We define the degree β stacky loop space by +(4.21) +Xζ,β := [V ss +β (ζ)/G]. +The stacky loop space in the above definition is the analogue of the stacky loop space in orbifold quasimap +theory [CCK, Section 4.2], which can be viewed as the orbifold version of Givental’s toric map space [Gi98, Section +5]. +We have +V ss +β (ζ) = +� +I∈Amin +ζ +Vβ,I +where +Vβ,I = {(u1, . . . , un+κ) ∈ Vβ | ui ̸= 0 if i ∈ I}. +Definition 4.11 (obstruction bundle and obstruction sheaf). We define the degree β obstruction bundle by +Obζ,β = [ +� +V ss +β (ζ) × Wβ +� +/G]. +The degree β obstruction sheaf Obζ,β is the locally free sheaf of OXζ,β-modules on Xζ,β associated to the vector +bundle Obζ,β. +The obstruction bundle Obζ,β is a toric vector bundle over the smooth toric DM stack Xζ,β, and +Obζ,β = Spec +� +SymOb∨ +ζ,β +� +. +Let TXζ,β be the tangent sheaf of Xζ,β. The two-term complex +(4.22) +� +TXζ,β +0 +−→ Obζ,β +� +is a perfect tangent-obstruction complex [LT98] on Xβ,ζ. Taking the dual of (4.22), we obtain +(4.23) +� +Ob∨ +ζ,β +0 +−→ ΩXζ,β +� +which is a perfect obstruction theory [BF97] on Xζ,β. In particular, the tangent-obstruction complex (4.22) and the +perfect obstruction theory are objects in D(Xζ,β), the derived category of coherent sheaves on Xζ,β. The virtual +tangent bundle is +T vir +Xζ,β = TXζ,β − ObXζ,β ∈ K(Xζ,β). +Let TXζ,β/BG be the relative tangent bundle of the smooth map Xζ,β → BG. We have the following short exact +sequence of vector bundles on Xζ,β: +(4.24) +0 → Xζ,β × LC → TXζ,β/BG = [ +� +V ss +β (ζ) × Vβ +� +/G] → TXζ,β → 0. +Given I ∈ Amin +ζ +, σI′ is an n-dimensional cone in Σ. Define +Keff +I (X, ζ)0,1 +−→ +Box(σI′) = +� � +i∈I′ +aivi : ai ∈ [0, 1) ∩ Q +� +β +�→ +v(β) := +� +i∈I′ +{−⟨Di, β⟩ + qi +2 }vi. +22 + +The big torus �T = (C∗)n+κ acts on Vβ by +(�t1, . . . , �tn+κ) · (u1, . . . , un+κ) = (�t1u1, . . . , �tn+κun+κ). +This induces an action of T = �T/G ∼= (C∗)n (the flavor torus) on Xζ,β. The tangent sheaf TXζ,β and the obstruction +sheaf Obζ,β are T-equivariant locally free sheaves Xζ,β, so the the perfect obstruction theory is T-equivariant and +is an object in DT (Xζ,β), the derived category of T-equivariant coherent sheaves on Xζ,β, and T vir +Xζ,β ∈ KT (Xζ,β). +Define +V ◦ +β,I = {(u1, . . . , un+κ) ∈ Vβ | ui(1, 0) ̸= 0 if i ∈ I}. +Then V ◦ +β,I is a Zariski open dense subset of Vβ,I, and +V ss +β (ζ)◦ := +� +I∈Amin +ζ +V ◦ +β,I +is a Zariski dense open subset of V ss +β (ζ). Define +X◦ +ζ,β := [V ss +β (ζ)◦/G] +which is the open substack of Xζ,β. (Our notation X◦ +ζ,β is motivated by Okounkov’s notation in [Ok20], in which +QM◦ denotes the open subtack of QM where ∞ = [1, 0] is not a base point.) +Given I ∈ Amin +ζ +, σI′ is a top-dimensional (i.e. n-dimensional) cone in Σ. Recall that g(v) ∈ GI is the image of +v = � +i∈I′ aivi ∈ Box(σI′) under the bijection Box(σI′) → GσI′ . The g(v)-fixed subspace of V is +V g(v) = {x = (x1, . . . , xn+κ) ∈ V | xi = 0 if i ∈ I′ and ai /∈ Z}. +The connected component Xζ,v of the inertia stack IXζ associated to v is Xζ,v = +�� +V g(v) ∩ V ss(ζ) +� +/G +� +which is an +open dense substack of the Artin stack Xv = [V g(v)/G]. There is an evaluation map +ev∞ : Xζ,β −→ Xv(β) +[u1, . . . , un+κ] �→ [u1(1, 0), . . . , un+κ(1, 0)]. +Then +ev−1 +∞ (Xζ,v(β)) = X◦ +ζ,β +4.12. Torus actions and C∗ +q fixed points. In orbifold quasimap theory, the I-function is obtained by torus +localization on the stacky loop space, using the C∗-action on P1, the coarse moduli space of P[a, 1] where a is a +positive integer. We denote this C∗ by C∗ +q since it corresponds to C× +q in [Ok20]. For �T-equivariant parameters, we +use notation similar to that in [CIJ] and [GiV]. +KC∗q(•) = K(BC∗ +q) = Z[q±1], +K � +T (•) = K(B �T) = Z[Λ±1 +1 , . . . , Λ±1 +n+κ], +Let +z = c1(q) ∈ H2 +C∗ +q(•; Z), +λj = −c1(Λj) ∈ H2 +� +T (•; Z). +Then +H∗ +C∗q(•; Z) = H∗(BC∗ +q; Z) = Z[z], +H∗ +� +T (•; Z) = H∗(B �T; Z) = Z[λ1, . . . , λn+κ], +� +M = +n+κ +� +j=1 +Zλj. +Let deg(x) = a, deg(y) = 1. Then +C[x, y] = +∞ +� +m=0 +C[x, y]m +where C[x, y]m denote the degree m part of the graded ring C[x, y]. If m ∈ Z≥0 then +H0(P[a, 1], OP1[a,1](m)) = C[x, y]m = +⌊ m +a ⌋ +� +k=0 +Cxkym−ka, +H1(P[a, 1], OP1[a,1](m)) = 0. +Let C∗ +q act on P[a, 1] by q · [x, y] = [qx, y] = [x, q−1/ay], and on C[x, y] by q · x = x, q · y = q−1/ay. Given any +number r ∈ Q, let ⌊r⌋ be the unique integer such that ⌊r⌋ ≤ r < ⌊r⌋ + 1, and let {r} := r − ⌊r⌋ ∈ [0, 1). As an +23 + +element in KC∗q(•), +H0(P[a, 1], OP1[a,1](m)) − H1(P[a, 1], OP1[a,1](m)) += +q−{ m +a } +1 − q−1 + q− m +a +1 − q = +∞ +� +k=0 +q−{ m +a }−k − +∞ +� +k=0 +q− m +a −1−k += +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +⌊ m +a ⌋ +� +k=0 +q−{ m +a }−k, +m ≥ 0 +0, +−a ≤ m ≤ −1 +− +−⌊ m +a ⌋−1 +� +k=1 +q−{ m +a }+k, +m ≤ −a − 1 +For j = 1, . . . , n + κ, we define +(4.25) +dj(β) := ⟨Dj, β⟩ − qj +2 ∈ Q +Then +(4.26) +V j +β = +� +� +� +� +� +� +� +⌊dj(β)⌋ +� +k=0 +Λjq−{dj(β)}−k, +dj(β) ≥ 0; +0, +dj(β) < 0 +, +W j +β = +� +� +� +� +� +� +� +0, +dj(β) ≥ −1, +−⌊dj(β)⌋−1 +� +k=1 +Λjq−{dj(β)}+k, +dj(β) < −1. +(V j +β )C∗ +q = +� +C, +dj(β) ∈ Z≥0, +0, +otherwise. +Write +V = +n+κ +� +j=1 +Vj = V + +β ⊕ V − +β ⊕ V ⊥ +β +where Vj = SpecC[xj], +(4.27) +V + +β := +� +j∈[1..n+κ] +dj(β)∈Z≥0 +Vj, +V − +β := +� +j∈[1..n+κ] +dj(β)∈Z<0 +Vj, +V ⊥ +β := +� +j∈[1..n+κ] +dj(β)/∈Z +Vj. +Then +V + +β = V +C∗ +q +β , +V + +β ⊕ V − +β = V g(v(β)). +We define +Fβ := +�� +V + +β ∩ V ss +G (ζ) +�� +G +� +. +Then Fβ is a closed substack of Xζ,v(β). +ev∞ : X◦ +ζ,β → Xζ,v(β) +restricts to an isomorphism +ev∞ : +� +X◦ +ζ,β +�C∗ +q → Fβ. +We identify +� +X◦ +ζ,β +�C∗ +q with Fβ under the above isomorphism. +If I ∈ Amin +ζ +and β ∈ Keff(Xζ, wζ) then the torus fixed point pI is contained in Fβ if and only of β ∈ Keff +I . +4.13. Virtual tangent and normal bundles. Since C∗ +q acts trivially on Fβ, it acts linearly on the fibers of any +C∗ +q-equivariant vector bundle on Fβ. If V is a C∗ +q-equivariant vector bundle over Fβ then +V = +� +d∈Z +Vd = V f ⊕ V m, +where Vd is the subbundle on which C∗ +q acts by weight d, and V m = +� +d̸=0 +Vd (resp. V f = V0) is the moving (resp. +fixed) part of V under the C∗ +q-action. Let +T 1 +β := TXζ,β +�� +Fβ , +T 2 +β := Obζ,β|Fβ . +Then T 1,f +β += TFβ is the tangent bundle of Fβ, and T 1,m +β += NFβ/X◦ +ζ,β is the normal bundle of Fβ in Xζ,β. +24 + +The virtual tangent bundle of Fβ is +T 1,f +β +− T 2,f +β += TFβ − 0 = TFβ. +Therefore, +[Fβ]vir = [Fβ]. +The virtual normal bundle of Fβ is defined to be +(4.28) +N vir +β +:= T 1,m +β +− T 2,m +β +∈ KC∗q× � +T (Fβ). +Let +ιβ→v(β) : Fβ → Xζ,v(β), +ιv : Xζ,v → Xζ, +ιβ = ιv(β) ◦ ιβ→v(β) : Fβ → Xζ +be inclusion maps. +Proposition 4.12. +(4.29) +N vir +β += ι∗ +β→v(β) � +N vir +β +where +(4.30) +� +N vir +β += +n+κ +� +j=1 +ι∗ +v(β)(U +� +T +j )−1� ∞ +� +k=0 +q−dj(β)+k − +∞ +� +k=0 +q{−dj(β)}+k� ++ +� +dj(β)∈Z<0 +ι∗ +v(β)(U +� +T +j )−1 +∈ KC∗q× � +T +� +Xζ,v(β) +� +. +Proof. +T 1,m +β += +� +dj(β)≥0 +ι∗ +β(U +� +T +j )−1� +� +k∈Z +0≤k 0. +Remark 5.3. The formula (5.4) is a multidimensional inverse Mellin transform (see e.g. [Ts]) of Γe2π√−1⟨t,σ⟩, +where the G-character t ∈ L∨ is identified with the line bundle Lt ∈ KG(V ) through the natural isomorphism. +Due to Proposition 5.8 +Remark 5.4. The right hand side of (5.4) converges in the domain +(5.5) +Ut := +� +B ∈ gR | |⟨B + t, ν⟩| < +� +i +|⟨Di, ν⟩|/4 for all ν ∈ gR\{0} +� +. +In particular, this holds if B = −t, so that Ut is non-empty for any character t. This condition is quite restrictive, +in particular none of the branes in K(MF([V/G], w)) in the the mirror quintic example satisfies this condition. +Expansion in phases. Now we turn to the description of the disk partition function component in the phases of +the GLSM. +Let C be a maximal cone of the secondary fan. The set of minimal anticones is defined as: +(5.6) +Amin +C +:= {I ⊂ [1..n + κ] | C ⊂ ∠I, |I| is minimal} +where ∠I is defined as in Section 2.3. The latter condition is equivalent to |I| = κ. To formulate our version of +Higgs-Coulomb Correspondence (Theorem 5.6), we recall some notations from Definition 2.7. For a fixed minimal +anticone I ∈ Amin +C +, the G-characters {Di : i ∈ I} form a Q-basis of L∨ +Q. We define {D∗,I +i +: i ∈ I} to be the dual +Q-basis of LQ. +Definition 5.5 (chamber hemisphere partition function). Let C be a maximal cone of the secondary fan. Define +the chamber hemisphere partition function +(5.7) +ZD2(Lt)C := (−1)κ +� +I∈Amin +C +���� +Λκ +a=1ξ∗ +a +Λκ +a=1Dia +���� +� +m∈(Z≥0)κ +� +i′∈I′ +Γ(⟨Di′, σm⟩ + αi′) +� +i∈I +(−1)mi +mi! +exp(⟨θ + 2π +√ +−1t, σm⟩), +where m = (mi1, . . . , miκ), I = {i1, . . . , iκ}, and σm = − � +i∈I(mi + αi)D∗,I +i +. +Theorem 5.6 (Higgs-Coulomb correspondence). Let UC := � +I∈Amin +C +UI ⊂ C, where UI is defined in Proposi- +tion A.3. UC is open and nonempty and if ζ ∈ UC and B ∈ Ut then we have the equality +(5.8) +ZD2(Lt) = ZD2(Lt)C +Proof. Under the assumptions in the theorem all the integrals and series in (5.8) converge due to Proposition 5.8 +and Proposition A.3. +Now we turn to the proof of the equality of the right hand side and left hand side. Our strategy is to deform the +integration contour in (5.4) while separating the contributions from different torus fixed points. +First we need to introduce some notations to work with contours. Let πi +mi denote a hyperplane ⟨Di, σ⟩+αi = −mi +and +π{i1,...,il} +{mi1,...,mil} := +l� +k=1 +{⟨Dik, σ⟩ + αik = −mik} . +Below we will use the following notations: Ik = {i1, . . . , ik} ⊂ {1, . . . , n + κ} such that Di1, . . . , Dik are linearly +independent if not stated otherwise. Ior +k stands for an ordering (i1, . . . , ik) of Ik. In addition mIk = (mi1, . . . , mik) ∈ +(Z≥0)k. +Given such Ik consider the following exact sequence: +1 → GIk → G +Di1,...,Dik +−→ +HIk ≃ (C∗)k → 1, +where HIk := G/GIk. +We also denote the corresponding Lie algebras by gIk (of dimension κ − k) and hIk +(of dimension k). +Canonically h∨ +Ik = ⟨Di1, . . . , Dik⟩. +Note that πIk +mIk is an affine space parallel to gIk so that +πg→hIk (πi +mi) is a hyperplane in hIk whenever i ∈ Ik. We will denote these hyperplanes by the same symbol. If +k = κ, then πIk +mIk is just a point and GIk = GσI′ +k is finite. +Given a point p ∈ πIk +mIk that is not contained in the other hyperplanes consider an analytic neighbourhood U +that does not intersect other hyperplanes and let [p] and U0 = πg→hIk (U) be their projections to the quotient hIk. +2The condition specifies a chamber in the hyperplane arrangement of pole hyperplanes of the integrand. The fact that such a chamber +is nonempty is not immediate. For the mirror quintic we can choose δ = (�100 +a=1 s1a)cξ101 − c �100 +a=1 ξa, where c is a small positive +number, 0 < 500c < 1. +33 + +The linear independence condition implies that �k +l=1 πilmil is a simple normal crossing divisor for any mIk +with the center at [p], so U0\ �k +l=1 πilmil is homotopically equivalent to (S1)k. +Let CIor +k +0 ([p]) denote a generator +in Hk(U0\ �k +l=1 πilmil ) oriented by the form (2π√−1)−kdDi1/Di1 ∧ . . . ∧ dDik/Dik, that is in the basis of (complex) +linear functions on hIk given by zl = Dil we have +(5.9) +1 +(2π√−1)k +� +C +Ior +k +0 +([p]) +Λk +l=1 +dzl +zl += 1. +Let also CIor +k +0 (p) denote a generator of +Hk(U\ +k� +l=1 +πilmil ) +projecting to CIor +k +0 ([p]). Let Ω ∈ Λκ−k(gIk)∨ +R be a volume form. Such a form defines an orientation of √−1(gIk)R. +This is canonically the Lie algebra of (GIk)comp, the maximal compact subgroup of GIk. +We define a cycle +CIor +k +Ω (p) ∈ Hκ(g\Polar, |ℑ(σ)| ≫ 0) +as a class of CIor +k +0 (p) + √−1(gIk)R oriented by the form (2π√−1)−kΛk +l=1dDil/Dil ∧ Ω. In the definition above Polar +stands for the polar divisor of the integrand, that is �n+κ +i=1 +� +mi≥0 πi +mi. If k = κ, then Ω ∈ R. If Ω = 1 we usually +omit it from the notation. +Remark 5.7. We will be considering integrals of the form +� +C +Ior +k +Ω +(p) dσ Γ · e⟨θ,σ⟩. Such integrals depend only on the +homology class of the cycle in the group +Hκ(g\ +n+κ +� +i=1 +� +m≥0 +πi +mi, |ℑ(σ)| ≫ 0). +In particular, two cycles are homologous if they are related by a smooth homotopy that leaves the set |ℑ(σ)| ≫ 0 +invariant and does not cross polar hyperplanes. +Lemma 5.8. Let CIor +k +Ω (p) be as above, k < κ and +(5.10) +|⟨B, ν⟩| < 1 +4 +n+κ +� +i=1 +⟨Di, ν⟩ for all ν ∈ gIk\{0}. +Then the integral +(5.11) +� +C +Ior +k +Ω +(p) +dσ Γ · e⟨θ,σ⟩ +is absolutely convergent. +In particular it is always convergent if B ∈ g⊥ +Ik. +We show that Γ · e⟨θ,σ⟩ uniformly +exponentially decays as |σ| → ∞ on the integration cycle. +Proof. We present the proof of this statement in Appendix A. +□ +Lemma 5.9 (One plane crossing). Let Ior +k = (Ior +k−1, i) such that gIk−1 ̸= gIk−1∪{i}, Ω be a volume form on (gIk−1)R +and p ∈ πIk +mIk be generic (does not intersect other polar hyperplanes). Pick f ∈ gIk−1\gIk. +(5.12) +C +Ior +k−1 +Ω +(p + εf) = CIor +k +ιf Ω(p) + C +Ior +k−1 +Ω +(p − εf), +where ιfΩ is a κ − k form on (gIk)R and ε small enough. +Proof. Let D(z, r) denote a disk with center at z ∈ C and radius r. Consider a linear coordinate system σ1, . . . , σκ +on g with the center at p such that CIor +k +Ω (r + εf) can be represented by the cycle +(∂D(0, α))k−1 × (ε + +√ +−1R) × ( +√ +−1Rκ−k) +for for some α ∈ R\{0} such that f = ∂/∂σk. Then the form Ω is equal to c dσk ∧ · · · ∧ dσκ, where c ∈ R\{0}. This +cycle is homotopic to the union +(∂D(0, α))k−1 × (−ε + +√ +−1R) × ( +√ +−1Rκ−k) +34 + +oriented by the same form and +(∂D(0, α))k × ( +√ +−1Rκ−k) +oriented by the form c dσk+1 ∧ . . . ∧ dσκ = ιfΩ. The homotopy is nontrivial only in the k-th component, where it +is shown in Figure 1. +ε +−ε +σk +Figure 1. Basic contour deformation +□ +Applying the lemma several times we get the following statement: +Lemma 5.10. Consider Ior +k−1, a volume form Ω on (gIk)R and p ∈ πIk−1 +mIk−1 generic (does not intersect other polar +hyperplanes) and f ∈ gIk−1\ � +i, gIk−1̸=gIk−1∪{i} gIk−1∪{i}. +(5.13) +C +Ior +k−1 +Ω +(p) = − +� +i/∈Ik−1 +� +mi≥0 +CIor +k +ιf Ω({p + [0, N]f} ∩ πi +mi) + C +Ior +k−1 +Ω +(p + Nf), +where ιfΩ is a κ − k form on (gIk)R and the cycle is empty if the intersection is. We note that only i such that +gIk−1 ̸= gIk−1∪{i} enter the right hand side. +Remark 5.11. Later we consider integrals of the form +� +C +dσ Γ · e⟨θ,σ⟩, +C ∈ Hκ(g\Polar, |ℑ(σ)| ≫ 0). Let {Cα}α∈A ⊂ Hκ(g\Polar, |ℑ(σ)| ≫ 0). If A is an infinite set, then by abuse of +notations when we write an expression of the form +(5.14) +� +α∈A +Cα = C +we actually mean +(5.15) +� +α∈A +� +Cα +dσ Γ · e⟨θ,σ⟩ = +� +C +dσ Γ · e⟨θ,σ⟩, +when the left hand side converges. We use this notation to simplify already bulky notations. +Corollary 5.12. Let p, Ω, N be as in Lemma 5.10 then there exist a constant const that does not depend on p and +f such that if ⟨ζ, f⟩ < const then +(5.16) +C +Ior +k−1 +Ω +(p) = − +� +i/∈Ik−1 +� +mi≥0 +CIor +k +ιf Ω({p + R≥0f} ∩ πi +mi). +Proof. Lemma 5.10 implies +(5.17) +� +C +Ior +k−1 +Ω +(p) +dσ Γ · e⟨θ,σ⟩ = − +� +i/∈Ik−1 +� +mi≥0 +� +C +Ior +k +ιf Ω({p+[0,N]f}∩πimi) +dσ Γ · e⟨θ,σ⟩ + +� +C +Ior +k−1 +Ω +(p+Nf) +dσ Γ · e⟨θ,σ⟩. +Using Proposition A.5 with p → p + Nf we can estimate the last term on the right hand side: +(5.18) +� +C +Ior +k−1 +Ω +(p+Nf) +dσ Γ · e⟨θ,σ⟩ ≤ e−const|p+Nf|, +35 + +In particular, the integral approaches 0 as N → ∞. Now the corollary follows from taking the limit N → ∞. +□ +Cdx∧dy(p) +C(1) +f ⊥(q) +C(2,1) +1 +((−4, −4)) +C(1,2) +1 +((−4, −1)) +gR +Figure 2. Contour deformations in projection to gR. +Figure 2 shows two consecutive applications of Corollary 5.12 in a simple example. The plane is a real part +gR of the two-dimensional Lie algebra g ≃ C2. +The picture shows only 2 series of polar divisors located at +x = −1, −2, −3, . . . and y = −1, −2, −3, . . .. The initial contour Cdx∧dy(p) is a purely imaginary contour depicted +by a green dot. +Its projection to the real Lie algebra is just a point. +Direction of deformation f is a dashed +green line. After first application of Corollary 5.12 the contour transforms into two series of copies of R × S1 +denoted by purple intervals. Indeed, projections of these contours to gR are intervals intersecting the corresponding +polar hyperplanes. Each of this contour is deformed parallel to the respective polar hyperplane applying Corol- +lary 5.12 again. The contours deform to a number of T 2 ≃ (S1)2 winding around each crossing of polar hyperplanes. +Let us introduce some more notations for cones of different dimensions in g. Given I such that {Di}i∈I form a +basis of g∨ consider the cone +(5.19) +∠∗,ζ +I += { +� +i∈I +sign(⟨ζ, D∗,I +i +⟩)ciD∗,I +i +| ci ≥ 0}, +and its subcones: +(5.20) +∠∗,ζ +J⊂I = ∠∗,ζ +I +∩ gJ. +The signs are chosen so that ⟨ζ, σ⟩ ≥ 0 for any σ ∈ ∠∗,ζ +J⊂I. We also define +signζ +J⊂I = (−1)#{i∈I\J | ⟨ζ,D∗,I +i +⟩<0}. +In particular, if I is a minimal anticone (which is the main case of interest) then ∠∗,ζ +I += ∠∗ +I, where ∠∗ +I ⊂ gR is the +dual cone of ∠I ⊂ g∨ +R, and signζ +J⊂I = 1 for all J ⊂ I. We also have ∠∗,ζ +∅⊂I = ∠∗,ζ +I +and ∠∗,ζ +I⊂I = 0. +Further recall that +Keff +I += +� � +i∈I +miD∗,I +i +��� mi ∈ Z≥0 +� +and define +Keff +I (α) := − +� +i∈I +αiD∗,I +i +− Keff +I += +� +− +� +i∈I +(αi + mi)D∗,I +i +��� mi ∈ Z≥0 +� +. +36 + +If αi = qi/2 then Keff +I (α) = −Keff +I (Xζ, wζ). If p ∈ πIk +mIk then +p−∠∗ζ +Ik⊂I = +� +− +� +i∈Ik +(αi+mi)D∗,I +i ++ +� +i∈I\Ik +(⟨Di, p⟩−ci)D∗,I +i +��� ci ∈ R, ci ≥ 0 if ⟨ζ, D∗,I +i +⟩ > 0 and ci ≤ 0 if ⟨ζ, D∗,I +i +⟩ < 0 +� +. +The following lemma is the main technical lemma of the proof. It relates integrals over (S1)k × Rκ−k to sums +of cycles over (S1)κ in some κ − k-dimensional cone by recursively applying Corollary 5.12 and checking that only +the contributions in the corresponding cone survive. +Lemma 5.13. Consider Ior +k , a volume form Ω on (gIk)R and p ∈ πIk +mIk generic (does not intersect other polar +hyperplanes). Then +(5.21) +CIor +k +Ω (p) = (−1)κ−k � +I,Ik⊂I +signζ +Ik⊂I +� +q∈(p−∠∗,ζ +Ik⊂I)∩Keff +I (α) +sign +�Λk +l=1Dil ∧ Ω +Λκ +l=1Dil +� +CIor(q). +We note that we can describe the set (p − ∠∗,ζ +Ik⊂I) ∩ Keff +I (α) explicitly as follows. Let −ni′ := ⌊⟨Di′, p⟩ + αi′⌋ ∈ Z for +i′ ∈ I\Ik. Then +(p − ∠∗,ζ +Ik⊂I) ∩ Keff +I (α) = +� +− +� +i∈I +(αi + mi)D∗,I +i +��� i ∈ I\Ik =⇒ mi ∈ Z and +� +ni ≤ mi +if ⟨ζ, D∗,I +i +⟩ > 0, +0 ≤ mi < ni +if ⟨ζ, D∗,I +i +⟩ < 0. +� +. +Proof. The right hand side is convergent due to Proposition A.3 since the series in the right hand side is a subseries +of the one considered in the proposition. +We prove the lemma by recursion in k. First, let k = κ − 1. We apply Corollary 5.12 where f ∈ (gIκ−1)R ≃ R +such that ⟨ζ, f⟩ < 0. Ω ∈ (gIκ−1)∨ +R ≃ g∨ +R/ +� +⊕i∈Iκ−1RDi +� +, so that for each i /∈ Iκ−1 one can write Ω = ciDi (where +right hand side is a representative). In particular, ιfΩ = ci⟨Di, f⟩, where +ci = Λκ−1 +l=1 Dil ∧ Ω +Λκ−1 +l=1 Dil ∧ Di +. +Furthermore, sign(⟨Di, f⟩) = sign(⟨ζ, D∗,I +i +⟩) = signζ +Iκ−1⊂I. So we have +(5.22) +C +Ior +κ−1 +Ω +(p) = − +� +i/∈Iκ−1 +� +mi≥0 +C(Ior,i) +ci⟨Di,f⟩({p + R≥0f} ∩ πi +mi). +Thus the theorem in this case follows. +We remark that the conditions in Corollary 5.12 are stricter than the convergence condition of the theorem, but +the statement is about equality of the analytic functions is some domain, so it extends to the maximal domain of +the mutual convergence. +Consider the general case 0 ≤ k < κ − 1. We choose a generic f ∈ (gIk)R ≃ Rκ−k such that ⟨ζ, f⟩ < 0 and apply +Corollary 5.12 +(5.23) +CIor +k +Ω (p) = − +� +i/∈Ik +� +mi≥0 +C(Ior +k ,i) +ιf Ω +({p + [0, N]f} ∩ πi +mi) + CIor +k +Ω (p + Nf). +The sum over mi ∈ Z restricts to a set +(5.24) +� +ni ≤ mi, ⟨Di, f⟩ > 0, +0 ≤ mi < ni, ⟨Di, f⟩ < 0, +where again −ni = ⌊⟨Di, p⟩ + αi⌋. In particular, if I is a minimal anticone then the second case does not appear. +Now we can use the recursion step for each term in the double sum on the right hand side. +(5.25) +C(Ior +k ,i) +ιf Ω +(˜p = {p + [0, N]f} ∩ πi +mi) = +� +I,Ik∪{i}⊂I +signζ +Ik∪{i}⊂I +� +q∈(˜p−∠∗,ζ +Ik∪{i}⊂I)∩Keff +I (α) +sign +�Λk +l=1Dil ∧ Di ∧ ιfΩ +Λκ +l=1Dil +� +CIor(q) +Let us analyze when q ∈ Keff +I (α) appears in the expansion of the formula (5.32) after applying (5.25). Such a +term appears when there exist i ∈ I′ +k, mi ∈ Z such that +(5.26) +q ∈ {p + [0, N]f} ∩ πi +mi − ∠∗,ζ +Ik∪{i}⊂I. +37 + +But � +i∈I′ +k ∠∗,ζ +Ik∪{i}⊂I = ∂∠∗,ζ +Ik⊂I. So we can rewrite equation (5.26) as +q ∈ p + cf − ∂∠∗,ζ +Ik⊂I +for some c > 0, or +(5.27) +q − cf ∈ p − ∂∠∗,ζ +Ik⊂I. +In other words, for each intersection of the ray q − R≥0f with the boundary of the cone p − ∠∗,ζ +Ik⊂I we have a +term with the cycle centered at q in the expansion of (5.32). Number of such intersections depends on whether +q ∈ p − ∠∗,ζ +Ik⊂I or not. +(1) q ∈ p − ∠∗,ζ +Ik⊂I. Scalar product of ζ with all the inward normals to ∂∠∗,ζ +Ik is by definition positive, f is a +linear combination of these normals, and ⟨ζ, f⟩ < 0. Then at least one coefficient of the linear combination +is negative, and q − R≥0f has one intersection point with p − ∂∠∗,ζ +Ik⊂I. The total contribution to the sum is +(5.28) +signζ +Ik∪{i}⊂Isign +�Λk +l=1Dil ∧ Di ∧ ιfΩ +Λκ +l=1Dil +� +CIor(q), +where +Ω = cDi ∧ Λκ +l=k+2Dil + · · · +and +ιfΩ = c⟨Di, f⟩Λκ +l=k+2Dil + · · · , +where dots denote the possible terms that vanish in the formula (5.28). Plugging this expression for the +sign, we get the equality +(5.29) +signζ +Ik∪{i}⊂Isign +�Λk +l=1Dil ∧ Di ∧ ιfΩ +Λκ +l=1Dil +� += signζ +Ik⊂Isign +�Λk +l=1Dil ∧ Ω +Λκ +l=1Dil +� +. +(2) q /∈ p − ∠∗,ζ +Ik⊂I. There are either 0 or two intersection points because the scalar product of f with at least +one inward normal to ∂∠∗,ζ +Ik⊂I is negative. The total contribution to the formula (5.32) is +(5.30) +� +signζ +Ik∪{i}⊂Isign +�Λk +l=1Dil ∧ Di ∧ ιfΩ +Λκ +l=1Dil +� ++ signζ +Ik∪{j}⊂Isign +�Λk +l=1Dil ∧ Dj ∧ ιfΩ +Λκ +l=1Dil +�� +CIor(q). +Here we can write +Ω = cDi ∧ Dj ∧ Λκ +l=k+3Dil + · · · , +where dots denote irrelevant terms again. In the first summand we have +ιfΩ = c⟨Di, f⟩Dj ∧ Λκ +l=k+3Dil + · · · , +and in the second one +ιfΩ = −c⟨Dj, f⟩Di ∧ Λκ +l=k+3Dil + · · · . +So the total coefficient is +(5.31) +signζ +Ik∪{i}sign(c⟨Dj, f⟩) + signζ +Ik∪{j}sign(c⟨Di, f⟩) = += signζ +Ik +� +sign(c⟨ζ, D∗I +j ⟩⟨Dj, f⟩) + sign(c⟨ζ, D∗,I +i +⟩⟨Di, f⟩) +� +. +In the last formula ⟨ζ, D∗,I +i +⟩Di and ⟨ζ, D∗,I +j ⟩Dj are inward normals of the cone ∠∗,ζ +Ik⊂I. However, we know that +the ray q − R≥0f intersects the cone p − ∠∗,ζ +Ik⊂I with inward pointing first intersection and outward pointing +second one, so the signs in the second line of the formula (5.31) are different and the total contribution +vanishes. +Collecting contributrions from all the points q ∈ Keff +I (α) for all I using the statement above we obtain the +statement of the lemma. The picture is shown in our toy example (we suppose that ⟨ζ, D∗{1,2} +1 +⟩, ⟨ζ, D∗{1,2} +2 +⟩ > 0) +in Figure 2. +□ +Using Lemma 5.13 in the case when CIor +Ω (p) = Cdσ(δ) (where we remind that our choice is dσ = Λκ +a=1ξ∗ +a) we write +(5.32) +� +Cdσ(δ) +dσ Γ · e⟨θ,σ⟩ = = (−1)κ +� +I∈Amin +C +� +q∈Keff +I (α) +sign +� Λκ +a=1ξ∗ +a +Λκ +a=1Dia +� � +CIor(q) +dσ Γ · e⟨θ,σ⟩. +Let us compute each term in the right hand side. +38 + +Lemma 5.14. +(5.33) +� +CIor(q) +dσ Γ · e⟨θ,σ⟩ = (2π +√ +−1)κ Λκ +a=1ξ∗ +a +Λκ +a=1Dia +� +i′∈I′ +Γ(⟨Di′, σm⟩ + αi′) +� +i∈I +(−1)mi +mi! +e⟨θ,σm⟩. +Proof. Each integral is a κ-dimensional torus around a simple normal crossing divisor and q = − � +i∈I(αi+mi)D∗,I +i +. +Let {σI +i }i∈I denote a set of coordinates on g corresponding to linear functions {Di}i∈I on g. Then we have +(5.34) +� +CIor(q) +dσ Γ · e⟨θ,σ⟩ = (2π +√ +−1)κ Res +CIor(q) dσ Γ · e⟨θ,σ⟩. +Next we use +(5.35) +dσ = Λκ +a=1ξ∗ +a +Λκ +a=1Dia +dσI +so that +(5.36) +Res +CIor(q) dσ Γ · e⟨θ,σ⟩ = Λκ +a=1ξ∗ +a +Λκ∗a +a=1Dia +κ +� +a=1 +Res +σI +ia→−αia−mia +dσI +ia Γ · e⟨θ,σ⟩ = += Λκ +a=1ξ∗ +a +Λκ∗a +a=1Dia +� +i′∈I′ +Γ(⟨Di′, σm⟩ + αi′) +� +i∈I +(−1)mi +mi! +e⟨θ,σm⟩, +where σm = q. +□ +Substituting this expression into (5.32) we prove the proposition. +□ +Corollary 5.15. Power series in (5.8) for different stability chambers are analytic continuations of each other in the +parameter ζ = ℜ(θ). The analytic continuation is through the integral representation of the disk partition function. +This analytic continuation can be understood as a particular case of the Crepant Transformation Conjecture [Ru99, +Ru02,Ru06] for toric Deligne-Mumford stacks proved in [CIJ]. +5.3. Disk partition functions and wall-crossing. We defined the chamber disk partition function components +corresponding to the phases of the GLSM (cones of the secondary fan). The wall-crossing is given by the analytic +continuation formula (5.4). +We would like to extend it to wall-crossing for arbitrary branes. +Let B = � +t ctLt ∈ K([V/G]). +Then for +different characters t1 and t2 the Higgs-Coulomb correspondence (5.8) hold for different domains B ∈ Ut1 and +B ∈ Ut2 correspondingly. It could happen that Ut1 ∩ Ut2 = ∅. Then we need another description for analytic +continuation. This description is given in Definition 5.18 for general one wall crossing. +First of all, we need to explain the wall-crossing setup that we use. We follow the one stated, for example, +in [BH2]. +Consider two maximal cones C+, C− of the secondary fan which are adjacent along a codimension one wall +h⊥ ⊂ g∨. Let h := (h⊥)⊥ ⊂ g and pick its integral generator h ∈ L, h = Ch. Also define hi := ⟨Di, h⟩. Then +�n+κ +i=1 hi = 0 since �n+κ +i=1 Di = 0 (Calabi-Yau condition). +In the terminology of the reference [BH2] h = (h1, . . . , hn+κ) defines the circuit that is associated to the wall- +crossing. We define +I± := {i | ± hi > 0}. +The circuit itself is {vi : i ∈ I+ ∪ I−}. +Let I0 ⊂ (I+ ∪ I−)′. If I = I0 ⊔ {i0} is a minimal anticone of C± where i0 ∈ I±, then I0 ⊔ {i±} is an anticone of +C± for each i± ∈ I±. Modification along the circuit is obtained by replacing all the cones of the form I0 ⊔ {i+} by +the anticones I0 ⊔ {i−}. We denote by A0 +C0 the set of all such I0. +Following [BH2] we call all the anticones of this form essential anticones. We have +Amin +C± = Aess +C± ⊔ Anoness +C± +. +Lemma 5.16. Nonessential anticones always contain at least one element from both I+ and I−. +Proof. Any minimal anticone must contain at least one number from I+ ∪ I− for dimensional reasons. If it does +not contain numbers from I−, then it is essential if it contains exactly one number from I+ and if it contains more +than 1, then the cones C± are adjacent along the wall of higher codimension. +□ +39 + +Remark 5.17. We use anticones instead of the cones because they are better adjusted to partition functions even +though they are completely interchangeable. Each minimal anticone corresponds to a torus fixed point of the corre- +sponding toric variety. +Definition 5.18 (Wall hemisphere partition function). Let C0 = C+ ∩ C− be a codimension one cone of the +secondary fan and +(5.37) +|⟨B + t, h⟩| < +� +i +|⟨Di, h⟩|/4. +For each Iκ−1 ∈ A0 +C0 and mIκ−1 ∈ (Z≥0)κ−1 we choose p(mIκ−1) ∈ πIκ−1 +mIκ−1 generic with the property that cardinality +of (p(m) ∓ R≥0h) ∩ Keff +Iκ−1∪{i±}(α) is bounded by c|m| for all i± ∈ I± (this property is satisfied if, for example, all +p(m) are on the same hyperplane) 3. +Define the wall hemisphere partition function corresponding to the wall C0: +(5.38) +ZD2(Lt)C0 := +� +Iκ−1∈A0 +C0 +Zess +Iκ−1(Lt) + +� +I∈Anoness +C± +ZI(Lt), +where +(5.39) +ZI(Lt) = +� +q∈Keff +I (α) +sign +� +dσ +Λκ +a=1Dia +� � +CIor(q) +dσ Γ · e⟨θ+2π√−1t,σ⟩, +I ∈ Anoness +C± +. +and +(5.40) +Zess +J (Lt) = +� +m∈(Z≥0)κ−1 +� +−sign +� +dσ +Λκ−1 +a=1 Dja ∧ Ω +� � +CJor +Ω +(p(m)) +dσ Γ · e⟨θ+2π√−1t,σ⟩+ ++ +� +i±∈I+∪I− +� +q∈(p(m)∓R≥0h)∩Keff +J∪{i±}(α) +sign +� +dσ +Λκ +a=1Dia +� � +CIor(q) +dσ Γ · e⟨θ+2π√−1t,σ⟩ +� +, +where Ω is any volume form in (gJ)R. Note that in (5.40) the sum in the second sum is finite. +Remark 5.19. There seems to be an infinite number of choices of points p(m) in the definition of the wall hemi- +sphere partition function. This is due to the fact that in the general case the cones in g corresponding to essential +anticones in the adjacent phases intersect and one cannot split the corresponding effective classes uniformly. As we +will see below the definition is independent of these choices. +Remark 5.20 (Grade restriction rule). The formula (5.37) is called the grade restriction rule since it puts a +restriction of the character t (grading of the brane). Disk partition function components can be analytically continued +directly along the walls of the secondary fan if the grade restriction rule is satisfied as is stated in the following +theorem. +Theorem 5.21 (Wall-crossing). In the setting of Definition 5.18 there exists a connected open set UC0 ⊂ gR such +that UC0 ∩ UC+ and UC0 ∩ UC− are nonempty and the wall hemisphere partition function converges for all ζ ∈ UC0, +B satisfies (5.37) and for all ζ ∈ UC0 ∩ UC± it is equal to the chamber hemisphere partition function +(5.41) +ZD2(Lt)C0 = ZD2(Lt)C± +In particular, the wall hemisphere partition function does not depend on the choices in the definition. +Proof. The convergence of the non-essential part of the right hand side is established in Proposition A.3. +Consider the essential part. Let ζ ∈ C+ ∪ C−. Then for each J we can write ζ = � +j∈J ζJ +j Dj, where ζJ +j > 0. +Note that all Dj are in the wall hyperplane (Rh)⊥, so ⟨Dj, h⟩ = 0 and for any c1 > 0 we can choose ζ far enough in +the interior of C+ ∪ C− such that ⟨ζ, p(m)/|p(m)|⟩ < −c1 for all m. Moreover, the same inequality holds for all q +in (5.40). Then we can use Proposition A.5 to estimate the integral summands and the residue summands in (5.40) +by e−c2|p(m)| and e−c2|q| respectively. The number of summands for each m is bounded by c3|m| because condition +q ∈ (p(m) ∓ R≥0h) ∩ Keff +J ⊔ {i±} is linear in m if p(m) are chosen as in the definition 5.18. Therefore, The essential +summand Zess +J (t) is bounded by � +mJ∈(Z≥0)κ−1 c3|mJ|e−c2dist(πJ +mJ ,0) < ∞. Moreover, the convergence is an open +condition on ζ, so it must hold in an open neighbourhood This finishes the proof of convergence. +3This property is rather technical and can be made even weaker. The meaning of this property is that p(m) must not go to infinity in +the direction of ±h too fast as |m| → ∞ +40 + +Consider an essential integral term in (5.40). If ζ ∈ C+ we can apply Corollary 5.12: +(5.42) +CJor +Ω (p(m)) = − +� +i±∈I± +� +mi±≥0 +C(Jor,i±) +ιhΩ +({p(m) + R≥0h} ∩ πi± +mi± ). +By slight abuse of notation we can write Ω = ci±Di± ∈ (gJ)∗ +R, so ιhΩ = ±ci±. So +(5.43) +sign +� +dσ +Λκ−1 +a=1 Dja ∧ Ω +� += sign +� +ci± +dσ +Λκ−1 +a=1 Dja ∧ Di± +� +, +and therefore +(5.44) +− sign +� +dσ +Λκ−1 +a=1 Dja ∧ Ω +� +CJor +Ω (p(m)) = ± +� +i± +� +q∈(p(m)+R≥0h)∩Keff +J∪{i±}(α) +sign +� +dσ +Λκ−1 +a=1 Dja ∧ Di± +� +C(Jor,i±)(q). +Combining this with the second term from (5.40) we obtain: +(5.45) +Zess +J (Lt) = +� +i+ +ZJ∪{i+}(Lt). +In the same way if ζ ∈ C− we compute +(5.46) +Zess +J (Lt) = +� +i− +ZJ∪{i−}(Lt). +The theorem is proved. 4 +□ +Appendix A. Convergence of multivariate hypergeometric functions +This is mostly known due to many people: [Ho] or, in the more systematic exposition [?], [?] and others. The +authors were not able to find some of the results, so we provide a short overview of the subject here. We use the +notations of the main part of the paper, particularly Section 5. +Let θ = ζ + 2π√−1B and σ = τ + √−1ν represent complex variables on g∨ and g correspondingly and Di ∈ +L∨, i ≤ n + κ be a collection of the vectors that spans g∨ over C such that �n+κ +i=1 Di = 0 (Calabi-Yau condition). +Let α ∈ Cn+κ be a generic vector and +(A.1) +Γ = Γ(σ) = +n+κ +� +i=1 +Γ(⟨Di, σ⟩ + αi). +We remind thatthe secondary fan Σ2 is defined as a fan with Σ2(1) = {Di}n+κ +i=1 and whose maximal cones are all +possible intersections of ∠I of dimension κ. +First of all we need some basic results about the gamma function. The Stirling approximation: +(A.2) +Γ(x + iy) = (2π)1/2zz−1/2e−z(1 + O( 1 +|z|)), +|Arg(z)| < π − α +where α > 0, and the Landau notation is +f(z) = O( 1 +|z|) is equivalent to lim sup +|z|→∞ +|z||f(z)| < ∞ +In particular, +(A.3) +|Γ(x + iy)| = (2π)1/2|z|x−1/2e−xe−yArg(z)(1 + O( 1 +|z|)), +|Arg(z)| < π − α. +Now let Sδ = R\ �∞ +n=0(−n − δ, −n + δ) be a domain in R separated from poles of the gamma function by a small +positive constant δ. +4We remark about the convergence again, the condition in Corollary 5.12 is stricter than in the theorem, but equality of analytic +functions extends to the maximal domain where both converge. +41 + +Lemma A.1. Let z = x + √−1y and x ∈ Sδ. Then +(A.4) +Γ(x) < const · |x|x− 1 +2 e−x. +(A.5) +|Γ(z)| < const · |z|x− 1 +2 e−min(x,0)e− π|y| +2 . +In addition, if x ∈ Sδ ∩ K, where K is a compact set, then +(A.6) +|Γ(z)| < const(|y| + 1)x−1/2e− π|y| +2 . +Proof. By Equation (A.3), there exist positive constants c1, c2 such that +(A.7) +c1|z|x− 1 +2 e−xe−yArg(z) < |Γ(z)| < c2|z|x− 1 +2 e−xe−yArg(z), +x > δ. +If x > 0, then x > δ since x ∈ Sδ. We can write Arg(z) = Arctan(y/x) = sign(y)π/2 − Arctan(x/y), so we have +−yArg(z) = −y(sign(y)π/2 − Arctan(x/y)) < π|y| +2 ++ x, +since Arctan(x/y) < x/y. Therefore, +(A.8) +|Γ(z)| < c2|z|x− 1 +2 e− π|y| +2 , +x > δ. +If x < 0, then | sin(πx)| > sin δ since x ∈ Sδ. We use the reflection formula and (A.7): +(A.9) +|Γ(z)| = +π +|Γ(1 − z) sin(πz)| < π +c1 +|1 − z|−(1−x)+1/2e1−xeyArg(1−z)| sin(πz)|−1, +x < 0. +If x < 0 then |1 − z| > |z|, |1 − z|x−1/2 < |z|x−1/2, and |Arg(1 − z)| < π +2 , so +(A.10) +|Γ(z)| < πe +c1 +|z|x−1/2e−xe +π|y| +2 | sin(πz)|−1. +The first formula (A.4) follows from (A.7) and (A.10) for y = 0. +We have sin(πz) = sin(π(x + √−1y)) = sin(πx) cosh(πy) + √−1 cos(πx) sinh(πy), +| sin(πz)| ≥ | sin(πx) cosh(πy)| > sin(πδ)eπy + e−πy +2 +> 1 +2 sin(πδ)eπ|y|. +Finally, we use this to simplify (A.10): +(A.11) +|Γ(z)| < const · |z|x−1/2e−xe +−π|y| +2 +. +Collecting the formulas (A.8) and (A.11) we get the inequality (A.5): +(A.12) +|Γ(z)| ≤ const · |z|x−1/2e− min(x,0)e−π|y|/2, +x ∈ Sδ, +where the constant depends on δ. +The third formula (A.6) follows from (A.12) and (|y| + const)x ∼ |y|x, y → ∞. +□ +We also recall the multivariate Cauchy-Hadamard theorem. +Theorem A.2 (Cauchy-Hadamard). Let +(A.13) +� +m∈(Z≥0)κ +cmzm. +The series (absolutely) converges in the polydisk with the multiradii r = (r1, . . . , rκ) if +(A.14) +lim +N→∞ supm,|m|=N(cmrm)1/|m| ≤ 1 +and such polydisk is maximal if the left hand side is equal to the right hand side. +Proposition A.3. +(1) The domain of convergence of +(A.15) +ZI = +� +m∈(Z≥0)κ +� +i′∈I′ +Γ(⟨Di′, σm⟩ + αi′) +� +i∈I +(−1)mi +mi! +exp(⟨θ, σm⟩), +is non-empty. Moreover, if the series converges at ζ0, then it also converges if ζ ∈ {ζ0} − ∠I. +42 + +(2) The domain of convergence of the series is UI + √−1Rκ, where UI ⊂ Rκ is defined by the constraints +(A.16) +⟨ζ + log Ψ(σ), σ⟩ = +� +i∈I +(⟨ζ, D∗,I +i +⟩ + log Ψi(σ))σi > 0, σ ∈ ∠∗ +I, +where Ψ(σ) = (Ψ1(σ), . . . , Ψκ(σ)) is the Horn vector defined below in the proof. +Proof. In this proof we work in the basis {D∗,I +i +}i∈I on g and the corresponding coordinate system on g and g∨. +That is if f ∈ g and f ∨ ∈ g∨, we write f = � +i fiD∗,I +i +, f ∗ = � +i f ∗ +i1Di, where fi := ⟨Di, f⟩ and f ∗ +i := ⟨f ∗, D∗I +i ⟩. +Let sI +i′i := ⟨Di′, D∗I +i ⟩. Consider the asymptotics of the series (A.15). We write the series as +(A.17) ZI = +� +m∈(Z≥0)κ +� +i′∈I′ +Γ(− +� +i +sI +i′i(mi + αi) + αi′) +� +i∈I +(−1)mi +mi! +exp(⟨θ, σm⟩) = exp(− +� +i +θiαi) +� +m∈(Z≥0)κ +cmzm, +where zm = �κ +i=1 zmi +i += � +i exp(−miθi). For generic α arguments of the gamma functions are uniformly separated +from Z≤0 by a number δ > 0 because sI +i′i are rational numbers. +Therefore we can apply Lemma A.1 and write the upper bound on the series expansion coefficients: +(A.18) +|cm| ≤ const · +� +i′∈I′(sI +i′m + ci′)(sI +i′m+ci′−1/2) +� +i∈I mmi−1/2 +i +, +where sI +i′m = +� +i∈I +sI +i′imi and ci′ = +� +i∈I +sI +i′iαi + αi′. +Thus we can write +(A.19) +lim +N→∞ +sup +m,|m|≥N +(cmrm)1/|m| ≤ +≤ lim +N→∞ +sup +m,|m|≥N +� +const +� +i′∈I′ +(sI +i′m + ci′)ci′−1/2 � +i +m−1/2 +i +· exp +� +−⟨ζ, m⟩ − +� +i′∈I′ +sI +i′m log(sI +i′m + ci′) − +� +i +mi log mi +��1/|m| +, +where |m| denotes any norm on Rn. +We define the Horn vector Ψ(σ) := � +i Ψi(σ)Di, where +(A.20) +Ψi(σ) := +σi +� +i′∈I′(sI +i′σ)sI +i′i . +In particular, log Ψ(σ) = � +i log Ψi(σ)Di, where log Ψi(σ) = log(σi) − � +i′∈I′ sI +i′i log(sI +i′σ). +We note that even +though log Ψi(σ) is not defined if any of σi, sI +i′σ = 0, the sum � +i σi log Ψi(σ) = ⟨log Ψ(σ), σ⟩ can be defined as a +limit. +Under the assumption of the proposition ζ = ℜ(θ) satisfies the equation +(A.21) +⟨ζ + log Ψ(σ), σ⟩ > 0 +for all σ ∈ Rκ\{0}, so we have +(A.22) +− ⟨ζ, m⟩ − +� +i′∈I′ +sI +i′m log(sI +i′m + ci′) − +� +i∈I +mi log mi ≤ +≤ −⟨ζ, m⟩ − +� +i′∈I′ +sI +i′m log(sI +i′m + ci′) − +� +i∈I +mi log mi + ⟨ζ + log Ψ(m), m⟩ = − +� +i′∈I′ +sI +i′m log(1 + ci′/(sI +i′m)), +where by slight abuse of notation we identify m = � +i∈I miD∗,I +i +. Consider the function x log(1 + c/x). As x → 0 we +have x log(1 + c/x) ∼ x log(c/x) → 0, and when x → ∞ then log(1 + c/x) = c/x + O(1/x2), so x log(1 + c/x) → c. +Therefore the last expression in (A.22) is bounded from above and below, so we can write +(A.23) +lim +N→∞ +sup +m,|m|≥N +(cmrm)1/|m| ≤ lim +N→∞ supm,|m|≥N exp(const +� +i′∈I′ +(sI +i′m + ci′)ci′−1/2 � +i∈I +m−1/2 +i +)1/|m| ≤ 1. +so the series (A.15) converges by the multivariate Cauchy-Hadamard theorem. If for some σ the inequality (A.21) +is not satisfied, then it is not satisfied for all m proportional to σ and the limit in (A.23) is greater than 1. This +proves the second claim of the proposition. +� +i Ψi(σ)σi is bounded and separated from 0 on the unit sphere, so � +i log Ψi(σ) is bounded on the same domain +below by a constant −N. First claim of the proposition is proved by choosing ζ = N � +i D∗,I +i +. +□ +43 + +Notation A.4. Below we work with estimates including many constants whose exact values are of no importance +and can be rather cumbersome. Notations const, consti denote various such constants. +Below we present the proof of Lemma 5.8 and Corollary 5.12. +Proof of Lemma 5.8. We use the formula (A.6) from Lemma A.1 to estimate the integrand: +(A.24) +| +n+κ +� +i=1 +Γ(⟨Di, σ⟩ + αi)e⟨θ,σ⟩| < const · +� +i/∈Ik +(|⟨Di, ℑ(σ)⟩| + 1)⟨Di,ℜ(σ)⟩+αi exp +� +�−2π⟨B, ℑ(σ)⟩ − π/2 +� +i/∈Ik +⟨Di, ℑ(σ)⟩ +� +� , +for σ on the integration contour. +We notice that condition (5.10) implies that +|⟨B, ν⟩| − 1 +4 +n+κ +� +i=1 +⟨Di, ν⟩ < −const · |ν|, +for some positive const since the expression is homogeneous of degree 1 in |ν|. +Therefore, expression in the exponential in (A.24) is bounded above by −c|ℑ(σ)| for some constant c > 0 (in +any norm on g) under the assumption of the lemma, so the absolute value of the integrand is bounded by an +exponentially decaying function. +□ +Proposition A.5. Let k < κ, Ior +k = (i1, . . . , ik) be such that {Di}i∈Ik are linearly independent. Consider a cycle +CIor +Ω (p) where p ∈ πIk +mIk and is separated from all other polar hyperplanes by positive number δ. Let B be such that +the integral +(A.25) +Int(p) := +� +C +Ior +Ik +Ω +(p) +dσ Γ · e⟨θ,σ⟩ +converges absolutely. There exist constants c1, c2 > 0 such that if ⟨ζ, p/|p|⟩ < −c1 then |Int(p)| < e−c2|p|, |p| ≫ 0. +Proof. Let us fix I and work in the basis given by {Di}i∈I. Using Fubini theorem we write +(A.26) +Int(p) = ± +� +Rκ−k +� +(S1)k dσ +n+κ +� +i=1 +Γ · e⟨θ,σ⟩ = += const +� +i∈Ik +(−1)mi +mi! +� +Rκ−k Ω +� +i/∈Ik +Γ(⟨Di, p + +√ +−1ℑ(σ)⟩ + αi)e⟨θ,σ⟩. +Let pi := ⟨Di, p⟩ and yi := ℑ(⟨Di, σ⟩), |y| := |ℑ(σ)|. Consider the main asymptotics of the Gamma functions +without argument shift by α: +(A.27) +n+κ +� +i=1 +|pi|−pi := exp +� +�− +� +i/∈Ik +pi log |pi| + +� +i∈Ik +(mi + αi) log(mi + αi) +� +� , +where the left hand side is defined as a limit if any pi = 0. Calabi-Yau condition �n+κ +i=1 Di = 0 implies that (A.27) +is a homogeneous function of degree 0 in p. We use it to simplify the asymptotics of the integrand: +(A.28) +� +i∈Ik +m−mi +i +� +i/∈Ik +|pi + αi + +√ +−1yi|pi � +i +|pi|−pi = += +� +i∈Ik +(mi + αi)αiexp +� +�� +i/∈Ik +pi log |1 + (αi + +√ +−1yi)/pi| + +� +i∈Ik +mi log(1 + αi/mk) +� +� ≤ +≤ econst·|p|exp +� +�� +i/∈Ik +pi log |1 + (αi + +√ +−1yi)/pi| +� +� , |p| → ∞. +In the last inequality we used the fact that (mi+αi)αi is bounded by a polynomial in |p| and mi log(1+αi/mi) < αi. +Analogously to the proof of Lemma 5.8 we have +(A.29) +e−2π⟨B,ℑ(σ)⟩−� +i/ +∈Ik π|yi|/2 ≤ e−const2|y|. +44 + +Now we apply Lemma A.1 to (A.25): +(A.30) +� +|pi|−pi|Int(p)| ≤ +� +i +|pi|−pi � +i∈Ik +m−mi +i +e− � +i∈Ik mie⟨ζ,p⟩× +× +� +Rκ−k Ω +� +i/∈Ik +|pi + αi + +√ +−1yi|pi+αi−1/2e− � +i/ +∈Ik min(pi+αi,0)e−2π⟨B,ℑ(σ)⟩−� +i/ +∈Ik π|yi|/2 ≤ +≤ econst4|p|+⟨ζ,p⟩e +� +i αi +� +Rκ−k Ω +� +i/∈Ik +|pi + αi + +√ +−1yi|αi−1/2exp +� +�� +i/∈Ik +pi log |1 + (αi + +√ +−1yi)/pi| +� +� e−const2|y|, +where in the last line we used (A.28) and (A.29). +We use the obvious inequality (a+b)x ≤ (2a)x+(2b)x for a, b > 0, x ∈ R\{0} to write |a+√−1b|x = (a2+b2)x/2 ≤ +2x/2(|a|x + |b|x). Furthermore, if pi < 0, +(A.31) +|1 + (αi + +√ +−1yi)/pi|pi < |1 + αi/pi|pi. +This estimate does not depend on y and is subexponential in p if pi is separated from −αi, so we can ignore these +terms in the estimate. Then +(A.32) +� +i/∈Ik,αi−1/2>0 +|pi + αi + +√ +−1yi|αi−1/2exp +� +� +� +i/∈Ik,pi>0 +pi log |1 + (αi + +√ +−1yi)/pi| +� +� < +< 2 +� +i/ +∈Ik (pi+αi−1/2)/2 +� +i/∈Ik,αi−1/2>0 +� +|pi + αi|αi−1/2 + |yi|αi−1/2� +� +i/∈Ik,pi>0 +((1 + |αi/pi|)pi + |yi|pi/|pi|pi) ≤ +≤ 2 +� +i/ +∈Ik (pi+αi−1/2)/2 +� +i/∈Ik,αi−1/2>0 +� +|pi + αi|αi−1/2 + |yi|αi−1/2� +� +i/∈Ik,pi>0 +(e|αi| + |yi|pi/|pi|pi), +where in the last inequality we use (1 + c/x)x = ex log(1+c/x) < ec, c, x > 0. The first product in the last line has +polynomial behaviour in pi so it is bounded by a polynomial in |p|. Also, factors 2pi are bounded by econst|p|. Let +us focus on the only non-trivial terms containing |yi|pi/ppi +i . We drop powers of |yi|αi−1/2 since they will produce +the same exponential asymptotics. In order to prove the claim of the proposition we prove that each integral +(A.33) +� +Rκ−k Ω +� +i∈J⊂I′ +k +|yi|pi/ppi +i e−const2|y| +is bounded above by econst|p|. Let ρ = |y|. Then in the polar coordinates on Rκ−k we write +(A.34) +� +Rκ−k Ω +� +i∈J⊂I′ +k +|y|pi/ppi +i e−const2|y| = const +� +i +p−pi +i +� ∞ +0 +dρ ρκ−k−1+� +i pie−const2ρ = += const · const +k−κ−� +i pi +2 +� +i +p−pi +i +Γ( +� +i +pi + κ − k). +If � +i pi is small, then so is the integral. Otherwise we can use the Stirling approximation. Γ(� +i pi+κ−k)/Γ(� +i pi) +is bounded by a polynomial in |p|, so we need to prove that +Γ( +� +i +pi) +� +i +p−pi +i +< econst|p|. +Stirling approximation implies: +(A.35) +Γ( +� +i +pi) < const( +� +i +pi) +� +i pi−1/2e− � +i pi. +Lemma A.6. Let a1, . . . , ak > 0 and a = �k +i=1 ai > 0. Then +(A.36) +aa � +i +a−ai +i +≤ ka +Proof. Equation (A.36) is equivalent to +(A.37) +k +� +i=1 +�ai +a +�− ai +a ≤ k +45 + +Define f : [0, ∞)k → R by +f(b1, . . . , bk) = +��k +i=1 b−bi +i +if b1, . . . , bk > 0, +0 +if bi = 0 for some i. +Then f is continuous, and is smooth on (0, ∞)k. +Let ∆k := {(b1, . . . , bk) ∈ R : bi ≥ 0, �k +i=1 bi = 1} be the +(k −1)-simplex. Then f is positive in the interior of ∆k and is zero on ∂∆k. Using Lagrange multiplier we compute +that +max +(b1,...,bk)∈∆k f(b1, . . . , bk) = f(1 +k , . . . , 1 +k ) = k. +Therefore, (A.37) holds whenever a1, . . . , ak > 0 and a = �k +i=1 ai. +□ +Applying Lemma A.6 we find that +(A.38) +Γ( +� +i +pi) +� +i +p−pi +i +≤ econst(� +i pi) ≤ econst|p|. +Returning to (A.30) we find +(A.39) +� +i +|pi|−pi|Int(p)| ≤ e⟨ζ,p⟩econst|p|, |p| → ∞, +or +(A.40) +|Int(p)| < e⟨ζ,p⟩+� +i pi log |pi|+const|p| = e(⟨ζ,p/|p|⟩+� +i pi/|p| log |pi|/|p|)+const)|p|, +where we used that � +i |pi|−pi is homogeneous of degree 0. 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Zhou, “K-theoretic quasimap wall-crossing,” arXiv:2012.01401. +Konstantin Aleshkin, Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA +Email address: aleshkin@math.columbia.edu +Chiu-Chu Melissa Liu, Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA +Email address: ccliu@math.columbia.edu +48 + diff --git a/FtE3T4oBgHgl3EQftQtg/vector_store/index.faiss b/FtE3T4oBgHgl3EQftQtg/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..70d7598d57bc978e773c787a121bbf4ebeec76c2 --- /dev/null +++ b/FtE3T4oBgHgl3EQftQtg/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a2fb58b792a33a2739fec2086ac1cb3359f8a514e171ed3dd6383e1bf31d7cfe +size 3604525 diff --git a/GtE1T4oBgHgl3EQf_QYB/vector_store/index.pkl b/GtE1T4oBgHgl3EQf_QYB/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7a00e8a9d4648f41e1cc0f58ec102f07eac09caf --- /dev/null +++ b/GtE1T4oBgHgl3EQf_QYB/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8dfae2eef69dc0b06b146767bf008623e2b6716325a20c92b8fbc668cbe5f8e5 +size 105686 diff --git a/H9E1T4oBgHgl3EQfFgPu/content/tmp_files/2301.02904v1.pdf.txt b/H9E1T4oBgHgl3EQfFgPu/content/tmp_files/2301.02904v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e861dc42b8209a2bd085b0a3fa73fd9fb396e6b0 --- /dev/null +++ b/H9E1T4oBgHgl3EQfFgPu/content/tmp_files/2301.02904v1.pdf.txt @@ -0,0 +1,1040 @@ +Sensitivity analysis for transportability in multi-study, +multi-outcome settings +Ngoc Q. Duong, Amy J. Pitts, Soohyun Kim, Caleb H. Miles +Department of Biostatistics, Mailman School of Public Health, Columbia University +Abstract +Existing work in data fusion has covered identification of causal estimands when +integrating data from heterogeneous sources. These results typically require additional +assumptions to make valid estimation and inference. However, there is little literature +on transporting and generalizing causal effects in multiple-outcome setting, where the +primary outcome is systematically missing on the study level but for which other +outcome variables may serve as proxies. We review an identification result developed +in ongoing work that utilizes information from these proxies to obtain more efficient +estimators and the corresponding key identification assumption. We then introduce +methods for assessing the sensitivity of this approach to the identification assumption. +Keywords: Causal inference, Data fusion, External validity, Generalizability, Missing data, +Proxy variable +arXiv:2301.02904v1 [stat.ME] 7 Jan 2023 + +1 +Introduction +Research in clinical medicine and public health is often concerned with estimating the effect +of some treatment in a specific target population. However, even in a randomized clinical +trial, which is considered the gold-standard study design, ensuring external validity remains +a challenge. This can be due to a variety of reasons, including non-random sampling, overly +stringent exclusion criteria, or an ill-defined target population of interest (Tan et al., 2022; +Kennedy-Martin et al., 2015). +Meta-analysis of summary statistics is a commonly used +tool to synthesize and generalize findings from published study-level summary statistics, +but tends to rely on strong, often implausible assumptions. An alternative approach that +allows for more control over the nuances and heterogeneity across studies is to combine +individual-level data, when available, from multiple studies, each of which may contain +insufficient information to address a given scientific question by itself, but which collectively +have the power to do so. There has been a growing body of work on generalizability and +transportability methods, which can help address the problem of external validity of the +effect estimates from integrating individual level data across studies. +Generalizability concerns the setting where the study population is a subset of the target +population of interest while transportability addresses the setting where the study popula- +tion is partially or completely external to the target population (Degtiar and Rose, 2023). +Specifically, generalizability typically involves extending the causal effect estimate derived +from a study as long as the covariates in the study population and the target population +have common support (Gechter, 2015; Tipton, 2014). On the other hand, transportability +entails extrapolating the effect estimated from a study in which some primary outcome of +interest is observed to a population represented by a sample in which the outcome is not +measured. +Existing methodologies involve directly transporting some estimated causal effect, e.g., +the average treatmemt effect (ATE), from studies where the outcomes are observed to other +studies with missing outcomes or across heterogeneous study designs and settings (Barein- +1 + +boim and Pearl, 2016; Dong et al., 2020; Pearl and Bareinboim, 2014; H¨unermund and +Bareinboim, 2019), or to some broader target population (Dahabreh et al., 2020a,b; Lesko +et al., 2017; Westreich et al., 2017). When considering multiple studies, it is often the case +that one will observe different outcomes at follow up. However, existing methods do not +take advantage of these other potentially correlated and informative outcome variables mea- +sured at follow-up, which could potentially be leveraged to achieve large efficiency gains. +Existing outcome proxy-blind methods typically rely on an assumption of homogeneous con- +ditional potential outcome means for valid transportation of estimation from one population +to another. Sensitivity analysis strategies have been proposed to study the extent to which +the violation of these assumptions will affect the estimations and inferences drawn (Nguyen +et al., 2017; Dahabreh and Hern´an, 2019; Dahabreh et al., 2022). +In ongoing work, we have developed a new strategy to more efficiently estimate the +ATE from integrated data across multi-outcome studies, with inconsistent availability of the +primary outcome of interest at the study level. The proposed methodology takes advantage +of the availability of follow-up measurements of potential correlates of the main outcome +to yield more precise estimate of the causal effects. +In this article, we consider the key +common outcome regression (or conditional exchangeability for study selection) assumption +for transportability while leveraging these outcome proxies, which differs slightly from the +common outcome regression assumption that has been traditionally used for transportability. +We discuss the resulting bias when this assumption is not met, and develop methodology for +sensitivity analysis to the violation of this assumption. +The remainder of the article is organized as follows. In Section 2, we discuss identification +of the average treatment effect in the multi-study, multi-outcome setting. In Section 3, we +discuss the bias incurred by violations of the key conditional exchangeability assumption. In +Section 4, we compare the conditional exchangeability assumption in our setting with that +used in settings that do not leverage outcome proxies. In Section 5, we develop methods +for sensitivity analysis for when our assumption is violated. We demonstrate the empirical +2 + +performance of our proposed methods in a simulation study in Section 6, and conclude with +a discussion in Section 7. +2 +Data integration for studies with primary outcome +missing systematically +2.1 +Study and data setting +In this setting, we let A be the treatment indicator, W be a set of covariates that are +commonly observed across studies, Y be the primary outcome variable, the set {T1, . . . , Tk} +be all the potential outcome proxies measured at follow-up in any study, and Js be the +study-specific subset of {T1, . . . , Tk} that is measured in study s. +Suppose there are S +studies that are ordered such that for each s in the first s∗ studies, we observe the set of +variables (Y, A, Js, W), while for each s in the remaining S − s∗ studies, only the subset +(A, Js, W) are observed. In other words, Y is systematically missing in the latter set of +studies. Unlike the standard setup in other works concerning effect transportability that +only involves (Y, A, W), we introduced the use of Ts, where Ts ⊂ Js is some user-specified +subset of Js for each study s. Ts could be chosen based on availability and subject matter +knowledge and must be chosen such that they are observed in at least one of the studies +{1, 2, . . . , s∗}. +Studies can be randomized experiments or observational; however, we will not consider +scenarios in which some studies are randomized experiments and others are observational in +this work. Then the study-specific average treatment effect and conditional average treat- +ment effect can be written as: +ATE(s) = E(Y1 − Y0 | S = s) +CATE(w, s) = E(Y1 − Y0 | W = w, S = s). +3 + +Accordingly, we can define the overall average treatment effect and conditional average treat- +ment effect as: +ATE = +S +� +s=1 +ATE(s) +CATE(w) = E(Y1 − Y0 | W = w) +where the weights can be user-specified such that � +s πs = 1. For instance, one can choose +πs = P(S = s), or the marginal probability of being in each study. Alternatively, we could +define ATE = EQW,SCATE(W, S) for a user-specified, known distribution QW,S of W and +S. +Since Y is not measured in s ∈ {s∗ +1, . . . , S}, we cannot directly estimate the ATE and +CATE using data from these studies alone. Our purpose is to transport the ATE from the +first s∗ studies where Y is observed, to the remaining S − s∗ studies while also leveraging +the information from the outcome proxy set Ts to improve efficiency. For ease of notation, +let σs be a subset of the first s∗ studies in which both Y and Ts are observed. We can then +use this information from the studies that form σs to estimate the outcome regression that +will allow us to transport the causal effects to study s. In this setting, we have shown in +ongoing, not-yet-published work that the ATE can be nonparametrically identified as: +ΨATE = +s∗ +� +s=1 +πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s} ++ +S +� +s=s∗+1 +πsE[E{E(Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s} +(1) +− E{E(Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s} | S = s]. +The terms in the first sum are simply the standard identification formula for the (study- +specific) average treatment effects when Y is observed. The second sum is identified since +it only depends on the distribution of Y in the studies in σs, i.e., in which Y is actually +4 + +observed. +Here, we introduced a modification to how transportability has traditionally been done by +incorporating information from a set of outcomes measured at follow-up that are correlated +with the main outcome of interest. +2.2 +Assumptions for Identification of the ATE +This derivation ATE can be nonparametrically identified given the assumptions that are +standard for identification for ATE when outcomes are all observed: +Assumption 1 (Positivity). P(A = 1 | W = w) > 0 for all w with positive probability. +Assumption 2 (Consistency). Y = AY1 + (1 − A)Y0. +Assumption 3 (Within-study conditional exchangeability). +E[Y a | W, A, S = s] = E[Y a | W, S = s] for all s. +The validity of our estimator relies on a fourth assumption that allows for the transporta- +tion of the effect across studies: +Assumption 4 (Common outcome regression (proxy-aware version)). +E(Y | Ts, W, A = a, S = s) = E(Y | Ts, W, A = a, S ∈ σs) for all s. +This is a missing at random (MAR)-type assumption, where S can in a sense be thought +of as a missingness indicator, since missingness is systematic by study. +We can also introduce a fifth assumption that is not necessary for identification, but +allows for more borrowing of information across studies, which can help with efficiency: +Assumption 5 (Common distribution of outcome proxies). Ts ⊥ S | W, A for all s. +5 + +This implies the distribution of Ts conditional on treatment assignment and baseline +covariates is the same across studies. Under this additional assumption, the identification +result simplifies to: +ATE = +S +� +s=1 +πsE[E{E(Y | Ts, W, A = 1, S ∈ σs) | W, A = 1} +− E{E(Y | Ts, W, A = 0, S ∈ σs) | W, A = 0} | S = s]. +In ongoing work, we have developed a simple substitution estimator that involves replac- +ing each expectation with a regression-based estimate and the outer expectation with an +empirical mean. +For the outcome proxy-blind approach, in addition to the first three standard internal +validity assumptions, Assumption 4 is replaced by a slightly different mean outcome ex- +changeability assumption: across studies assumption (exchangeability over S) (Dahabreh +and Hern´an, 2019; Lesko et al., 2017): +Assumption 6 (Common outcome regression (proxy-blind version)). +E(Y | W, A = a, S = s) = E(Y | W, A = a, S ∈ σs) for all s. +Assumption 4 differs from Assumption 6 by additionally conditioning on Ts for each study +s. Assumptions 4 and 5 together imply Assumption 6. In this article, we will only consider +sensitivity analysis for the violation of Assumption 4. +When Assumption 5 is violated, +the ATE estimator based on Assumption 4 (i.e., the substitution estimator based on the +identification formula (1)) will remain consistent. +6 + +3 +Characterizing the bias resulting from violation of +the identification assumption +The validity of ΨATE is dependent on the key assumption 4. This assumption requires no +heterogeneity in the conditional outcome means given treatment, covariates, and outcomes +proxies between studies with and without missing outcome (Y ) data. This allows for trans- +portation of the conditional outcome means, and correspondingly, the ATE and CATE, +estimable from one study to others. +In practice, this could be a strong assumption to make while also untestable using ob- +served data. For instance, in previous unpublished work, we estimated the average treatment +effect of cognitive remediation (CR) therapy on Social Behavioral Scale (SBS) score, a mea- +sure for social functioning, using harmonized data from three trials in the NIMH Database +of Cognitive Training and Remediation Studies (DoCTRS) database. However, the degree +of effectiveness of CR, especially on functional and occupational outcomes, was less evident +and has been suggested to vary depending on the setting in which the treatment was admin- +istered (Barlati et al., 2013; Combs et al., 2008; McGurk et al., 2007; Wykes et al., 2007, +2011). When this assumption is violated, the substitution estimators described in the pre- +vious section will be biased. Therefore, we examine two strategies for sensitivity analysis in +order to examine the robustness of estimates under varying degrees of assumption violation. +To quantify the degree of violation, let the bias functions be defined as: +u(A = 1, Ts, W) = E(Y | Ts, W, A = 1, S = s) − E(Y | Ts, W, A = 1, S ∈ σs), +u(A = 0, Ts, W) = E(Y | Ts, W, A = 0, S = s) − E(Y | Ts, W, A = 0, S ∈ σs) +(2) +7 + +Then, equation (1) when assumption 4 is violated instead becomes: +ATE = +s∗ +� +s=1 +πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s) ++ +S +� +s=s∗+1 +πsE [E {E (Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s)} +−E {E (Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s)} | S = s] ++ +S +� +s=s∗+1 +πsE[E {u (A = 1, Ts, W) | W, A = 1, S = s} +− E {u (A = 0, TS, W) | W, A = 0, S = s} | S = s], +where the last sum is not identified. Then, the study-specific bias for study s is: +E [E {u (A = 1, Ts, W) | W, A = 1, S = s} − E {u (A = 0, Ts, W) | W, A = 0, S = s} | S = s] += E[δ∗(W)|S = s]. +(3) +By rearranging terms, δ∗(W) can be alternatively written as: +E [E (Y | Ts, W, A = 1, S = s) − E (Y | TS, W, A = 1, s ∈ σs) | W, A = 1, S = s] +− E [E (Y | Ts, W, A = 0, S = s) − E (Y | Ts, W, A = 0, s ∈ σs) | W, A = 0, S = s] += E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) +− {E [E (Y | Ts, W, A = 1, s ∈ σs) | W, A = 1, S = s] +− E [E (Y | Ts, W, A = 0, s ∈ σs) | W, A = 0, S = s]}. +(4) +The latter term cannot be simplified unless Assumption 5 holds. +8 + +4 +Comparison with bias functions in settings without +incorporation of follow-up surrogate outcomes +In recent work, Dahabreh and Hern´an (2019) developed sensitivity analysis for transportabil- +ity considering a similar setting of two types of studies with and without missing outcomes. +In the base case, there are two studies considered (missingness of the outcome variable de- +noted by a binary indicator S). To describe this setting using our notation, we simply have +σ0 = σ1 = {1} (i.e., study S = 1 with the observed outcome of interest is used to impute +the conditional outcome means for study S = 0). Equivalently, for ease of interpretation in +the base case, let S = 1 and S = 0 denote the study where the primary outcome of interest +is observed and not observed, respectively. +In the setting where the model used to impute conditional potential outcomes does not +utilize information from Ts, Dahabreh and Hern´an (2019) define: +u(A = a, W) = E[Y | A = a, W, S = 1] − E[Y | A = a, W, S = 0]. +The difference between these bias functions can then be obtained as: +δ(W) = u(A = 1, W) − u(A = 0, W) += E[Y 1 − Y 0 | W, S = 1] − E[Y 1 − Y 0 | W, S = 0] +This expression can be qualitatively expressed as the difference in the conditional average +treatment effects between the two studies. This qualitative interpretation can aid in concep- +tualizing and thinking about more appropriate values and range for sensitivity parameters +when examining robustness of the results. More specifically, assuming higher levels of the +outcome are preferred, if we believe the participants in studies with missing outcomes benefit +less from treatment, then true δ can be assumed to be positive and vice versa (Dahabreh and +Hern´an, 2019). Since our bias functions are conditional on the set of proxy outcomes, the +9 + +term δ∗(W) in (4) unfortunately cannot be reduced further to a more interpretable statistical +entity. When we take Ts to be the empty set, the bias function δ∗(W) reduces to the same +expression. +5 +Accounting for violation of the common outcome re- +gression assumption through sensitivity analyses +We consider two scenarios in which we assume the bias terms u(A = 1, Ts, W) and u(A = +0, Ts, W) to be 1) constants and 2) bounded functions of the outcome proxies and/or baseline +covariates. The first scenario involves making a stronger assumption about the bias terms. +On the other hand, the second scenario requires weaker assumptions but allow them to be +non-constant. +5.1 +Bias functions assumed to be some fixed values +Although it might be more reasonable to assume that the bias functions are dependent on +some baseline covariates, for ease of implementation of sensitivity analysis, one can also +suppose they are constant. When u(A = 1, Ts, W) and u(A = 0, Ts, W) are independent of +the baseline covariates W and the outcome proxy set Ts, the conditional expectations of the +bias functions, and in turn, the term δ∗(W) in (3), reduce to: +δ = u1 − u0, where δ, u1, and u0 ∈ R +(5) +The sensitivity analysis involves correcting for the above-mentioned bias term by adding it +back to the identification formula ΨATE, which relies on the common outcome regression +assumption. +10 + +ATE = +s∗ +� +s=1 +πsE {E (Y | W, A = 1, S ∈ σs) − E (Y | W, A = 0, S ∈ σs) | S = s} ++ +S +� +s=s∗+1 +πsE [E {E (Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s} +− E {E (Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s} | S = s] + +S +� +s=s∗+1 +πs (u1 − u0) +=ΨATE + +S +� +s=s∗+1 +πs (u1 − u0) +(6) +where u1 and u0 are scalars. +In practice, the true bias term would be unknown. Thus, one strategy is to propose a +grid of sensitivity parameters that covers the potential range of values in which the true bias +term might fall. This grid of sensitivity parameters can be specified using subject-matter +knowledge. We can then adjust for the bias term in the estimation step by adding back +the different sensitivity parameters to the estimated ATE using our proposed method. This +also allows for observation of the behavior of the estimated ATE as we vary the sensitivity +parameters. +5.2 +Bounded covariate-dependent bias functions +One might also believe that the bias term is not constant at all levels of the baseline covariates +and/or the outcome proxies. When the assumption of fixed-value bias terms is considered +too strong, but the functional forms for bias terms cannot be confidently determined from +existing knowledge of the data mechanism (as will typically be the case), one can still recover +some information about the true ATE without having to correctly specify the bias terms. If +we instead assume the bias terms to be some bounded functions, we can compute a bound +around the (na¨ıve) ATE estimate that contains the true ATE by varying the bounds of these +functions. This provides information on how far away the true ATE can be from the estimate +11 + +obtained constrained by the bounds of the bias term. +Identifying the bounds for the bias term can be expressed as maximizing and minimizing +the objective function: +E[E[u(A = 1, Ts, W) | W, A = 1, S = s] − E[u(A = 0, Ts, W) | W, A = 0, S = s] | S = s] +subject to the following constraints: +|u(A = 1, Ts = ts, W = w)| ≤ γ1 +|u(A = 0, Ts = ts, W = w)| ≤ γ0 +for all ts and w, which implies |E[u(A = 1, Ts, W) | W, A = 1, S = s]| ≤ γ1 and |E[u(A = +0, Ts, W) | W, A = 0, S = s]| ≤ γ0 where γ1, γ1 ∈ R+. +Then we have −(γ1 + γ0) ≤ u(A = 1, Ts, W) − u(A = 0, Ts, W) ≤ γ1 + γ0. If we have no +reason to suspect we know more about the bounds of one bias function than the other (as +will typically be the case), we may simply choose to specify a scalar sensitivity parameter γ +to be the maximum of γ1 and γ2, in which case we have −2γ ≤ u(A = 1, Ts, W) − u(A = +0, Ts, W) ≤ 2γ. +By equation (6) even though we do not know the form of the bias functions u(A = +1, Ts, W) and u(A = 0, Ts, W), we can partially recover the true ATE using the bounds +around the na¨ıve estimate: +ΨATE − 2 max(γ1, γ0) ≤ ATE ≤ ΨATE + 2 max(γ1, γ0) +ΨATE − 2γ ≤ ATE ≤ ΨATE + 2γ +(7) +If the bias functions are in fact bounded by some value smaller than or equal to our specified +values for the sensitivity bounds, the true ATE would fall between [ΨATE − 2γ, ΨATE + 2γ]. +Then, the true ATE is partially identified without assumptions about the functional form +12 + +of u(A = 1, Ts, W) and u(A = 0, Ts, W). One can then use the bootstrap standard error for +the substitution estimator of the identification formula (1) to determine the amount to add +and subtract from the upper and lower bounds, respectively, in order to produce confidence +intervals for the partial identification sets for each value of the sensitivity parameter. Since +the sensitivity bounds are a deterministic function of the sensitivity parameter, bootstrapping +need only be done once. +6 +Simulations +6.1 +Data generating mechanism +We consider the setting of two studies, with S = 1 indicating the study where the primary +outcome is available. +We generate random sample draws with sample size n = 100 for +both studies. The data generating mechanism is as follows. W, T0 come from independent +standard normal distributions, and T1 comes from a normal distribution with mean and +variance of 1. Then +T = I(A = 1) × T1 + I(A = 0) × T0 +Y 0 = −4T0 + W + ϵ0 +Y 1 = 4T1 + W + ϵ1 +Y = I(A = 1) × Y1 + I(A = 0) × Y0 +where ϵ1, ϵ0 ∼ N(0, 1). +Via these specifications, T fully mediates the relationship between A and Y (direct effect +from A to Y is constrained to be 0). As a result, the true ATE = 4. This is also a more +basic setting in which the vector T is observed in all studies. +Due to the nature of the DoCTRS database, which is comprised of randomized clinical +trials, in our base setting, we specified the marginal probability P(A = 1) = 0.5, represent- +13 + +ing random treatment assignment. This treatment assignment satisfies the positivity and +exchangeability assumption. +Specifically, to incorporate the difference in conditional outcome means between the +two types of studies, in studies missing the outcome, we added constant bias terms to +the counterfactual outcomes Y0 and Y1. Similar to the data generating step, we preserved +the observed counterfactual outcome from the corresponding treatment assignment, which +satisfies the consistency assumption. By (5), we have: +Y 0 +S=1 = Y 0 +S=0 + u0 +Y 1 +S=1 = Y 1 +S=0 + u0 + δ +(8) +for u0 ∈ {−3, 0, 3}, δ ∈ {−2, 0, 2}. +Then the bias reduces to a single parameter δ, since it is no longer a function of u0 when +computing the ATE: +E(Y 1 − Y 0 | S = 1) = E(Y 1 − Y 0 | S = 0) + δ +(9) +In the case where the bias term is a function of baseline covariates and surrogate outcome, +we had the following specification for the true bias: +u0 = b0 × sin (Ts + W) +u1 = b1 × +exp(Ts+W) +1+exp(Ts+W) +for b0 ∈ {2, 3, 4} and b1 ∈ {1, 2, 3}. +6.2 +Adjusting for sensitivity parameter in estimation step +In the presence of non-zero bias, when the value of the sensitivity parameter δ is specified +such that it is equal to true δ, the ATE estimate after bias adjustment tends to be closer to +14 + +the true ATE after compared to before. In addition, the corresponding 95% CIs are expected +to cover the true ATE 95% of the times. Although coverage probability can be examined +more in a more robust fashion using bootstrapped confidence intervals across all simulations, +in Fig. 1, 2, and A.1-A.4, the 95% CIs covers the true ATE at the value of the sensitivity +parameter that reflects the degree of assumption violation all but one instance, which is in +line with our expectations. +Scenario 1. When the bias terms are assumed to be constants, a natural approach +would be to specify a two-dimensional grid of sensitivity parameters for both scalars u0 and +u1. However, by (8), it is equivalent to specifying u0 (or u1) and δ. In fact, since the u0 (or +u1) as constant terms cancel out during adjustment, it is sufficient to specify one sensitivity +parameter δ (9). We also note that δ being 0 does not necessarily imply assumption 4 is +met, since the bias terms u0 and u1 could cancel exactly. +To implement sensitivity analysis, we follow the steps: +1. Specify a grid of sensitivity parameters δ. +The grid should be reasonably wide to +contain true δ. +2. Estimate the na¨ıvely transported ATE using the identification result in (1) +3. Sequentially add the values in the sensitivity parameter grid to the na¨ıvely estimated +ATE, using the result in (6) to obtain the bias-corrected ATE estimates. +We then plotted the bias-corrected estimates under different sensitivity parameters against +the true ATE. Additionally, we bootstrapped the bias-corrected estimates to obtain the 95% +confidence intervals and explore coverage across different values of u0 and δ. +Scenario 2. When we want to make minimal assumptions about the functional form of +the bias, we can still perform sensitivity analysis on the true ATE using the following steps: +1. Specify a grid of sensitivity parameters called γ that potentially include the upper and +lower bounds of the true bias functions +15 + +2. Computed the “na¨ıve” ATE estimate using the identification result in (1) +3. Construct the upper and lower bound around the estimated ATE using (7) where γ is +replaced with the sensitivity parameters. +We also plot the na¨ıve ATE estimates and the bounds around these estimates at each value +of the sensitivity parameters. In practice, the bias functions are of course unknown and +cannot be estimated from observed data. Therefore, when specifying the grid of sensitivity +parameters, the analyst needs to employ subject matter knowledge about the data generating +mechanism to select values of δ and γ. +We then explore the behavior of the adjusted estimators via simulations. In the first case, +we focused on the general unbiasedness of the correctly-adjusted point estimate for both the +overall ATE and ATE among studies with missing outcomes, as well as the 95% CI coverage +across degrees of assumption violation (i.e., across values of true u0 and δ). In the second +case, we looked for correct bounding of the true ATE. +6.3 +Simulation Results +6.3.1 +Bias terms as constants +We examine the estimates produced by our method under the different degrees of violation +of assumption 4, before and after taking into account the specified sensitivity parameter. +Figure 1 shows the estimates (95% CI) for the true overall ATE using our method under +varying magnitudes and directions of the bias terms from one single simulation. +16 + +Figure 1: Sensitivity-parameter-adjusted ATE estimate shown against the true overall ATE +across values of the true bias and sensitivity parameter; n=100 for each study, 95% CI +constructed from 1000 bootstrap samples. When sensitivity parameter δ = 0, the adjusted +estimate corresponds to the unadjusted estimate. Horizontal dotted line shows the true ATE +given true δ; vertical dotted line indicates sensitivity parameter δ equals true δ +In the presence of non-zero bias, when the value of the sensitivity parameter δ is specified +such that it is equal to true δ, the ATE estimate after bias adjustment tends to be closer to +the true ATE after compared to before. In addition, the corresponding 95% CIs are expected +to cover the true ATE 95% of the times. Although coverage probability can be examined +more in a more robust fashion using bootstrapped confidence intervals across all simulations, +in Fig. 1, 2, and A.1-A.4, the 95% CIs covers the true ATE at the value of the sensitivity +parameter that reflects the degree of assumption violation all but one instance, which is in +17 + +True = -2 +True = 0 +True θ = 2 +UU +2 +Estimate +4 +JU +-2 +2 +Sensitivity parameterline with our expectations. +Figure 2: +Sensitivity-parameter-adjusted ATE estimates shown against the true study- +specific ATE in the study in which the outcome is unobserved across values of the true +bias and sensitivity parameter; n=100 for each study, 95% CI constructed from 1000 boot- +strap samples. When sensitivity parameter δ = 0, the adjusted estimate corresponds to the +unadjusted estimate. Horizontal dotted line shows the true study-specific ATE given true δ; +vertical dotted line indicates sensitivity parameter δ equals true δ +Figure 2 shows similar results for the study-specific ATE estimates in the study with +missing outcomes (before and after bias adjustment) from the same simulated data. Com- +pared to the results in Figure 1, after adjustment using the correct sensitivity parameters, +the 95% CIs contain the true ATE more frequently than the CIs of the unadjusted estimates +in the study with missing primary outcome. Figure 2 also shows an example where infer- +18 + +True = -2 +True = 0 +True = 2 +10.0 - +7.5 +5.0 +uo += +2.5 +-3 +0.0 +10.0 +7.5 +Estimate +rue +5.0 +uo +2.5 +II +- +0.0 +7.5 +5.0 - +2.5 +II +3 +0.0 +-2.5 +2 +0 +2 +2 +0 +2 +2 +Sensitivity parameterence is sensitive to the violation of our assumption at a magnitude of δ between -1 and -2 +(u0 = −3, bottom left panel), between which the 95% CI changes from not containing to +zero to containing zero. +When we increased the sample size (n=200 and n=500), we saw general reductions in the +errors of these single estimates (Figures A.1, A.3). In most cases, even when there is error +in the adjusted estimates, the 95% CI bootstrap confidence intervals provide good coverage +(Figures 1, A.1, A.3). The reduction in error and improved coverage are more pronounced +when estimating the study-specific effect in the study with missing outcomes than in the +overall ATE combining the two studies (Figures A.2, A.4). +We also ran 1000 simulations under the same data generating mechanism and obtained +the unadjusted and sensitivity-parameter-adjusted estimates for each simulation. We then +showed the mean and 2.5th and 97.5th quantiles of these estimates under each combination +of the true bias values. We can see that when averaged across 1000 simulations, the adjusted +estimates closely approximate the true ATE (Figures 3, 4) when the true value of δ is used +for the sensitivity parameter. +19 + +Figure 3: Sensitivity-parameter-adjusted ATE estimates shown against the true overall ATE +across values of the true bias sensitivity parameter; mean, 2.5th and 97.5th quantiles obtained +from 1000 simulations. When sensitivity parameter δ = 0, the adjusted estimate corresponds +to the unadjusted estimate. Horizontal dotted line shows the true overall ATE given true δ; +vertical dotted line indicates sensitivity parameter δ equals true δ +20 + +True = -2 +True = 0 +True = 2 +6 +5 +. +True +4 +uO = 3 +2 +6 +Estimate +150 +True uO = 0 +4 +2 +6 +5 +. +. +True uO = 3 +4 +2 +2 +1 +0 +2 +-2 +-1 +0 +" +2 +-2 +1 +0 +2 +Sensitivity parameter Figure 4: Sensitivity-parameter-adjusted ATE estimates shown against the true ATE in the +study with missing outcome across values of the true bias and sensitivity parameter; mean, +2.5th and 97.5th quantiles obtained from 1000 simulations. When sensitivity parameter δ += 0, the adjusted estimate corresponds to the unadjusted estimate. Horizontal dotted line +shows the true study-specific ATE given true δ; vertical dotted line indicates sensitivity +parameter δ equals true δ +When approximate sensitivity parameters δ are used (δ ∈ {−1, 1} when true δ ∈ {−2, 2}), +the middle 95% values of adjusted estimates also cover the true ATE whereas those of +unadjusted estimates do not (Figure 4). +Figure 5 compares the errors in the estimates and sensitivity of associated inferences +between the outcome proxy-blind method of Dahabreh et al. (2020b); Lesko et al. (2017) +and our proposed method across 1000 simulations. +21 + +True = -2 +True = 0 +True = 2 +8 +6 +True uO = -3 +4 +2 +0 +8 +6 +Estimate +True uO = 0 +4 +2 +0 +8 +6 +True uO = 3 +4 +2 +0 +2 +U +2 +-2 +1 +2 +0 +2 +Sensitivity parameterFigure 5: Sensitivity-parameter-adjusted ATE estimates obtained from our proposed method +and the outcome proxy-blind method; mean, 2.5th and 97.5th quantiles obtained from 1000 +simulations. When sensitivity parameter δ = 0, the adjusted estimate corresponds to the +unadjusted estimate. Horizontal dotted line shows the true overall ATE given true δ; vertical +dotted line indicates sensitivity parameter δ equals true δ +The distributions of the estimates from both methods are centered on the true parameter. +However, the estimates tend to be more precise when we utilize the information from the +outcome proxy (as demonstrated through the narrower 2.5th-97.5th quantile range). The +efficiency gains have implications for the sensitivity analysis, since resulting inferences are +not as sensitive given analogous magnitude in violation of the identification assumption 4. +Assumption 4 implies both u0 and u1 equal 0. As a result, the true δ also equals 0. +This suggests transportation of the conditional potential outcome means, and in turn, the +22 + +True = -2 +True = 0 +True θ = 2 +6 +5 +3 +6 +4 +3 +6 +5 +2 +2 +2 +-2 +2 +Sensitivityparameter +Outcome regression method +Proposed methodconditional average treatment effects, can be done without incurring bias (vertical middle +panes, figure 3). We also observed that, when δ is 0, regardless of the values of u0 (and +u1), there is also no bias (vertical middle panes, figure 3) in the unadjusted estimator. In +both cases, no bias correction would be necessary, and incorporating a non-zero δ sensitivity +parameter will actually introduce bias to the estimate. +6.3.2 +Bias terms as bounded functions +When the sensitivity parameter γ is greater or equal to max{γ0, γ1} for the true function +bounds γ0 and γ1, the bounds always include the true ATE when the bias functions are +bounded by γ0 and γ1 (Figure 6). +23 + +Figure 6: ATE estimates with sensitivity bounds shown against the true overall ATE across +values of the true bias and sensitivity parameter. When sensitivity parameter γ = 0, the +bounds collapse to a point estimate. Blue horizontal dotted line shows the true study-specific +ATE given true bias functions +Although this approach requires minimal assumptions about the bias functional form, it +can also be conservative since the true bias functions are unlikely to evaluate to the bounds +across the domain of the functions. For instance, the bottom three panels of Figure 6 show +that when the sensitivity parameter γ is greater than or equal to max(true γ0, true γ1), +while the bounds on the estimate contain the true ATE, they also contains the null value +zero as well. On the other hand, these bounds do not rely on an assumption of constant bias +functions, which we may often have no reason to believe. Here, we demonstrated through +24 + +True y 1 = 1 +True 1 = 2 +True y 1 = 3 +U +Estimated ATEwith +4 +n +8. +Sensitivity parameter ysimulations that sensitivity analysis with relaxed and more credible assumptions can still +provide helpful information about the parameter of interest. However, when the bounds +are too narrow or too wide, sensitivity analysis using bounded bias functions might not be +accurate (i.e., not containing the true parameter) or useful (i.e., containing the null value +when the truth is non-null), respectively. +7 +Discussion +In this paper, we discussed a data integrative method that utilizes information from avail- +able proxies of the outcome of interest measured at follow-up for efficiency gains. We then +presented two sensitivity analysis strategies specific to this approach for causal effect trans- +portation when the identification assumption is violated. Our modification to the identifica- +tion of the ATE in (1) allows for more efficient estimators given sufficiently strong outcome +proxies. As a result, our bias functions also have similar, yet distinct interpretations than +the bias functions of Dahabreh and Hern´an (2019). +When the bias terms are assumed to be constants, we can obtain different bias-adjusted +point estimates based on our specification of the sensitivity parameters. Additionally, via +obtaining the 95% bootstrap confidence interval for the bias-adjusted estimates, we can +examine the robustness of inferences made using our method under varying magnitudes of +assumption violation. Specifically, beyond certain values of the sensitivity parameters, the +95% CI will cross the null value 0. These are the degrees of violation that can affect inferences +(where the 95% CI suggest a change from significant results to non-significant results). +We also proposed sensitivity analysis using bounded bias functions as an alternative when +one believes the assumption of a fixed-value bias term is too strong. This approach allows +for inferences with minimal assumptions about the unobserved bias functions but can still +provide useful information about the parameter of interest. Due to fewer assumptions being +made, the results are more conservative and robust, hence more reasonable and credible. +25 + +Specifically, although we are unable to obtain a point estimate, sensitivity analysis using +bounded bias functions can still be informative in the sense of providing information about +the general direction of the parameter of interest (beneficial or harmful). This method is +generally more conservative if the bounds on the functions are not close to their extreme +values, if the bias functions are generally not close to their extreme values, or if there is a +large difference between the extrema of the two bias functions. +Correct specification of the bias functions would allow for more precise and informative +estimation of the true ATE. However, since they are generally unknown and non-estimable +from observed data, sensitivity analysis will typically be the realistic course of action. +When conducting sensitivity analysis, the analyst can start off by specifying a wide grid +of the sensitivity parameter and examining the behaviors of the point estimates and 95% +CI (first approach) as well as bounds around the estimates (second approach). They can +then search for the “critical” sensitivity parameters that still suggest rejection of the null +hypothesis, i.e., the 95% CI (in the first case) and bounds around the estimate (in the +second case) that do not contain 0. It can be determined if greater bias is plausible by using +background knowledge of the data generating mechanism or further hypothesizing about +such mechanism. 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A meta-analysis +of cognitive remediation for schizophrenia: Methodology and effect sizes. The American +Journal of Psychiatry, 168(5):472–485. 7 +Wykes, T., Reeder, C., Landau, S., Everitt, B., Knapp, M., Patel, A., and Romeo, R. (2007). +Cognitive remediation therapy in schizophrenia: Randomised controlled trial. The British +Journal of Psychiatry, 190(5):421–427. 7 +29 + +A +Additional figures +Figure A.1: Sensitivity-parameter-adjusted ATE estimates shown against the true overall +ATE across values of the true bias and sensitivity parameter; n=200 for each study, 95% CI +constructed from 1000 bootstrap samples. When sensitivity parameter δ = 0, the adjusted +estimate corresponds to the unadjusted estimate. +Horizontal dotted line shows the true +overall ATE given true δ; vertical dotted line indicates sensitivity parameter δ equals true δ +30 + +True = -2 +True = 0 +True θ = 2 +6 +6 +stimate +5 +rue +uo +4 += +- +3 +6 +5 +4 +3 +3 +2 +0 +-2 +0 +2 +2 +2 +Sensitivity parameterFigure A.2: Sensitivity-parameter-adjusted ATE estimates shown against the true ATE in +the study with missing outcome across values of the true bias and sensitivity parameter; +n=200 for each study, 95% CI constructed from 1000 bootstrap samples. When sensitivity +parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate. Horizontal +dotted line shows the true study-specific ATE given true δ; vertical dotted line indicates +sensitivity parameter δ equals true δ +31 + +True = -2 +True = 0 +True θ = 2 +7.5 +5.0 +rue +uo += +2.5 +0.0 +8 +6 +Estimate +True +4 +uo +II +- +0 +8 +6. +True uO = +4 +2.* +3 +0 +2 +0 +2 +-2 +0 +2 +Sensitivity parameterFigure A.3: Sensitivity-parameter-adjusted ATE estimates shown against the true overall +ATE across values of the true bias and sensitivity parameter; n=500 for each study, 95% CI +constructed from 1000 bootstrap samples. When sensitivity parameter δ = 0, the adjusted +estimate corresponds to the unadjusted estimate. +Horizontal dotted line shows the true +overall ATE given true δ; vertical dotted line indicates sensitivity parameter δ equals true δ +32 + +True = -2 +True = 0 +True θ = 2 +6 +5 +4 +3 +=3 +2 +6 +5 +Estimate +True +4 +uo += +3 +2 +6 +True uO +4 +II +3 +0 +-2 +2 +Sensitivity parameter Figure A.4: Sensitivity-parameter-adjusted ATE estimates shown against the true ATE in +the study with missing outcome across values of the true bias and sensitivity parameter; +n=500 for each study, 95% CI constructed from 1000 bootstrap samples. When sensitivity +parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate. Horizontal +dotted line shows the true study-specific ATE given true δ; vertical dotted line indicates +sensitivity parameter δ equals true δ +33 + +True = -2 +True = 0 +True θ = 2 +6 +True +4 +u0 = -3 +2 +0. +8 +Estimate +6 +True uO = 0 +4 +-T +- +0 +8 +I +6. +True uO = +4 +2 +2 +0 +-2 +0 +2 +2 +2 +Sensitivity parameterB +Derivation of the sensitivity analysis formula when +Assumption 4 is violated +ATE = +s∗ +� +s=1 +πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s} ++ +S +� +s=s∗+1 +πsE [E {E (Y | Ts, W, A = 1, S = s) | W, A = 1, S = s} +− E {E (Y | Ts, W, A = 0, S = s) | W, A = 0, S = s} | S = s] += +s∗ +� +s=1 +πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s} ++ +S +� +s=s∗+1 +πsE [E {E (Y | Ts, W, A = 1, S ∈ σs) + u (A = 1, Ts, W) | W, A = 1, S = s} +− E {E (Y | Ts, W, A = 0, S ∈ σs) + u (A = 0, Ts, W) | W, A = 0, S = s} | S = s] += +s∗ +� +s=1 +πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s) ++ +S +� +s=s∗+1 +πsE [E {E (Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s)} +−E {E (Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s)} | S = s] ++ +S +� +s=s∗+1 +πsE[E {u (A = 1, Ts, W) | W, A = 1, S = s} +− E {u (A = 0, Ts, W) | W, A = 0, S = s} | S = s] +34 + diff --git a/H9E1T4oBgHgl3EQfFgPu/content/tmp_files/load_file.txt b/H9E1T4oBgHgl3EQfFgPu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab30d53a122a62f5a7a750190f108c7eef97b5d3 --- /dev/null +++ b/H9E1T4oBgHgl3EQfFgPu/content/tmp_files/load_file.txt @@ -0,0 +1,663 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf,len=662 +page_content='Sensitivity analysis for transportability in multi-study, multi-outcome settings Ngoc Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Duong, Amy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Pitts, Soohyun Kim, Caleb H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Miles Department of Biostatistics, Mailman School of Public Health, Columbia University Abstract Existing work in data fusion has covered identification of causal estimands when integrating data from heterogeneous sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' These results typically require additional assumptions to make valid estimation and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' However, there is little literature on transporting and generalizing causal effects in multiple-outcome setting, where the primary outcome is systematically missing on the study level but for which other outcome variables may serve as proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We review an identification result developed in ongoing work that utilizes information from these proxies to obtain more efficient estimators and the corresponding key identification assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We then introduce methods for assessing the sensitivity of this approach to the identification assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Keywords: Causal inference, Data fusion, External validity, Generalizability, Missing data, Proxy variable arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='02904v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='ME] 7 Jan 2023 1 Introduction Research in clinical medicine and public health is often concerned with estimating the effect of some treatment in a specific target population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' However, even in a randomized clinical trial, which is considered the gold-standard study design, ensuring external validity remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This can be due to a variety of reasons, including non-random sampling, overly stringent exclusion criteria, or an ill-defined target population of interest (Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Kennedy-Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Meta-analysis of summary statistics is a commonly used tool to synthesize and generalize findings from published study-level summary statistics, but tends to rely on strong, often implausible assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' An alternative approach that allows for more control over the nuances and heterogeneity across studies is to combine individual-level data, when available, from multiple studies, each of which may contain insufficient information to address a given scientific question by itself, but which collectively have the power to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' There has been a growing body of work on generalizability and transportability methods, which can help address the problem of external validity of the effect estimates from integrating individual level data across studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Generalizability concerns the setting where the study population is a subset of the target population of interest while transportability addresses the setting where the study popula- tion is partially or completely external to the target population (Degtiar and Rose, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Specifically, generalizability typically involves extending the causal effect estimate derived from a study as long as the covariates in the study population and the target population have common support (Gechter, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Tipton, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' On the other hand, transportability entails extrapolating the effect estimated from a study in which some primary outcome of interest is observed to a population represented by a sample in which the outcome is not measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Existing methodologies involve directly transporting some estimated causal effect, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', the average treatmemt effect (ATE), from studies where the outcomes are observed to other studies with missing outcomes or across heterogeneous study designs and settings (Barein- 1 boim and Pearl, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Pearl and Bareinboim, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' H¨unermund and Bareinboim, 2019), or to some broader target population (Dahabreh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Lesko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Westreich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When considering multiple studies, it is often the case that one will observe different outcomes at follow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' However, existing methods do not take advantage of these other potentially correlated and informative outcome variables mea- sured at follow-up, which could potentially be leveraged to achieve large efficiency gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Existing outcome proxy-blind methods typically rely on an assumption of homogeneous con- ditional potential outcome means for valid transportation of estimation from one population to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Sensitivity analysis strategies have been proposed to study the extent to which the violation of these assumptions will affect the estimations and inferences drawn (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Dahabreh and Hern´an, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Dahabreh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In ongoing work, we have developed a new strategy to more efficiently estimate the ATE from integrated data across multi-outcome studies, with inconsistent availability of the primary outcome of interest at the study level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The proposed methodology takes advantage of the availability of follow-up measurements of potential correlates of the main outcome to yield more precise estimate of the causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In this article, we consider the key common outcome regression (or conditional exchangeability for study selection) assumption for transportability while leveraging these outcome proxies, which differs slightly from the common outcome regression assumption that has been traditionally used for transportability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We discuss the resulting bias when this assumption is not met, and develop methodology for sensitivity analysis to the violation of this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The remainder of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In Section 2, we discuss identification of the average treatment effect in the multi-study, multi-outcome setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In Section 3, we discuss the bias incurred by violations of the key conditional exchangeability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In Section 4, we compare the conditional exchangeability assumption in our setting with that used in settings that do not leverage outcome proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In Section 5, we develop methods for sensitivity analysis for when our assumption is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We demonstrate the empirical 2 performance of our proposed methods in a simulation study in Section 6, and conclude with a discussion in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 2 Data integration for studies with primary outcome missing systematically 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='1 Study and data setting In this setting, we let A be the treatment indicator, W be a set of covariates that are commonly observed across studies, Y be the primary outcome variable, the set {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' , Tk} be all the potential outcome proxies measured at follow-up in any study, and Js be the study-specific subset of {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' , Tk} that is measured in study s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Suppose there are S studies that are ordered such that for each s in the first s∗ studies, we observe the set of variables (Y, A, Js, W), while for each s in the remaining S − s∗ studies, only the subset (A, Js, W) are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In other words, Y is systematically missing in the latter set of studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Unlike the standard setup in other works concerning effect transportability that only involves (Y, A, W), we introduced the use of Ts, where Ts ⊂ Js is some user-specified subset of Js for each study s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Ts could be chosen based on availability and subject matter knowledge and must be chosen such that they are observed in at least one of the studies {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' , s∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Studies can be randomized experiments or observational;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' however, we will not consider scenarios in which some studies are randomized experiments and others are observational in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Then the study-specific average treatment effect and conditional average treat- ment effect can be written as: ATE(s) = E(Y1 − Y0 | S = s) CATE(w, s) = E(Y1 − Y0 | W = w, S = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 3 Accordingly, we can define the overall average treatment effect and conditional average treat- ment effect as: ATE = S � s=1 ATE(s) CATE(w) = E(Y1 − Y0 | W = w) where the weights can be user-specified such that � s πs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' For instance, one can choose πs = P(S = s), or the marginal probability of being in each study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Alternatively, we could define ATE = EQW,SCATE(W, S) for a user-specified, known distribution QW,S of W and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Since Y is not measured in s ∈ {s∗ +1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' , S}, we cannot directly estimate the ATE and CATE using data from these studies alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Our purpose is to transport the ATE from the first s∗ studies where Y is observed, to the remaining S − s∗ studies while also leveraging the information from the outcome proxy set Ts to improve efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' For ease of notation, let σs be a subset of the first s∗ studies in which both Y and Ts are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We can then use this information from the studies that form σs to estimate the outcome regression that will allow us to transport the causal effects to study s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In this setting, we have shown in ongoing, not-yet-published work that the ATE can be nonparametrically identified as: ΨATE = s∗ � s=1 πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s} + S � s=s∗+1 πsE[E{E(Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s} (1) − E{E(Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s} | S = s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The terms in the first sum are simply the standard identification formula for the (study- specific) average treatment effects when Y is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The second sum is identified since it only depends on the distribution of Y in the studies in σs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', in which Y is actually 4 observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Here, we introduced a modification to how transportability has traditionally been done by incorporating information from a set of outcomes measured at follow-up that are correlated with the main outcome of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='2 Assumptions for Identification of the ATE This derivation ATE can be nonparametrically identified given the assumptions that are standard for identification for ATE when outcomes are all observed: Assumption 1 (Positivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' P(A = 1 | W = w) > 0 for all w with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Assumption 2 (Consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Y = AY1 + (1 − A)Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Assumption 3 (Within-study conditional exchangeability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' E[Y a | W, A, S = s] = E[Y a | W, S = s] for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The validity of our estimator relies on a fourth assumption that allows for the transporta- tion of the effect across studies: Assumption 4 (Common outcome regression (proxy-aware version)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' E(Y | Ts, W, A = a, S = s) = E(Y | Ts, W, A = a, S ∈ σs) for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This is a missing at random (MAR)-type assumption, where S can in a sense be thought of as a missingness indicator, since missingness is systematic by study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We can also introduce a fifth assumption that is not necessary for identification, but allows for more borrowing of information across studies, which can help with efficiency: Assumption 5 (Common distribution of outcome proxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Ts ⊥ S | W, A for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 5 This implies the distribution of Ts conditional on treatment assignment and baseline covariates is the same across studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Under this additional assumption, the identification result simplifies to: ATE = S � s=1 πsE[E{E(Y | Ts, W, A = 1, S ∈ σs) | W, A = 1} − E{E(Y | Ts, W, A = 0, S ∈ σs) | W, A = 0} | S = s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In ongoing work, we have developed a simple substitution estimator that involves replac- ing each expectation with a regression-based estimate and the outer expectation with an empirical mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' For the outcome proxy-blind approach, in addition to the first three standard internal validity assumptions, Assumption 4 is replaced by a slightly different mean outcome ex- changeability assumption: across studies assumption (exchangeability over S) (Dahabreh and Hern´an, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Lesko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2017): Assumption 6 (Common outcome regression (proxy-blind version)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' E(Y | W, A = a, S = s) = E(Y | W, A = a, S ∈ σs) for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Assumption 4 differs from Assumption 6 by additionally conditioning on Ts for each study s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Assumptions 4 and 5 together imply Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In this article, we will only consider sensitivity analysis for the violation of Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When Assumption 5 is violated, the ATE estimator based on Assumption 4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', the substitution estimator based on the identification formula (1)) will remain consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 6 3 Characterizing the bias resulting from violation of the identification assumption The validity of ΨATE is dependent on the key assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This assumption requires no heterogeneity in the conditional outcome means given treatment, covariates, and outcomes proxies between studies with and without missing outcome (Y ) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This allows for trans- portation of the conditional outcome means, and correspondingly, the ATE and CATE, estimable from one study to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In practice, this could be a strong assumption to make while also untestable using ob- served data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' For instance, in previous unpublished work, we estimated the average treatment effect of cognitive remediation (CR) therapy on Social Behavioral Scale (SBS) score, a mea- sure for social functioning, using harmonized data from three trials in the NIMH Database of Cognitive Training and Remediation Studies (DoCTRS) database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' However, the degree of effectiveness of CR, especially on functional and occupational outcomes, was less evident and has been suggested to vary depending on the setting in which the treatment was admin- istered (Barlati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Combs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' McGurk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Wykes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', 2007, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When this assumption is violated, the substitution estimators described in the pre- vious section will be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Therefore, we examine two strategies for sensitivity analysis in order to examine the robustness of estimates under varying degrees of assumption violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' To quantify the degree of violation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' let the bias functions be defined as: u(A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W) = E(Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) − E(Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S ∈ σs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' u(A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W) = E(Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) − E(Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S ∈ σs) (2) 7 Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' equation (1) when assumption 4 is violated instead becomes: ATE = s∗ � s=1 πsE{E(Y | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) − E(Y | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) | S = s) + S � s=s∗+1 πsE [E {E (Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S ∈ σs) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s)} −E {E (Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S ∈ σs) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s)} | S = s] + S � s=s∗+1 πsE[E {u (A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s} − E {u (A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' TS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s} | S = s],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' where the last sum is not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Then, the study-specific bias for study s is: E [E {u (A = 1, Ts, W) | W, A = 1, S = s} − E {u (A = 0, Ts, W) | W, A = 0, S = s} | S = s] = E[δ∗(W)|S = s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' (3) By rearranging terms, δ∗(W) can be alternatively written as: E [E (Y | Ts, W, A = 1, S = s) − E (Y | TS, W, A = 1, s ∈ σs) | W, A = 1, S = s] − E [E (Y | Ts, W, A = 0, S = s) − E (Y | Ts, W, A = 0, s ∈ σs) | W, A = 0, S = s] = E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) − {E [E (Y | Ts, W, A = 1, s ∈ σs) | W, A = 1, S = s] − E [E (Y | Ts, W, A = 0, s ∈ σs) | W, A = 0, S = s]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' (4) The latter term cannot be simplified unless Assumption 5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 8 4 Comparison with bias functions in settings without incorporation of follow-up surrogate outcomes In recent work, Dahabreh and Hern´an (2019) developed sensitivity analysis for transportabil- ity considering a similar setting of two types of studies with and without missing outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In the base case, there are two studies considered (missingness of the outcome variable de- noted by a binary indicator S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' To describe this setting using our notation, we simply have σ0 = σ1 = {1} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', study S = 1 with the observed outcome of interest is used to impute the conditional outcome means for study S = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Equivalently, for ease of interpretation in the base case, let S = 1 and S = 0 denote the study where the primary outcome of interest is observed and not observed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In the setting where the model used to impute conditional potential outcomes does not utilize information from Ts, Dahabreh and Hern´an (2019) define: u(A = a, W) = E[Y | A = a, W, S = 1] − E[Y | A = a, W, S = 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The difference between these bias functions can then be obtained as: δ(W) = u(A = 1, W) − u(A = 0, W) = E[Y 1 − Y 0 | W, S = 1] − E[Y 1 − Y 0 | W, S = 0] This expression can be qualitatively expressed as the difference in the conditional average treatment effects between the two studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This qualitative interpretation can aid in concep- tualizing and thinking about more appropriate values and range for sensitivity parameters when examining robustness of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' More specifically, assuming higher levels of the outcome are preferred, if we believe the participants in studies with missing outcomes benefit less from treatment, then true δ can be assumed to be positive and vice versa (Dahabreh and Hern´an, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Since our bias functions are conditional on the set of proxy outcomes, the 9 term δ∗(W) in (4) unfortunately cannot be reduced further to a more interpretable statistical entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When we take Ts to be the empty set, the bias function δ∗(W) reduces to the same expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 5 Accounting for violation of the common outcome re- gression assumption through sensitivity analyses We consider two scenarios in which we assume the bias terms u(A = 1, Ts, W) and u(A = 0, Ts, W) to be 1) constants and 2) bounded functions of the outcome proxies and/or baseline covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The first scenario involves making a stronger assumption about the bias terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' On the other hand, the second scenario requires weaker assumptions but allow them to be non-constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='1 Bias functions assumed to be some fixed values Although it might be more reasonable to assume that the bias functions are dependent on some baseline covariates, for ease of implementation of sensitivity analysis, one can also suppose they are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When u(A = 1, Ts, W) and u(A = 0, Ts, W) are independent of the baseline covariates W and the outcome proxy set Ts, the conditional expectations of the bias functions, and in turn, the term δ∗(W) in (3), reduce to: δ = u1 − u0, where δ, u1, and u0 ∈ R (5) The sensitivity analysis involves correcting for the above-mentioned bias term by adding it back to the identification formula ΨATE, which relies on the common outcome regression assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 10 ATE = s∗ � s=1 πsE {E (Y | W, A = 1, S ∈ σs) − E (Y | W, A = 0, S ∈ σs) | S = s} + S � s=s∗+1 πsE [E {E (Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s} − E {E (Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s} | S = s] + S � s=s∗+1 πs (u1 − u0) =ΨATE + S � s=s∗+1 πs (u1 − u0) (6) where u1 and u0 are scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In practice, the true bias term would be unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Thus, one strategy is to propose a grid of sensitivity parameters that covers the potential range of values in which the true bias term might fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This grid of sensitivity parameters can be specified using subject-matter knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We can then adjust for the bias term in the estimation step by adding back the different sensitivity parameters to the estimated ATE using our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This also allows for observation of the behavior of the estimated ATE as we vary the sensitivity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='2 Bounded covariate-dependent bias functions One might also believe that the bias term is not constant at all levels of the baseline covariates and/or the outcome proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When the assumption of fixed-value bias terms is considered too strong, but the functional forms for bias terms cannot be confidently determined from existing knowledge of the data mechanism (as will typically be the case), one can still recover some information about the true ATE without having to correctly specify the bias terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' If we instead assume the bias terms to be some bounded functions, we can compute a bound around the (na¨ıve) ATE estimate that contains the true ATE by varying the bounds of these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This provides information on how far away the true ATE can be from the estimate 11 obtained constrained by the bounds of the bias term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Identifying the bounds for the bias term can be expressed as maximizing and minimizing the objective function: E[E[u(A = 1, Ts, W) | W, A = 1, S = s] − E[u(A = 0, Ts, W) | W, A = 0, S = s] | S = s] subject to the following constraints: |u(A = 1, Ts = ts, W = w)| ≤ γ1 |u(A = 0, Ts = ts, W = w)| ≤ γ0 for all ts and w, which implies |E[u(A = 1, Ts, W) | W, A = 1, S = s]| ≤ γ1 and |E[u(A = 0, Ts, W) | W, A = 0, S = s]| ≤ γ0 where γ1, γ1 ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Then we have −(γ1 + γ0) ≤ u(A = 1, Ts, W) − u(A = 0, Ts, W) ≤ γ1 + γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' If we have no reason to suspect we know more about the bounds of one bias function than the other (as will typically be the case), we may simply choose to specify a scalar sensitivity parameter γ to be the maximum of γ1 and γ2, in which case we have −2γ ≤ u(A = 1, Ts, W) − u(A = 0, Ts, W) ≤ 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' By equation (6) even though we do not know the form of the bias functions u(A = 1, Ts, W) and u(A = 0, Ts, W), we can partially recover the true ATE using the bounds around the na¨ıve estimate: ΨATE − 2 max(γ1, γ0) ≤ ATE ≤ ΨATE + 2 max(γ1, γ0) ΨATE − 2γ ≤ ATE ≤ ΨATE + 2γ (7) If the bias functions are in fact bounded by some value smaller than or equal to our specified values for the sensitivity bounds, the true ATE would fall between [ΨATE − 2γ, ΨATE + 2γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Then, the true ATE is partially identified without assumptions about the functional form 12 of u(A = 1, Ts, W) and u(A = 0, Ts, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' One can then use the bootstrap standard error for the substitution estimator of the identification formula (1) to determine the amount to add and subtract from the upper and lower bounds, respectively, in order to produce confidence intervals for the partial identification sets for each value of the sensitivity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Since the sensitivity bounds are a deterministic function of the sensitivity parameter, bootstrapping need only be done once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 6 Simulations 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='1 Data generating mechanism We consider the setting of two studies, with S = 1 indicating the study where the primary outcome is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We generate random sample draws with sample size n = 100 for both studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The data generating mechanism is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W, T0 come from independent standard normal distributions, and T1 comes from a normal distribution with mean and variance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Then T = I(A = 1) × T1 + I(A = 0) × T0 Y 0 = −4T0 + W + ϵ0 Y 1 = 4T1 + W + ϵ1 Y = I(A = 1) × Y1 + I(A = 0) × Y0 where ϵ1, ϵ0 ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Via these specifications, T fully mediates the relationship between A and Y (direct effect from A to Y is constrained to be 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' As a result, the true ATE = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This is also a more basic setting in which the vector T is observed in all studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Due to the nature of the DoCTRS database, which is comprised of randomized clinical trials, in our base setting, we specified the marginal probability P(A = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5, represent- 13 ing random treatment assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This treatment assignment satisfies the positivity and exchangeability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Specifically, to incorporate the difference in conditional outcome means between the two types of studies, in studies missing the outcome, we added constant bias terms to the counterfactual outcomes Y0 and Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Similar to the data generating step, we preserved the observed counterfactual outcome from the corresponding treatment assignment, which satisfies the consistency assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' By (5), we have: Y 0 S=1 = Y 0 S=0 + u0 Y 1 S=1 = Y 1 S=0 + u0 + δ (8) for u0 ∈ {−3, 0, 3}, δ ∈ {−2, 0, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Then the bias reduces to a single parameter δ, since it is no longer a function of u0 when computing the ATE: E(Y 1 − Y 0 | S = 1) = E(Y 1 − Y 0 | S = 0) + δ (9) In the case where the bias term is a function of baseline covariates and surrogate outcome, we had the following specification for the true bias: u0 = b0 × sin (Ts + W) u1 = b1 × exp(Ts+W) 1+exp(Ts+W) for b0 ∈ {2, 3, 4} and b1 ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='2 Adjusting for sensitivity parameter in estimation step In the presence of non-zero bias, when the value of the sensitivity parameter δ is specified such that it is equal to true δ, the ATE estimate after bias adjustment tends to be closer to 14 the true ATE after compared to before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In addition, the corresponding 95% CIs are expected to cover the true ATE 95% of the times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Although coverage probability can be examined more in a more robust fashion using bootstrapped confidence intervals across all simulations, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 1, 2, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='4, the 95% CIs covers the true ATE at the value of the sensitivity parameter that reflects the degree of assumption violation all but one instance, which is in line with our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Scenario 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When the bias terms are assumed to be constants, a natural approach would be to specify a two-dimensional grid of sensitivity parameters for both scalars u0 and u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' However, by (8), it is equivalent to specifying u0 (or u1) and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In fact, since the u0 (or u1) as constant terms cancel out during adjustment, it is sufficient to specify one sensitivity parameter δ (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We also note that δ being 0 does not necessarily imply assumption 4 is met, since the bias terms u0 and u1 could cancel exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' To implement sensitivity analysis, we follow the steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Specify a grid of sensitivity parameters δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The grid should be reasonably wide to contain true δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Estimate the na¨ıvely transported ATE using the identification result in (1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Sequentially add the values in the sensitivity parameter grid to the na¨ıvely estimated ATE, using the result in (6) to obtain the bias-corrected ATE estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We then plotted the bias-corrected estimates under different sensitivity parameters against the true ATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Additionally, we bootstrapped the bias-corrected estimates to obtain the 95% confidence intervals and explore coverage across different values of u0 and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When we want to make minimal assumptions about the functional form of the bias, we can still perform sensitivity analysis on the true ATE using the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Specify a grid of sensitivity parameters called γ that potentially include the upper and lower bounds of the true bias functions 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Computed the “na¨ıve” ATE estimate using the identification result in (1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Construct the upper and lower bound around the estimated ATE using (7) where γ is replaced with the sensitivity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We also plot the na¨ıve ATE estimates and the bounds around these estimates at each value of the sensitivity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In practice, the bias functions are of course unknown and cannot be estimated from observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Therefore, when specifying the grid of sensitivity parameters, the analyst needs to employ subject matter knowledge about the data generating mechanism to select values of δ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We then explore the behavior of the adjusted estimators via simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In the first case, we focused on the general unbiasedness of the correctly-adjusted point estimate for both the overall ATE and ATE among studies with missing outcomes, as well as the 95% CI coverage across degrees of assumption violation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', across values of true u0 and δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In the second case, we looked for correct bounding of the true ATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='3 Simulation Results 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='1 Bias terms as constants We examine the estimates produced by our method under the different degrees of violation of assumption 4, before and after taking into account the specified sensitivity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Figure 1 shows the estimates (95% CI) for the true overall ATE using our method under varying magnitudes and directions of the bias terms from one single simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 16 Figure 1: Sensitivity-parameter-adjusted ATE estimate shown against the true overall ATE across values of the true bias and sensitivity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' n=100 for each study, 95% CI constructed from 1000 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Horizontal dotted line shows the true ATE given true δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' vertical dotted line indicates sensitivity parameter δ equals true δ In the presence of non-zero bias, when the value of the sensitivity parameter δ is specified such that it is equal to true δ, the ATE estimate after bias adjustment tends to be closer to the true ATE after compared to before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In addition, the corresponding 95% CIs are expected to cover the true ATE 95% of the times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Although coverage probability can be examined more in a more robust fashion using bootstrapped confidence intervals across all simulations, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 1, 2, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='4, the 95% CIs covers the true ATE at the value of the sensitivity parameter that reflects the degree of assumption violation all but one instance, which is in 17 True = -2 True = 0 True θ = 2 UU 2 Estimate 4 JU 2 2 Sensitivity parameterline with our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Figure 2: Sensitivity-parameter-adjusted ATE estimates shown against the true study- specific ATE in the study in which the outcome is unobserved across values of the true bias and sensitivity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' n=100 for each study, 95% CI constructed from 1000 boot- strap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Horizontal dotted line shows the true study-specific ATE given true δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' vertical dotted line indicates sensitivity parameter δ equals true δ Figure 2 shows similar results for the study-specific ATE estimates in the study with missing outcomes (before and after bias adjustment) from the same simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Com- pared to the results in Figure 1, after adjustment using the correct sensitivity parameters, the 95% CIs contain the true ATE more frequently than the CIs of the unadjusted estimates in the study with missing primary outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Figure 2 also shows an example where infer- 18 True = -2 True = 0 True = 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 uo = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5 Estimate rue 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 uo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5 II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5 II 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5 2 0 2 2 0 2 2 Sensitivity parameterence is sensitive to the violation of our assumption at a magnitude of δ between -1 and -2 (u0 = −3, bottom left panel), between which the 95% CI changes from not containing to zero to containing zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When we increased the sample size (n=200 and n=500), we saw general reductions in the errors of these single estimates (Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In most cases, even when there is error in the adjusted estimates, the 95% CI bootstrap confidence intervals provide good coverage (Figures 1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The reduction in error and improved coverage are more pronounced when estimating the study-specific effect in the study with missing outcomes than in the overall ATE combining the two studies (Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We also ran 1000 simulations under the same data generating mechanism and obtained the unadjusted and sensitivity-parameter-adjusted estimates for each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We then showed the mean and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th quantiles of these estimates under each combination of the true bias values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We can see that when averaged across 1000 simulations, the adjusted estimates closely approximate the true ATE (Figures 3, 4) when the true value of δ is used for the sensitivity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 19 Figure 3: Sensitivity-parameter-adjusted ATE estimates shown against the true overall ATE across values of the true bias sensitivity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' mean, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th quantiles obtained from 1000 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Horizontal dotted line shows the true overall ATE given true δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' vertical dotted line indicates sensitivity parameter δ equals true δ 20 True = -2 True = 0 True = 2 6 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' True 4 uO = 3 2 6 Estimate 150 True uO = 0 4 2 6 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' True uO = 3 4 2 2 1 0 2 2 1 0 " 2 2 1 0 2 Sensitivity parameter Figure 4: Sensitivity-parameter-adjusted ATE estimates shown against the true ATE in the study with missing outcome across values of the true bias and sensitivity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' mean, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th quantiles obtained from 1000 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Horizontal dotted line shows the true study-specific ATE given true δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' vertical dotted line indicates sensitivity parameter δ equals true δ When approximate sensitivity parameters δ are used (δ ∈ {−1, 1} when true δ ∈ {−2, 2}), the middle 95% values of adjusted estimates also cover the true ATE whereas those of unadjusted estimates do not (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Figure 5 compares the errors in the estimates and sensitivity of associated inferences between the outcome proxy-blind method of Dahabreh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Lesko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' (2017) and our proposed method across 1000 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 21 True = -2 True = 0 True = 2 8 6 True uO = -3 4 2 0 8 6 Estimate True uO = 0 4 2 0 8 6 True uO = 3 4 2 0 2 U 2 2 1 2 0 2 Sensitivity parameterFigure 5: Sensitivity-parameter-adjusted ATE estimates obtained from our proposed method and the outcome proxy-blind method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' mean, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th quantiles obtained from 1000 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Horizontal dotted line shows the true overall ATE given true δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' vertical dotted line indicates sensitivity parameter δ equals true δ The distributions of the estimates from both methods are centered on the true parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' However, the estimates tend to be more precise when we utilize the information from the outcome proxy (as demonstrated through the narrower 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th-97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5th quantile range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The efficiency gains have implications for the sensitivity analysis, since resulting inferences are not as sensitive given analogous magnitude in violation of the identification assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Assumption 4 implies both u0 and u1 equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' As a result, the true δ also equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This suggests transportation of the conditional potential outcome means, and in turn, the 22 True = -2 True = 0 True θ = 2 6 5 3 6 4 3 6 5 2 2 2 2 2 Sensitivityparameter Outcome regression method Proposed methodconditional average treatment effects, can be done without incurring bias (vertical middle panes, figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We also observed that, when δ is 0, regardless of the values of u0 (and u1), there is also no bias (vertical middle panes, figure 3) in the unadjusted estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' In both cases, no bias correction would be necessary, and incorporating a non-zero δ sensitivity parameter will actually introduce bias to the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='2 Bias terms as bounded functions When the sensitivity parameter γ is greater or equal to max{γ0, γ1} for the true function bounds γ0 and γ1, the bounds always include the true ATE when the bias functions are bounded by γ0 and γ1 (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 23 Figure 6: ATE estimates with sensitivity bounds shown against the true overall ATE across values of the true bias and sensitivity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter γ = 0, the bounds collapse to a point estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Blue horizontal dotted line shows the true study-specific ATE given true bias functions Although this approach requires minimal assumptions about the bias functional form, it can also be conservative since the true bias functions are unlikely to evaluate to the bounds across the domain of the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' For instance, the bottom three panels of Figure 6 show that when the sensitivity parameter γ is greater than or equal to max(true γ0, true γ1), while the bounds on the estimate contain the true ATE, they also contains the null value zero as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' On the other hand, these bounds do not rely on an assumption of constant bias functions, which we may often have no reason to believe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Here, we demonstrated through 24 True y 1 = 1 True 1 = 2 True y 1 = 3 U Estimated ATEwith 4 n 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Sensitivity parameter ysimulations that sensitivity analysis with relaxed and more credible assumptions can still provide helpful information about the parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' However, when the bounds are too narrow or too wide, sensitivity analysis using bounded bias functions might not be accurate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', not containing the true parameter) or useful (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', containing the null value when the truth is non-null), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 7 Discussion In this paper, we discussed a data integrative method that utilizes information from avail- able proxies of the outcome of interest measured at follow-up for efficiency gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We then presented two sensitivity analysis strategies specific to this approach for causal effect trans- portation when the identification assumption is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Our modification to the identifica- tion of the ATE in (1) allows for more efficient estimators given sufficiently strong outcome proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' As a result, our bias functions also have similar, yet distinct interpretations than the bias functions of Dahabreh and Hern´an (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When the bias terms are assumed to be constants, we can obtain different bias-adjusted point estimates based on our specification of the sensitivity parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Additionally, via obtaining the 95% bootstrap confidence interval for the bias-adjusted estimates, we can examine the robustness of inferences made using our method under varying magnitudes of assumption violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Specifically, beyond certain values of the sensitivity parameters, the 95% CI will cross the null value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' These are the degrees of violation that can affect inferences (where the 95% CI suggest a change from significant results to non-significant results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' We also proposed sensitivity analysis using bounded bias functions as an alternative when one believes the assumption of a fixed-value bias term is too strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This approach allows for inferences with minimal assumptions about the unobserved bias functions but can still provide useful information about the parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Due to fewer assumptions being made, the results are more conservative and robust, hence more reasonable and credible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 25 Specifically, although we are unable to obtain a point estimate, sensitivity analysis using bounded bias functions can still be informative in the sense of providing information about the general direction of the parameter of interest (beneficial or harmful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' This method is generally more conservative if the bounds on the functions are not close to their extreme values, if the bias functions are generally not close to their extreme values, or if there is a large difference between the extrema of the two bias functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Correct specification of the bias functions would allow for more precise and informative estimation of the true ATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' However, since they are generally unknown and non-estimable from observed data, sensitivity analysis will typically be the realistic course of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When conducting sensitivity analysis, the analyst can start off by specifying a wide grid of the sensitivity parameter and examining the behaviors of the point estimates and 95% CI (first approach) as well as bounds around the estimates (second approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' They can then search for the “critical” sensitivity parameters that still suggest rejection of the null hypothesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', the 95% CI (in the first case) and bounds around the estimate (in the second case) that do not contain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' It can be determined if greater bias is plausible by using background knowledge of the data generating mechanism or further hypothesizing about such mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' If there is little or no evidence that the true bias functions exceed these critical sensitivity parameters, one can be more comfortable in concluding that the observed effect and associated 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Comparing clinical trial population representativeness to real-world populations: An external validity analysis encompassing 43,895,trials and 5,685,738 individuals across 989 unique drugs and 286 conditions in England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The Lancet Healthy Longevity, 3(10):e674–e689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 1 Tipton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' How generalizable is your experiment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' An index for comparing ex- perimental samples and populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Journal of Educational and Behavioral Statistics, 39(6):478–501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 1 Westreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Edwards, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Lesko, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Stuart, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', and Cole, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Trans- portability of trial results using inverse odds of sampling weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' American Journal of Epidemiology, 186(8):1010–1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 2 Wykes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Huddy, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Cellard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', McGurk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', and Czobor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A meta-analysis of cognitive remediation for schizophrenia: Methodology and effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The American Journal of Psychiatry, 168(5):472–485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 7 Wykes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Reeder, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Landau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Everitt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Knapp, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', Patel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=', and Romeo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Cognitive remediation therapy in schizophrenia: Randomised controlled trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' The British Journal of Psychiatry, 190(5):421–427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 7 29 A Additional figures Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='1: Sensitivity-parameter-adjusted ATE estimates shown against the true overall ATE across values of the true bias and sensitivity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' n=200 for each study, 95% CI constructed from 1000 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Horizontal dotted line shows the true overall ATE given true δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' vertical dotted line indicates sensitivity parameter δ equals true δ 30 True = -2 True = 0 True θ = 2 6 6 stimate 5 rue uo 4 = 3 6 5 4 3 3 2 0 2 0 2 2 2 Sensitivity parameterFigure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='2: Sensitivity-parameter-adjusted ATE estimates shown against the true ATE in the study with missing outcome across values of the true bias and sensitivity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' n=200 for each study, 95% CI constructed from 1000 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Horizontal dotted line shows the true study-specific ATE given true δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' vertical dotted line indicates sensitivity parameter δ equals true δ 31 True = -2 True = 0 True θ = 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 rue uo = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='0 8 6 Estimate True 4 uo II 0 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' True uO = 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' * 3 0 2 0 2 2 0 2 Sensitivity parameterFigure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='3: Sensitivity-parameter-adjusted ATE estimates shown against the true overall ATE across values of the true bias and sensitivity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' n=500 for each study, 95% CI constructed from 1000 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Horizontal dotted line shows the true overall ATE given true δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' vertical dotted line indicates sensitivity parameter δ equals true δ 32 True = -2 True = 0 True θ = 2 6 5 4 3 =3 2 6 5 Estimate True 4 uo = 3 2 6 True uO 4 II 3 0 2 2 Sensitivity parameter Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content='4: Sensitivity-parameter-adjusted ATE estimates shown against the true ATE in the study with missing outcome across values of the true bias and sensitivity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' n=500 for each study, 95% CI constructed from 1000 bootstrap samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' When sensitivity parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Horizontal dotted line shows the true study-specific ATE given true δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' vertical dotted line indicates sensitivity parameter δ equals true δ 33 True = -2 True = 0 True θ = 2 6 True 4 u0 = -3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' 8 Estimate 6 True uO = 0 4 T 0 8 I 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' True uO = 4 2 2 0 2 0 2 2 2 Sensitivity parameterB Derivation of the sensitivity analysis formula when Assumption 4 is violated ATE = s∗ � s=1 πsE{E(Y | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) − E(Y | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) | S = s} + S � s=s∗+1 πsE [E {E (Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s} − E {E (Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s} | S = s] = s∗ � s=1 πsE{E(Y | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) − E(Y | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) | S = s} + S � s=s∗+1 πsE [E {E (Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S ∈ σs) + u (A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s} − E {E (Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S ∈ σs) + u (A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s} | S = s] = s∗ � s=1 πsE{E(Y | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) − E(Y | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s) | S = s) + S � s=s∗+1 πsE [E {E (Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S ∈ σs) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s)} −E {E (Y | Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S ∈ σs) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s)} | S = s] + S � s=s∗+1 πsE[E {u (A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s} − E {u (A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' W) | W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' A = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQfFgPu/content/2301.02904v1.pdf'} +page_content=' S = s} | S = s] 34' 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0000000000000000000000000000000000000000..f04df70d9563a73d519158b6928a4bc019389b0b Binary files /dev/null and b/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf differ diff --git a/JNAyT4oBgHgl3EQfTfcu/content/tmp_files/2301.00105v1.pdf.txt b/JNAyT4oBgHgl3EQfTfcu/content/tmp_files/2301.00105v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e35d68687386d54daeec0c03424d6998a7d2537 --- /dev/null +++ b/JNAyT4oBgHgl3EQfTfcu/content/tmp_files/2301.00105v1.pdf.txt @@ -0,0 +1,379 @@ +arXiv:2301.00105v1 [math.ST] 31 Dec 2022 +Inequality for the variance of an asymmetric loss +Naoya Yamaguchi, Yuka Yamaguchi and Maiya Hori +January 3, 2023 +Abstract +We assume that the forecast error follows a probability distribution which is +symmetric and monotonically non-increasing on non-negative real numbers, and if +there is a mismatch between observed and predicted value, then we suffer a loss. +Under the assumptions, we solve a minimization problem with an asymmetric loss +function. In addition, we give an inequality for the variance of the loss. +1 +Introduction +Let ˆy be a predicted value of an observed value y. In this paper, we make the assump- +tions (I) and (II): +(I) The prediction error z := ˆy − y is the realized value of a random variable Z, whose +probability density function f(z) satisfies f(x) = f(−x) for x ∈ R and f(x) ≥ f(y) +for 0 ≤ x ≤ y. +(II) Let k1, k2 ∈ R>0. If there is a mismatch between y and ˆy, then we suffer a loss +L(z) := +� +k1z, +z ≥ 0, +−k2z, +z < 0. +Under the assumptions (I) and (II), we solve the minimization problem for the expected +value of L(Z + c): +C = arg min +c {E[L(Z + c)]}. +In addition, we give the following theorem. +Theorem 1. We have +V [L(Z + C)] ≤ V [L(Z)], +where equality holds only when C = 0; that is, when k1 = k2. +Theorem 1 is obtained by the following lemma. +Lemma 2. Suppose that a probability density function f(t) is monotonically non-increasing +on R≥0 and satisfies +� ∞ +0 f(t)dt = 1 +2. Then, for any x ≥ 0, we have +F(x) := 4 +� x +0 +f(t)dt +� ∞ +x +tf(t)dt − x +2 + 2x +�� x +0 +f(t)dt +�2 +≥ 0. +If f(t) is strictly decreasing, then F(x) > 0 holds for x > 0. Also, F(x) = 0 holds for +x ≥ 0 if and only if f(t) equals to the probability density function of a continuous uniform +distribution on R≥0. +1 + +These results are a generalization of the results of [5]. The paper [5] made the as- +sumptions (I’) and (II): +(I’) The prediction error z := ˆy − y is the realized value of a random variable Z, whose +probability density function is a generalized Gaussian distribution function (see, +e.g., [1], [2], and [3]) with mean zero +f(z) := +1 +2abΓ(a) exp +� +− +���z +b +��� +1 +a� +, +where Γ(a) is the gamma function and a, b > 0. +Assumption (I) is weaker than (I’). Thus, we assume a more general situation than in [5]. +In [5], under the assumptions (I’) and (II), the minimization problem for the expected +value of L(Z + c) is solved and the inequality V [L(Z + C)] ≤ V [L(Z)] is obtained. This +inequality is derived from the following inequality: For a, x > 0, we have +xaγ(a, x)2 − xaΓ(a)2 + 2γ(a, x)Γ(2a, x) > 0, +(1) +where +Γ(a) := +� +∞ +0 +ta−1e−tdt, +Γ(a, x) := +� +∞ +x +ta−1e−tdt, +γ(a, x) := +� x +0 +ta−1e−tdt. +Inequality (1) is the special case of Lemma 2 that f(z) is a generalized Gaussian distri- +bution function. +Assumptions (I) and (II) have a background in the procurement from an electricity +market. Suppose that we purchase electricity ˆy from an market, based on a forecast of the +electricity y that will be needed. This situation makes the assumption (I). If ˆy − y > 0, +then there is a waste of procurement fee proportional to ˆy − y. If y − ˆy > 0, then we are +charged with a penalty proportional to y − ˆy. This situation makes the assumption (II). +For details, see [4]. +2 +Proof of results +For c ∈ R, let sgn(c) := 1 (c ≥ 0); −1 (c < 0). From +� ∞ +0 f(z)dz = 1 +2, the expected value +of L(Z + c) and L(Z + c)2 are as follows: For any c ∈ R, +E[L(Z + c)] = (k1 + k2) +� ∞ +|c| +zf(z)dz + c(k1 − k2) +2 ++ |c|(k1 + k2) +� |c| +0 +f(z)dz, +E[L(Z + c)2] = (k2 +1 + k2 +2) +� ∞ +0 +z2f(z)dz + sgn(c)(k2 +1 − k2 +2) +� |c| +0 +z2f(z)dz ++ 2c(k2 +1 − k2 +2) +� ∞ +|c| +zf(z)dz + c2(k2 +1 + k2 +2) +2 ++ c|c|(k2 +1 − k2 +2) +� |c| +0 +f(z)dz. +Therefore, the expected value and the variance of L(Z) are as follows: +E[L(Z)] = (k1 + k2) +� ∞ +0 +zf(z)dz, +V[L(Z)] = (k2 +1 + k2 +2) +� ∞ +0 +z2f(z)dz − (k1 + k2)2 +�� ∞ +0 +zf(z)dz +�2 +. +2 + +We determine the value c that gives the minimum value of E[L(Z + c)]. From +d +dc E[L(Z + c)] = k1 − k2 +2 ++ sgn(c)(k1 + k2) +� |c| +0 +f(z)dz, +d2 +dc2 E[L(Z + c)] = (k1 + k2)f(c) ≥ 0, +we can see that E[L(Z + c)] has the minimum value at the zero point of +d +dc E[L(Z + c)]. +The zero point C satisfies the following equation: +k1 − k2 +2 ++ sgn(C)(k1 + k2) +� |C| +0 +f(z)dz = 0. +From this, C = 0 if and only if k1 = k2. Also, we have +E[L(Z + C)] = (k1 + k2) +� ∞ +|C| +zf(z)dz, +V[L(Z + C)] = (k2 +1 + k2 +2) +� ∞ +0 +z2f(z)dz − 2(k1 + k2)2 +� |C| +0 +f(z)dz +� |C| +0 +z2f(z)dz +− 4|C|(k1 + k2)2 +� |C| +0 +f(z)dz +� ∞ +|C| +zf(z)dz + C2(k1 + k2)2 +4 +− (k1 + k2)2 +�� ∞ +|C| +zf(z)dz +�2 +− C2(k1 + k2)2 +�� |C| +0 +f(z)dz +�2 +. +Let +G(x) := − +�� ∞ +0 +zf(z)dz +�2 ++ 2 +� x +0 +f(z)dz +� x +0 +z2f(z)dz + 4x +� x +0 +f(z)dz +� ∞ +x +zf(z)dz +− x2 +4 + +�� ∞ +x +zf(z)dz +�2 ++ x2 +�� x +0 +f(z)dz +�2 +. +Then, V [L(Z)] − V [L(Z + C)] = (k1 + k2)2G(C) holds. From G(0) = 0 and +d +dxG(x) = 4 +� x +0 +f(z)dz +� ∞ +x +zf(z)dz − x +2 + 2x +�� x +0 +f(z)dz +�2 ++ 2f(x) +� x +0 +z2f(z)dz + 2xf(x) +� ∞ +x +zf(z)dz, +if Lemma 2 is proved, then Theorem 1 is immediately obtained. We prove Lemma 2. +Proof of Lemma 2. Take any x ≥ 0. If f(x) = 0, then F(x) = 0 − x +2 + 2x · 1 +4 = 0. Below, +we consider the case that f(x) > 0. Let α := +� x +0 f(t)dt. For a function g = g(t) satisfying +f(x) ≥ g(t) ≥ 0 for x ≤ t and α + +� ∞ +x g(t)dt = 1 +2, we define a functional S(g) by +S(g) := +� ∞ +x +tg(t)dt. +Regarding S(g) as a solid with the bottom surface area +� ∞ +x g(t)dt = 1 +2 − α (constant), we +find that if we make g(t) as large as possible within the range where t is small, then S(g) +become smaller. Thus, the function g that minimizes S(g) is g(t) = u(t) defined by +u(t) := +� +f(x), +x ≤ t ≤ x + +1 +f(x) +� 1 +2 − α +� +, +0, +otherwise. +3 + +From +S(u) = +� ∞ +x +tu(t)dt = x +�1 +2 − α +� ++ +1 +2f(x) +� +α2 − α + 1 +4 +� +and α ≥ xf(x), we have +F(x) ≥ 4αS(u) − x +2 + 2xα2 += 4α +� +x +�1 +2 − α +� ++ +1 +2f(x) +� +α2 − α + 1 +4 +�� +− x +2 + 2xα2 +≥ 2xα − 4xα2 + 2x +� +α2 − α + 1 +4 +� +− x +2 + 2xα2 += 0. +Also, from this, if f(t) is strictly decreasing, then F(x) > 0 holds for x > 0. In addition, +f(t) is the function of the form +f(t) = +� +1 +2a, +0 ≤ t ≤ a, +0, +t > a +if and only if F(x) = 0 holds for x ≥ 0. +4 + +References +[1] Alex Dytso, Ronit Bustin, H. Vincent Poor, and Shlomo Shamai. Analytical prop- +erties of generalized gaussian distributions. Journal of Statistical Distributions and +Applications, 5(1):6, Dec 2018. +[2] Saralees Nadarajah. A generalized normal distribution. Journal of Applied Statistics, +32(7):685–694, 2005. +[3] Th. Subbotin. On the law of frequency of error. Recueil Math´ematique, 31:296–301, +1923. +[4] Naoya Yamaguchi, Maiya Hori, and Yoshinari Ideguchi. Minimising the expectation +value of the procurement cost in electricity markets based on the prediction error of +energy consumption. Pac. J. Math. Ind., 10:Art. 4, 16, 2018. +[5] Naoya Yamaguchi, Yuka Yamaguchi, and Ryuei Nishii. Minimizing the expected value +of the asymmetric loss function and an inequality for the variance of the loss. Journal +of Applied Statistics, 48(13-15):2348–2368, 2021. PMID: 35707067. +Faculty of Education, University of Miyazaki, 1-1 Gakuen Kibanadai-nishi, Miyazaki +889-2192, Japan +Email address, Naoya Yamaguchi: n-yamaguchi@cc.miyazaki-u.ac.jp +Email address, Yuka Yamaguchi: y-yamaguchi@cc.miyazaki-u.ac.jp +General Education Center, Tottori University of Environmental Studies, 1-1-1 +Wakabadai-kita, Tottori, 689-1111. Japan +Email address, Maiya Hori: m-hori@kankyo-u.ac.jp +5 + diff --git a/JNAyT4oBgHgl3EQfTfcu/content/tmp_files/load_file.txt b/JNAyT4oBgHgl3EQfTfcu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d172e8804febb15cadb41ebe0fcfbcdd5eb2312 --- /dev/null +++ b/JNAyT4oBgHgl3EQfTfcu/content/tmp_files/load_file.txt @@ -0,0 +1,94 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf,len=93 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='00105v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='ST] 31 Dec 2022 Inequality for the variance of an asymmetric loss Naoya Yamaguchi, Yuka Yamaguchi and Maiya Hori January 3, 2023 Abstract We assume that the forecast error follows a probability distribution which is symmetric and monotonically non-increasing on non-negative real numbers, and if there is a mismatch between observed and predicted value, then we suffer a loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Under the assumptions, we solve a minimization problem with an asymmetric loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' In addition, we give an inequality for the variance of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' 1 Introduction Let ˆy be a predicted value of an observed value y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' In this paper, we make the assump- tions (I) and (II): (I) The prediction error z := ˆy − y is the realized value of a random variable Z, whose probability density function f(z) satisfies f(x) = f(−x) for x ∈ R and f(x) ≥ f(y) for 0 ≤ x ≤ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' (II) Let k1, k2 ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' If there is a mismatch between y and ˆy, then we suffer a loss L(z) := � k1z, z ≥ 0, −k2z, z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Under the assumptions (I) and (II), we solve the minimization problem for the expected value of L(Z + c): C = arg min c {E[L(Z + c)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' In addition, we give the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' We have V [L(Z + C)] ≤ V [L(Z)], where equality holds only when C = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' that is, when k1 = k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Theorem 1 is obtained by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Suppose that a probability density function f(t) is monotonically non-increasing on R≥0 and satisfies � ∞ 0 f(t)dt = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Then, for any x ≥ 0, we have F(x) := 4 � x 0 f(t)dt � ∞ x tf(t)dt − x 2 + 2x �� x 0 f(t)dt �2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' If f(t) is strictly decreasing, then F(x) > 0 holds for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Also, F(x) = 0 holds for x ≥ 0 if and only if f(t) equals to the probability density function of a continuous uniform distribution on R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' 1 These results are a generalization of the results of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' The paper [5] made the as- sumptions (I’) and (II): (I’) The prediction error z := ˆy − y is the realized value of a random variable Z, whose probability density function is a generalized Gaussian distribution function (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=', [1], [2], and [3]) with mean zero f(z) := 1 2abΓ(a) exp � − ���z b ��� 1 a� , where Γ(a) is the gamma function and a, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Assumption (I) is weaker than (I’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Thus, we assume a more general situation than in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' In [5], under the assumptions (I’) and (II), the minimization problem for the expected value of L(Z + c) is solved and the inequality V [L(Z + C)] ≤ V [L(Z)] is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' This inequality is derived from the following inequality: For a, x > 0, we have xaγ(a, x)2 − xaΓ(a)2 + 2γ(a, x)Γ(2a, x) > 0, (1) where Γ(a) := � +∞ 0 ta−1e−tdt, Γ(a, x) := � +∞ x ta−1e−tdt, γ(a, x) := � x 0 ta−1e−tdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Inequality (1) is the special case of Lemma 2 that f(z) is a generalized Gaussian distri- bution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Assumptions (I) and (II) have a background in the procurement from an electricity market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Suppose that we purchase electricity ˆy from an market, based on a forecast of the electricity y that will be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' This situation makes the assumption (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' If ˆy − y > 0, then there is a waste of procurement fee proportional to ˆy − y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' If y − ˆy > 0, then we are charged with a penalty proportional to y − ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' This situation makes the assumption (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' For details, see [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' 2 Proof of results For c ∈ R, let sgn(c) := 1 (c ≥ 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' −1 (c < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' From � ∞ 0 f(z)dz = 1 2, the expected value of L(Z + c) and L(Z + c)2 are as follows: For any c ∈ R, E[L(Z + c)] = (k1 + k2) � ∞ |c| zf(z)dz + c(k1 − k2) 2 + |c|(k1 + k2) � |c| 0 f(z)dz, E[L(Z + c)2] = (k2 1 + k2 2) � ∞ 0 z2f(z)dz + sgn(c)(k2 1 − k2 2) � |c| 0 z2f(z)dz + 2c(k2 1 − k2 2) � ∞ |c| zf(z)dz + c2(k2 1 + k2 2) 2 + c|c|(k2 1 − k2 2) � |c| 0 f(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Therefore, the expected value and the variance of L(Z) are as follows: E[L(Z)] = (k1 + k2) � ∞ 0 zf(z)dz, V[L(Z)] = (k2 1 + k2 2) � ∞ 0 z2f(z)dz − (k1 + k2)2 �� ∞ 0 zf(z)dz �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' 2 We determine the value c that gives the minimum value of E[L(Z + c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' From d dc E[L(Z + c)] = k1 − k2 2 + sgn(c)(k1 + k2) � |c| 0 f(z)dz, d2 dc2 E[L(Z + c)] = (k1 + k2)f(c) ≥ 0, we can see that E[L(Z + c)] has the minimum value at the zero point of d dc E[L(Z + c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' The zero point C satisfies the following equation: k1 − k2 2 + sgn(C)(k1 + k2) � |C| 0 f(z)dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' From this, C = 0 if and only if k1 = k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Also, we have E[L(Z + C)] = (k1 + k2) � ∞ |C| zf(z)dz, V[L(Z + C)] = (k2 1 + k2 2) � ∞ 0 z2f(z)dz − 2(k1 + k2)2 � |C| 0 f(z)dz � |C| 0 z2f(z)dz − 4|C|(k1 + k2)2 � |C| 0 f(z)dz � ∞ |C| zf(z)dz + C2(k1 + k2)2 4 − (k1 + k2)2 �� ∞ |C| zf(z)dz �2 − C2(k1 + k2)2 �� |C| 0 f(z)dz �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Let G(x) := − �� ∞ 0 zf(z)dz �2 + 2 � x 0 f(z)dz � x 0 z2f(z)dz + 4x � x 0 f(z)dz � ∞ x zf(z)dz − x2 4 + �� ∞ x zf(z)dz �2 + x2 �� x 0 f(z)dz �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Then, V [L(Z)] − V [L(Z + C)] = (k1 + k2)2G(C) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' From G(0) = 0 and d dxG(x) = 4 � x 0 f(z)dz � ∞ x zf(z)dz − x 2 + 2x �� x 0 f(z)dz �2 + 2f(x) � x 0 z2f(z)dz + 2xf(x) � ∞ x zf(z)dz, if Lemma 2 is proved, then Theorem 1 is immediately obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' We prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Take any x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' If f(x) = 0, then F(x) = 0 − x 2 + 2x · 1 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Below, we consider the case that f(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Let α := � x 0 f(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' For a function g = g(t) satisfying f(x) ≥ g(t) ≥ 0 for x ≤ t and α + � ∞ x g(t)dt = 1 2, we define a functional S(g) by S(g) := � ∞ x tg(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Regarding S(g) as a solid with the bottom surface area � ∞ x g(t)dt = 1 2 − α (constant), we find that if we make g(t) as large as possible within the range where t is small, then S(g) become smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Thus, the function g that minimizes S(g) is g(t) = u(t) defined by u(t) := � f(x), x ≤ t ≤ x + 1 f(x) � 1 2 − α � , 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' 3 From S(u) = � ∞ x tu(t)dt = x �1 2 − α � + 1 2f(x) � α2 − α + 1 4 � and α ≥ xf(x), we have F(x) ≥ 4αS(u) − x 2 + 2xα2 = 4α � x �1 2 − α � + 1 2f(x) � α2 − α + 1 4 �� − x 2 + 2xα2 ≥ 2xα − 4xα2 + 2x � α2 − α + 1 4 � − x 2 + 2xα2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Also, from this, if f(t) is strictly decreasing, then F(x) > 0 holds for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' In addition, f(t) is the function of the form f(t) = � 1 2a, 0 ≤ t ≤ a, 0, t > a if and only if F(x) = 0 holds for x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' 4 References [1] Alex Dytso, Ronit Bustin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Vincent Poor, and Shlomo Shamai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Analytical prop- erties of generalized gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Journal of Statistical Distributions and Applications, 5(1):6, Dec 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' [2] Saralees Nadarajah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' A generalized normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Journal of Applied Statistics, 32(7):685–694, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' [3] Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Subbotin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' On the law of frequency of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Recueil Math´ematique, 31:296–301, 1923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' [4] Naoya Yamaguchi, Maiya Hori, and Yoshinari Ideguchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Minimising the expectation value of the procurement cost in electricity markets based on the prediction error of energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Pac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=', 10:Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' 4, 16, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' [5] Naoya Yamaguchi, Yuka Yamaguchi, and Ryuei Nishii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Minimizing the expected value of the asymmetric loss function and an inequality for the variance of the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Journal of Applied Statistics, 48(13-15):2348–2368, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' PMID: 35707067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Faculty of Education, University of Miyazaki, 1-1 Gakuen Kibanadai-nishi, Miyazaki 889-2192, Japan Email address, Naoya Yamaguchi: n-yamaguchi@cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='miyazaki-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='jp Email address, Yuka Yamaguchi: y-yamaguchi@cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='miyazaki-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='jp General Education Center, Tottori University of Environmental Studies, 1-1-1 Wakabadai-kita, Tottori, 689-1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content=' Japan Email address, Maiya Hori: m-hori@kankyo-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} +page_content='jp 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNAyT4oBgHgl3EQfTfcu/content/2301.00105v1.pdf'} diff --git a/KdE3T4oBgHgl3EQfAQmY/vector_store/index.faiss b/KdE3T4oBgHgl3EQfAQmY/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b2cbfd052b9f51f9adfaa9235a52e6c2720aa27d --- /dev/null +++ b/KdE3T4oBgHgl3EQfAQmY/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9cf7d47e8dd4c55f3d8b122070dc69e5360fb3eeb155161b885f33b33d6861ad +size 1441837 diff --git a/KtA0T4oBgHgl3EQfCv_p/content/tmp_files/2301.01995v1.pdf.txt b/KtA0T4oBgHgl3EQfCv_p/content/tmp_files/2301.01995v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3ff8c29aa1f5e9c9dd0489033129eec3d30095f --- /dev/null +++ b/KtA0T4oBgHgl3EQfCv_p/content/tmp_files/2301.01995v1.pdf.txt @@ -0,0 +1,2018 @@ +MNRAS 000, 1–15 (2022) +Preprint 6 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Exploring the Intrinsic Scatter of the Star-Forming Galaxy Main Sequence +at redshift 0.5 to 3.0 +Rongjun Huang 黄钅容钧 ID 1,2★, Andrew J. Battisti +ID 1,2†, Kathryn Grasha +ID 1,2,5‡, Elisabete da Cunha +ID 2,3, +Claudia del P Lagos ID 2,3, Sarah K. Leslie ID 2,4 and Emily Wisnioski ID 1,2 +1Research School of Astronomy and Astrophysics, Australian National University, Cotter Road, Weston Creek, ACT 2611, Australia +2ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia +3International Centre for Radio Astronomy Research, University of Western Australia, 35 Stirling Hwy., Crawley, WA 6009, Australia +4Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, The Netherlands +5Visiting Fellow, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA +Accepted 2023 January 3. Received 2022 December 22; in original form 2022 October 14 +ABSTRACT +Previous studies have shown that the normalization and scatter of the galaxy ‘main sequence’ (MS), the relation between star +formation rate (SFR) and stellar mass (𝑀∗), evolves over cosmic time. However, such studies often rely on photometric redshifts +and/or only rest-frame UV to near-IR data, which may underestimate the SFR and 𝑀∗ uncertainties. We use MAGPHYS+photo-z +to fit the UV to radio spectral energy distributions of 12,380 galaxies in the COSMOS field at 0.5 < 𝑧 < 3.0 and self-consistently +include photometric redshift uncertainties on the derived SFR and 𝑀∗. We quantify the effect on the observed MS scatter from +(1) photometric redshift uncertainties (which are minor) and (2) fitting only rest-frame ultraviolet to near-infrared observations +(which are severe). At fixed redshift and 𝑀∗, we find that the intrinsic MS scatter for our sample of galaxies is 1.4 to 2.6 times +larger than the measurement uncertainty. The average intrinsic MS scatter has decreased by 0.1 dex from 𝑧 = 0.5 to ∼ 2.0. At +low-𝑧, the trend between the intrinsic MS scatter and 𝑀∗ follows a functional form similar to an inverse stellar mass-halo mass +relation (SMHM; 𝑀∗/𝑀halo vs 𝑀∗), with a minimum in intrinsic MS scatter at log(𝑀∗/𝑀⊙) ∼ 10.25 and larger scatter at both +lower and higher 𝑀∗; while this distribution becomes flatter for high-𝑧. The SMHM is thought to be a consequence of feedback +effects and this similarity may suggest a link between galaxy feedback and the intrinsic MS scatter. These results favor a slight +evolution in the intrinsic MS scatter with both redshift and mass. +Key words: methods: observational, galaxies: evolution, galaxies: general, galaxies: star formation +1 INTRODUCTION +The galaxy main sequence (MS) describes the empirical relation +between the star formation rate (SFR) of galaxies and their stellar +masses (e.g., Daddi et al. 2007; Noeske et al. 2007; Speagle et al. +2014; Whitaker et al. 2014; Renzini & Peng 2015; Barro et al. 2017; +Leslie et al. 2020; Thorne et al. 2021). These studies find that galaxies +have higher SFR with increasing redshifts at a fixed stellar mass +(𝑀∗) in the earlier universe, and more massive galaxies have higher +SFRs at a fixed redshift. Some of these studies show a flattening or +turnover in the relationship at high masses (log(𝑀∗/𝑀⊙) > 10.5) +(e.g., Whitaker et al. 2014; Leslie et al. 2020; Thorne et al. 2021) and +suggest that this turnover is driven by the quenching of star formation +due to feedback processes. +The galaxy MS is a powerful tool for understanding and constrain- +ing the distribution and evolution of galaxies (Katsianis et al. 2020; +Curtis-Lake et al. 2021; Popesso et al. 2022; Daddi et al. 2022). +According to theories of galaxy feedback, the existence of a rela- +tively tight MS is thought to be mainly driven by the dynamical +★ E-mail: u6569836@anu.edu.au +† E-mail: andrew.battisti@anu.edu.au +‡ ARC DECRA Fellow +balance between inflows and outflows caused by self-regulated star- +formation and/or active galactic nuclei (AGN; Somerville & Davé +2015). Characterising this evolution is difficult because observations +only provide a single snapshot in time for each observed galaxy. +However, the evolution in the scatter, slope, and normalisation in the +MS of large statistical samples of star-forming (SF) galaxies with +cosmic time provides an indirect way to study galaxy evolution. The +width (or scatter) of the MS at a single redshift is thought to reflect +the burstiness of the average star formation history (e.g., Guo et al. +2013; Schreiber et al. 2015; Santini et al. 2017; Caplar & Tacchella +2019; Donnari et al. 2019; Katsianis et al. 2019; Matthee & Schaye +2019). Theories suggest that a small MS width (small scatter, e.g., +∼ 0.1 dex) is indicative of gradual, continuous star formation histo- +ries (SFHs). In contrast, large MS widths (large scatter, e.g., ∼ 0.4 +dex) are indicative of more bursty, stochastic SFHs (e.g., Tacchella +et al. 2016; Sparre et al. 2017). +The question around whether the intrinsic MS scatter is constant +or evolving is actively debated. Previous studies have found a time- +independent MS scatter (e.g., Daddi et al. 2007; Noeske et al. 2007; +Whitaker et al. 2012; Ciesla et al. 2014; Speagle et al. 2014; Pessa +et al. 2021), while others suggest it evolves with redshift (e.g., Kur- +czynski et al. 2016; Santini et al. 2017; Katsianis et al. 2019; Davies +© 2022 The Authors +arXiv:2301.01995v1 [astro-ph.GA] 5 Jan 2023 + +2 +R. Huang et al. +et al. 2022). As the width of MS is related to the SFH, the MS scatter +can provide useful constraints on the evolution of SF galaxies. For +example, a larger burstiness or stochasticity in the SFH can lead to +an increase in MS scatter, and this may change over cosmic time +(Matthee & Schaye 2019). +Improving our understanding of the galaxy MS and its scatter re- +quires using large samples of galaxies with accurate redshifts. How- +ever, it is observationally expensive to get spectroscopic redshifts for +every galaxy. A common solution is to instead use photometrically +derived redshifts (𝑧phot). Most previous studies of the galaxy MS have +relied on determining stellar masses and SFRs based on SED-fitting +at fixed photometric redshift 𝑧phot and ignore the uncertainty of the +𝑧phot (e.g., Speagle et al. 2014; Leslie et al. 2020; Thorne et al. 2021). +Studies that do not account for zphot uncertainty will systematically +underestimate the uncertainties in all distance-dependent parameters +(e.g., 𝑀∗ & SFR). +In this study, we use MAGPHYS+photo-z (Battisti et al. 2019) to +study the intrinsic scatter of the MS. The improvement in using +MAGPHYS+photo-z is that it sets 𝑧phot as an unknown quantity and +finds its probability distribution (Battisti et al. 2019). Hence, the un- +certainty in the 𝑧phot is incorporated into the overall uncertainty in the +derived physical properties of the galaxy. This allows us to examine +how much of the scatter in the MS is driven by measurement uncer- +tainty as opposed to true intrinsic MS scatter or other measurement +uncertainties. Simultaneously, MAGPHYS+photo-z also includes IR +information to resolve the effect of dust attenuation at UV-near-IR +wavelengths on the SED based on dust emission from mid-IR-radio, +which dramatically improves the accuracy of the derived properties, +particularly for SFRs (Battisti et al. 2019). Therefore, the unique as- +pect of MAGPHYS+photo-z is that it uses broadband photometry to +predict the best-fitting properties in a self-consistent manner, which +helps to mitigate potential biases on the derived values. +This paper is organised as follows: Section 2 introduces the data +and methods used in this study, Section 3 summarises our results, +and Section 4 compares our results with some previous observa- +tional studies and simulations and Section 5 outlines our conclu- +sions. Throughout this paper, the flat Lambda Cold Dark Matter +(ΛCDM) model is adopted by assuming the Hubble constant is 𝐻0 = +70km/s/Mpc and the mass density of the Universe is Ωm,0 = 0.3. +2 DATA AND METHODS +2.1 COSMOS Sample +The multi-wavelength observations of galaxies used in this study +come from two catalogues: the COSMOS2020 catalogue (Weaver +et al. 2022) and the COSMOS Super-deblended catalogue (Jin et al. +2018). The COSMOS2020 catalogue contains photometric data for +∼ 1 million sources in 13 filters from UV to near-IR (Weaver et al. +2022), and the COSMOS Super-deblended catalogue presents photo- +metric data for ∼ 200, 000 galaxies in 11 filters in the mid-IR, far-IR, +and radio (Jin et al. 2018). We cross-match the galaxies’ ID and select +a subsample of galaxies with SEDs that are sampled well enough to +constrain their stellar mass and (dust-corrected) SFRs robustly. To +achieve this, we use two criteria: (1) signal-to-noise ratio (𝑆/𝑁) of +𝑆/𝑁 > 3 in three or more UV–near-IR bands and (2) 𝑆/𝑁>3 in 2 or +more IR–radio bands. By virtue of criterion (2), all of the sources in +our sample have a match in the Super-deblended catalogue. +It is important to note that criterion (2) roughly translates into +a cut in SFR such that only galaxies above a certain SFR will be +included. This SFR threshold increases as a function of increasing +redshift (see Section 2.2.4). Additionally, AGNs are excluded based +on X-ray detections and IR & radio colour cuts (Seymour et al. 2008; +Donley et al. 2012; Kirkpatrick et al. 2013). This is done because +MAGPHYS+photo-z does not include AGN models, so the derived +properties are not accurate for these sources. Due to the limited +availability of data required for these AGN diagnostics, some AGNs +may not be identified and removed. Further details and references on +these cuts are in Section 3.3 of the Battisti et al. (2019). These selec- +tion criteria leave us with a photometric sample of 14,607 galaxies. +For later comparison (Section 3.1.1), only 3,873 of the whole 14,607 +galaxies have spectroscopic redshifts (𝑧spec). +2.2 Methods +2.2.1 MAGPHYS+photo-z +MAGPHYS fits the full SEDs of galaxies with known redshifts from +the ultraviolet to the radio (da Cunha et al. 2008, 2015) by com- +bining the emission from stellar populations with the attenua- +tion and re-emission of starlight by interstellar dust. The recent +MAGPHYS+photo-z extension, described in Battisti et al. (2019), ex- +tends the code to fit the SEDs of galaxies with unknown redshifts, and +constrain the photometric redshift simultaneously with other galaxy +physical properties. In practice, the code builds libraries of model +UV-to-radio SEDs at different redshifts and compares them with +the observed SEDs of galaxies, using a Bayesian method to obtain +the likelihood distributions of physical parameters such as redshifts, +stellar masses, and SFRs. There are two sets of libraries used in +MAGPHYS+photo-z: (1) an optical library that describes emissions +from stars, and (2) an infrared library that describes the emission +from dust. The optical library uses the spectral population synthesis +models of (e.g., Bruzual & Charlot 2003) and initial mass function +from (e.g., Chabrier 2003); while the infrared library consists of +models for PAHs and hot dust emitting in the mid-IR, and warm and +cold dust components in thermal equilibrium that emit in the far-IR +to submillimeter (da Cunha et al. 2008). These two sets of model +libraries maintain the balance of the energy absorbed by dust (via +attenuation in UV to near-IR) and the energy re-emitted by dust (via +thermal emission in mid-IR to sub-mm). Due to insufficient models +at 𝑧 < 0.4 to compare to based on the redshift prior that is adopted in +the MAGPHYS+photo-z, galaxies with 𝑧phot < 0.4 are not constrained +well. Therefore, we exclude galaxies with 𝑧phot < 0.4 after running +the code (Battisti et al. 2019). An example MAGPHYS+photo-z fit +is shown in Figure 1. We compare the distributions of photometric +redshifts of our sample with those of the full COSMOS2020 sample +in Figure 2. +We fit the SEDs of 14,607 galaxies with MAGPHYS+photo-z to +determine the M*, SFR, 𝑧phot and respective errors. We use the 𝜒2 +value of the best-fit model from MAGPHYS+photo-z as an indicator +of the goodness of fit. We fit the 𝜒2 distribution with a lognormal +function (see Figure 3) and convert the lognormal parameters 𝜇 and +𝜎 to the geometric parameters 𝜇geo and 𝜎geo. Finally, we perform +a 2𝜎 confidence cut (i.e., 𝜒2 < 𝜇geo + 2𝜎geo) to the histogram and +remove the high-𝜒2 cases. The remaining galaxies are reduced to +13,639, and their 𝜒2 ≤ 4.76. +However, some galaxies have problematic SEDs due to inconsis- +tencies in fluxes and/or upper limits between bands. In these cases, +MAGPHYS+photo-z derives large uncertainties of 𝑧phot and distance- +dependent parameters. In addition, some cases have multiple redshift +solutions (e.g., degeneracies in Lyman vs Balmer break position), +which can lead to multi-peaked solutions for distance-dependent de- +rived properties. We adopt the following selection criteria based on +MNRAS 000, 1–15 (2022) + +Intrinsic MS Scatter at 0.5 < 𝑧 < 3.0 +3 +Figure 1. Output of MAGPHYS+photo-z for one of the galaxies of our sample, COSMOS2020 ID 1587. The upper panel shows the best-fit SED (black curve), +the observed data (red square) and the predicted unattenuated SED (blue curve). The black open circle on the SED fitting curve is the corresponding model +photometry. The goodness of fit is presented by 𝜒2 in the upper right corner. The lower panel shows the likelihood distribution of 10 basic physical parameters: +𝑧phot, stellar mass (log[𝑀∗/𝑀⊙]), log[SFR/(𝑀⊙yr−1)], specific-SFR (log[sSFR/yr]), dust luminosity (log[𝐿dust/𝐿⊙]), dust mass (log[𝑀dust/𝑀⊙]), mass- +weighted stellar age (log[𝐴𝑔𝑒𝑚/yr]), V-band dust attenuation (𝐴𝑉 /mag), 2175 +◦ +𝐴 bump strength (𝐸′ +𝑏) and the effective dust temperature (𝑇𝑑𝑢𝑠𝑡/K) (Battisti +et al. 2019). +key parameters (i.e., 𝑧phot, 𝑀∗ and SFR): +���� +���� +𝜎(𝑧phot) +≤ 0.25 +𝜎(log(𝑀∗/𝑀⊙)) +≤ 0.3 +𝜎(log(SFR/𝑀⊙yr−1)) +≤ 0.3, +where 𝜎(𝑧phot), 𝜎(log(𝑀∗/𝑀⊙)) and 𝜎(log(SFR/𝑀⊙yr−1)) are +measurement uncertainties for 𝑧phot, 𝑀∗ and SFR, respectively. The +measurement uncertainty is calculated by half of difference between +upper and lower 1𝜎 (68%) boundary of probability distribution func- +tion (PDF) for each parameters derived by MAGPHYS+photo-z. We +restrict the measurement uncertainty on redshift based on the size of +our adopted redshift bins of 0.5 dex (i.e., 2 times larger than the un- +certainty boundary). The limits of uncertainties on 𝑀∗ and SFR are +set to 0.3 dex (roughly a factor of 2) because we want the measure- +ment uncertainties to be lower than the typical intrinsic MS scatter, +which is ∼0.3 dex (e.g., Daddi et al. 2007; Ciesla et al. 2014; Speagle +et al. 2014). These cuts remove 201 galaxies (1.5%) from our sample +and we are left with 13, 418 galaxies. +2.2.2 Reference Main Sequence Relation +Numerous studies have examined the nature of the galaxy MS (e.g., +Speagle et al. 2014; Johnston et al. 2015; Tomczak et al. 2016; +Pearson et al. 2018; Bisigello et al. 2018; Leslie et al. 2020; Thorne +et al. 2021). Tomczak et al. (2016) and Leslie et al. (2020) introduce +nonlinear fits to the MS. For our reference MS relation, we adopt +Leslie et al. (2020) which also used galaxies in the COSMOS field, +which has the form: +log(SFR(𝑀, 𝑡)/𝑀⊙yr−1) = 𝑆0 − 𝑎1𝑡 − log +� +1 + 10𝑀′ +𝑡 +10𝑀 +� +𝑀′ +𝑡 = 𝑀0 + 𝑎2𝑡, +(1) +MNRAS 000, 1–15 (2022) + +t +ID:1587 +Zfit = 0.81 +12 +x2 = 0.74 +(7/77)60] +1f +Resid. +1 +0 +100 +101 +102 +EOT +104 +10 +A/μm [obs-frame] +1DO +0.75 +0.86 ++0.02 +1.56 ++0.07 +-0.12 +-9.44+0.12 +-0.05 +0.02 +-0.14 +-0.13 +0.50 +kel +0.25 +- +0.DO +0 +2 +4 +12 +2 +-12 +-i0 +8 +12 +Zotat +lo-g[MMα ] +lo-g[SFR/(Mα yr-1]] +log[sSFR/yr-1] +og[LdrtfL ] +41.3510.90 +-0.35 +-0.10 +-0.10 +-0.07 +-7.90 +0.50 +ke +0.25 +- +0.0O +8 +1 +6 +8 +5 +115 +0.5 +152 +40 +8 +log[Mdr/M a] +lo-g[Ageur] +Avfmag +Eb' +Taust/K4 +R. Huang et al. +Figure 2. Normalized 𝑧phot histogram for the galaxies. The red histogram in- +dicates the distribution of photo-z’s from MAGPHYS+photo-z for the 14,607 +galaxies in our sample, while the blue histogram represents the parent distri- +bution of the whole 964,506 galaxies from COSMOS2020 catalogue (Weaver +et al. 2022), where the 𝑧phot are derived using LePhare (Arnouts et al. 2002; +Ilbert et al. 2006). We show the corresponding look-back time 𝑡lb on the top +axis. +Figure 3. Distribution of MAGPHYS+photo-z fit 𝜒2 of our 14,607 galaxies. +The black curve represents the normalized lognormal distribution function +fitting to the 𝜒2 histogram, while the vertical black dashed line indicates the +normal 2𝜎 confidence cut within 𝜒2 ≤ 4.76. +where 𝑆0 = 2.97+0.08 +−0.09, 𝑀0 = 11.16+0.15 +−0.16, 𝑎1 = 0.22+0.01 +−0.01, 𝑎2 = +0.12+0.03 +−0.02, 𝑀 is log(𝑀∗/𝑀⊙) and 𝑡 is the age of the universe in +Gyr. Leslie et al. (2020) separate their sample into two classes, ‘All’ +and ‘SF’ (‘Star-forming’). We adopt the ‘SF’ relation, which should +coincide more closely with the sample used in our study. The Leslie +et al. (2020) ‘SF’ sample applies a colour selection (NUV-r-J cut) +that will exclude ‘passive’ galaxies with low SFRs, which has a +similar role as our selection criterion described in next paragraph +(see Section 2.2.3). The probed steller mass range in Leslie et al. +(2020) is 9.0 ≲ 𝑀∗ ≲ 11.0 and redshift range is 0.3 < 𝑧 < 6. +They use radio data to derive SFRs, which provides a dust-unbiased +measurement of the SFR (Leslie et al. 2020). +Although the goal of this study is not to investigate the relation +between SFR and 𝑀∗, we note that the exact functional form of +the MS is still under debate (e.g., Katsianis et al. 2021; Leja et al. +2022). Different methods of estimating SFRs are thought to be the +primary reason for differences between studies (Katsianis et al. 2020). +Hence, despite using similar catalogues from the COSMOS field as +Leslie et al. (2020) that use radio continuum for robust SFRs (dust- +insensitive), there are some other systematic problems that can arise, +such as priors, metallicities, timescales, stellar masses, ages, etc. We +stress that the reference MS we show is intended only to guide the +eye and we do not use it for any selection cuts (i.e., to define ‘on’ vs +‘off’ the MS), which instead are based on sSFR (see Section 2.2.3). +Therefore, the choice of the reference MS has no impact on the results +of this study. +2.2.3 sSFR Selection +We also adopt a specific-SFR (sSFR = SFR/𝑀∗) cut to eliminate +quenched galaxies (see the comparison to U-V-J selection in Ap- +pendix A). These ‘passive’ galaxies form stars at a much lower rate +for a given stellar mass compared to SF galaxies (Renzini & Peng +2015). By definition, quenched galaxies have low sSFR values. The +purpose of the sSFR cut is to remove these red galaxies to avoid +overestimating the MS scatter. Since the sSFR of SF galaxies evolves +with cosmic time (Madau & Dickinson 2014), we adopt a redshift- +dependent cut1 in this selection criterion. In Figure 4, we use a linear +regression model fit to the median-3𝜎 values for sSFR bins (24 bins) +vs 𝑧phot and remove quenched galaxies, which we define as 3 − 𝜎 +outliers lying below the equation: +log(sSFR/yr−1) = 0.57𝑧phot − 11.60. +(2) +2.2.4 Influence of IR-selection on SFR- and mass-completeness +A galaxy’s SFR scales with the IR luminosity (𝐿IR) (Kennicutt & +Evans 2012). Due to this, the IR-selection criteria in our sample +only includes galaxies above a certain SFR (depending on redshift), +introducing an SFR bias. 𝐿IR in this paper represents the integrated +dust emission from both dust components in MAGPHYS+photo-z +over all wavelengths. As the luminosity distance (𝐷lum) increases, +the lowest SFR of the SF galaxies we can observe will increase +correspondingly2. Hence, the functional form for IR-selection in SFR +is similar to the relationship between luminosity and redshift: +log(SFRIR/𝑀⊙yr−1) ∝ log(𝐿IR) += log(4𝜋𝐹IR𝐷2 +lum) += log(𝛼𝐷2 +lum), +(3) +where 𝛼 is a constant factor determined by the data and 𝐷lum is +the luminosity distance in units of Mpc. By converting 𝐷lum to +𝑧phot and applying Equation 3 to log(SFR) 𝑣𝑠 𝑧phot, we obtain an +empirical estimate of our SFR limit with redshift based on the 1𝜎 +lower boundary of our population, and the constant factor of the +function 𝛼 = 1.50 × 10−7 corresponds to the lower boundary of the +68% (1𝜎) population enclosed curve (see Figure 5): +log(SFRIR/𝑀⊙yr−1) = log(1.50 × 10−7(𝐷lum/Mpc)2). +(4) +1 We explored adopting a constant cut at log(sSFR/yr−1) = −11, which +increases our sample by ∼100 galaxies, but this has a very small difference on +our results. We suspect that this phenomenon could be driven by IR-selection, +which is described in the next subsection. +2 This excludes the negative-k correction effect +MNRAS 000, 1–15 (2022) + +Look-back time tb (Gyr) +0.0 +7.71 +10.24 +11.95 +12.31 +12.55 +12.71 +12.83 +11.35 +0.8 +Samples in this study +0.7 +Full COSMOS2020 catalog +0.6 +Number density +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +2 +5 +1 +3 +4 +6 +0 +7 +8 +Zphot400 +- +μgeo=1.43, Ogeo=1.83 +350 +MAGPHYS x² histogram +300 - +250 +nt +no +200 +150 +100 +4.76 +50 +0 +4 +6 +0 +8 +10 +2Intrinsic MS Scatter at 0.5 < 𝑧 < 3.0 +5 +Figure 4. log(sSFR) vs z enclosed contour plot for the 13,071 selection +galaxies within 2𝜎 − 𝜒2 from redshifts 0.4 to 3.25. The colours ranging +from blue to yellow indicate the increasing number density. The number of +galaxies inside the enclosed blue and green curve is 68% (1𝜎) and 95% +(2𝜎) of the total population, respectively. The magenta points indicate the +median value of log(sSFR/yr−1), while the red points are the median-3𝜎 +values for 24 sSFR bins. The red line is the sSFR cut adopted in this study, +and there are 64 galaxies identified as quenched galaxies. Due to the mini- +mum timescale of star formation in MAGPHYS+photo-z, there is a maximum +value of log(sSFR/yr−1) ∼ −8, corresponding to the adopted SFR timescale +(100Myr), showing as a horizontal boundary in the diagram. +Figure 5. log(SFR) vs z enclosed contour plot for the 13,071 selection +galaxies within 𝜒2 selection from redshifts 0.4 to 3.25. The equation 4 is +plotted as the red curve in this diagram. +Due to the detection limits, we cannot trust our ability to detect +galaxies that are below Equation 4 in Figure 5. This SFR incom- +pleteness translates to an incompleteness on stellar mass (via galaxy +MS relation). We infer the corresponding mass-completeness thresh- +old at each redshift using the Leslie et al. (2020) MS relation. For +subsequent analysis, we will refer to samples above and below this +threshold as our mass-complete and mass-incomplete samples, re- +spectively. +3 RESULTS AND ANALYSIS +3.1 The Role of Redshift uncertainty and IR data on the +Measured Scatter of the MS +In this study, we use MAGPHYS+photo-z to constrain the stellar +masses and SFRs of our galaxies because it uses the full wave- +length range from UV to radio, and it constrains the photometric +redshifts jointly with the other physical parameters. Using the full +SED provides the tightest possible constraints on 𝑀∗ and SFR, thus +minimizing the main sequence scatter that is due to errors on these +parameters. Obtaining the photometric redshift at the same time al- +lows us to fold in the redshift error into the errors on 𝑀∗ and SFR. +This improves our ability to quantify the ‘observational’ scatter on +the main sequence and, in turn, characterise its intrinsic MS scatter. +In this section, we test the accuracy of MAGPHYS+photo-z and quan- +tify the influence that the redshift precision and inclusion of IR data +have on derived physical properties (for the COSMOS filter set). +3.1.1 Accuracy of 𝑧phot relative to 𝑧spec +To examine the 𝑧phot accuracy of MAGPHYS+photo-z, we use the +latest COSMOS master spectroscopic catalogue (curated by M. Sal- +vato for internal use within the COSMOS collaboration), which is +the same dataset used to originally test the code (Battisti et al. 2019). +There are spectroscopic redshifts, 𝑧spec, for 3,873 out of the 14,607 +galaxies in our sample. After applying the 𝜒2 cut, we obtain 3,724 +galaxies. Here we adopt some metrics defined in Section 4.1.1 of +Battisti et al. (2019) to estimate the accuracy of 𝑧phot. We find +𝜎NMAD = 0.086, 𝜂 = 4.2% and 𝑧bias = −0.002, where 𝜎NMAD +(normalized median absolute deviation) is known as the precision or +scatter of the data, 𝜂 characterises the fraction of catastrophic failures +and 𝑧bias represents the accuracy of the redshift (i.e. systematic devi- +ation or bias). The value of 𝑧bias is much smaller than 𝜎NMAD, and +hence we constrain the redshifts very well with the multiple UV to +radio bands. Since we use a similar database as the one used in Bat- +tisti et al. (2019), the results of 𝜎NMAD, 𝜂 and 𝑧bias should be similar. +As a comparison, these values calculated in Battisti et al. (2019) are +𝜎NMAD = 0.032, 𝜂 = 0.037, 𝑧bias = −0.004 for the COSMOS2015 +samples, respectively. +The upper panels of Figure 6 is a demonstration of 𝑧phot accuracy +of MAGPHYS+photo-z, which also shows a comparison between the +𝑀∗ and SFR derived from 𝑧phot and 𝑧spec. The median values of +differences for 𝑀∗ and SFR are 0.00 and 0.05 dex, respectively, +reflecting that MAGPHYS+photo-z does not affect the overall mea- +surement of 𝑀∗ and SFR. Therefore, we do not expect that relying +on 𝑧phot will introduce significant bias or dominate the uncertainty +of the other derived properties. +3.1.2 Uncertainties of 𝑧phot, 𝑀∗ and SFR +We characterise the measurement uncertainties of 𝑧phot, 𝑀∗ and SFR +for our sample of 13,418 galaxies in Figure 7. In our analysis, the +data are separated into five bins of 𝑀∗ with a width of ≥ 0.5 dex at a +specific redshift epoch. The uncertainty in 𝑀∗ in each bin is ∼ 0.06 +dex, while the 𝑧phot uncertainty is ∼ 0.05. Both uncertainties are ∼ 10 +times smaller than the bin size in this study (≥ 0.5 dex for log(𝑀∗) +bin and ∼ 0.5 for redshift epoch), therefore we do not anticipate +that these uncertainties will have a substantial impact on the derived +intrinsic scatter on the MS. The median value of SFR’s uncertainty +is 0.08 dex, which is comparable to the scatter of galaxies on the +MS (e.g., ∼ 0.2 dex; Speagle et al. 2014). Thus, when measuring +MNRAS 000, 1–15 (2022) + +log(sSFR/yr=1) +10 +68%enclosedcurve +95%enclosedcurve +sSFR cut: 0.57Zphot - 11.60 +-12 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Zphot3 +2 +log(SFR/Moyr-1) +0 +IRlimit:log(1.50e-07*D2. +um +68%enclosedcurve +95%enclosedcurve +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Zphot6 +R. Huang et al. +the intrinsic scatter of MS, we need to consider SFR’s uncertainty as +the component of the scatter in MS and remove it properly to obtain +the intrinsic MS scatter. Our method for removing this component is +described in Section 3.2. +3.1.3 Contribution of including IR data to the Uncertainties of 𝑀∗ +and SFR +IR wavelengths probe dust emission and provide information re- +garding the amount of dust-obscured star formation. By excluding +IR observations from the SED fits, we can determine the impact +of these bands on the uncertainties of 𝑀∗ and SFR. We rerun the +MAGPHYS+photo-z without fitting the observational data for filters +at wavelengths longer than IRAC2 (4.5um) for the same 14,607 galax- +ies. After rejecting the cases with bad fits (𝜒2 > 2𝜎), we compare +the uncertainties of 𝑧phot, 𝑀∗ and SFR derived from UV to near-IR +photo-z fitting to those results from fitting the full available SED in +the top panels of Figure 8. As expected (Battisti et al. 2022), the +non-IR fits tend to come with larger measurement uncertainties be- +cause fewer observations are available to constrain the models. For +𝑧phot and 𝑀∗, including the IR bands only leads to a relatively small +improvement (i.e., decrease) in the uncertainty. In contrast, for SFR, +the median uncertainty when IR bands are not included is nearly +2.5 times larger than that with the IR bands. This is because the IR +bands are important to distinguish the amount of dust-obscured star +formation. It is harder to accurately measure the intrinsic scatter of +the MS with the larger measurement error in SFR. Therefore, by +restricting the sample to sources where SED fitting can be performed +that include IR filters, we significantly reduce the amount of scatter +of the MS arising from measurement uncertainty to accurately con- +strain the intrinsic MS scatter. The lower panels of Figure 8 show the +difference in the values of 𝑧phot, 𝑀∗ and SFR with and without the +IR bands included. It can be seen that the median of the difference +remains close to zero as a function of each property suggesting that +there is minimal bias occurring as a result of the MAGPHYS+photo-z +priors. +3.2 Measuring the intrinsic MS scatter +We +divide +our +sample +into +6 +redshift +bins: +𝑧phot += +0.5, 1.0, 1.5, 2.0, 2.5 & 3.0, with widths of Δ𝑧 = 0.5 except for +the lowest bin, which spans 0.4-0.75 due to the limitation on the red- +shift prior for MAGPHYS+photo-z (Section 2.2.1). At each redshift, +we further divide the sample into 5 stellar mass bins. We determine +the log(SFR) dispersion (standard deviation) of the galaxies within +each bin relative to the median log(SFR) measurement uncertain- +ties. We set the following five bins for all selected galaxies according +to their mass: log(𝑀∗) < 9.5 log(𝑀⊙), 9.5 to 10.0 log(𝑀⊙), 10.0 +to 10.5 log(𝑀⊙), 10.5 to 11.0 log(𝑀⊙) and > 11.0 log(𝑀⊙). The +values adopted for each bin is the median log(SFR) and 𝑀∗ of +each group. We characterise the intrinsic MS scatter in the range of +0.4 < 𝑧phot < 3.25 (see Figure 2). The upper boundary (𝑧phot = 3.25) +corresponds to where our sample size dramatically decreases such +that we do not have enough sources to properly characterize the +MS scatter. Then we take the further selections of 2𝜎-𝜒2 and sSFR +cut (see Section 2.2.1 and 2.2.3) to reduce the effects of quiescent +galaxies on the intrinsic MS scatter. +For each bin, we assume that any excess in the SFR disper- +sion relative to the median measurement uncertainty in SFR from +MAGPHYS+photo-z is due to the intrinsic scatter of the MS relation. +We determine the intrinsic MS scatter by assuming the measured +scatter is a result of the measurement uncertainty and intrinsic MS +scatter being added in quadrature, which can be rearranged as: +log(𝜎int/𝑀⊙yr−1) += +√︃ +(log(𝜎tot/𝑀⊙yr−1))2 − (log(𝜎meas/𝑀⊙yr−1))2, +(5) +where 𝜎int, 𝜎tot, and 𝜎meas are intrinsic MS scatter of the galaxies, +the observed standard deviations, and the median MAGPHYS+photo-z +uncertainty, respectively. Figure 9 displays the galaxy MS for 0.5 ≲ +𝑧phot ≲ 3.0, showing that the dispersion in SFR of the galaxies are +significantly larger than the measurement uncertainties (representa- +tive error bars in lower-right of each panel). In each interval of 𝑀∗, +the size of the SFR intrinsic MS scatter is 0.15−0.39 dex larger than +the measurement uncertainty. The horizontal dashed lines in Figure +9 indicate the limit on SFR defined in Equation 4 and Figure 5 at the +median 𝑧phot for each bin. The vertical dashed lines correspond to the +stellar mass at this SFR from the reference MS relation. We define the +right half in each panel as the mass-complete area. There are 12,380 +galaxies (94.71%) in the mass-complete regions and 691 galaxies +(5.29%) in the mass-incomplete regions. We show the values of the +scatter terms for our mass-complete bins in Table 1. +We observe the following three trends (see the left panel of Figure +10 and Table 1). First, the median SFR measurement uncertainties +are always smaller than the intrinsic MS scatter of the SFR. The +minimum difference between intrinsic MS scatter and uncertainties +is 0.13 dex in the range of 1010 − 1010.5𝑀⊙ at 𝑧phot = 2.5, while the +maximum occurring at 𝑧phot = 0.5 for galaxies with log(𝑀∗/𝑀⊙) > +11.0 is 0.39 dex. Second, excluding the mass-incomplete regions, our +galaxies roughly follow the same observed main sequence as shown +in Leslie et al. (2020) (i.e. Equation 4), but with slightly lower SFRs +than the reference MS for most redshift bins. Third, the intrinsic +dispersion in SFR at a given mass tends to decrease as the 𝑧phot +increases at a fixed 𝑀∗. +The size of the 𝑧phot interval we selected may affect the behaviour +of the SFR intrinsic MS scatter evolution. The width of each 𝑧phot in +our criteria is Δ𝑧phot = 0.5 except 0.35 at 𝑧phot = 0.5. However, with +the increase of 𝑧phot, the cosmic time corresponding to the Δ𝑧phot is +decreasing because the look-back time 𝑡lb does not linearly increase +with 𝑧phot. As a result, this reducing length of the binning interval in +cosmic time with increasing 𝑧phot may affect the SFR intrinsic MS +scatter. To examine this issue, we adopt look-back time (𝑡lb) instead of +redshift as a more consistent way to measure the intrinsic MS scatter. +We convert the 𝑧phot into 𝑡lb and rearrange our sample of 13,071 +galaxies in 6 𝑡lb bins (4.9, 6.1, 7.3, 8.5, 9.7, 10.9Gyr) with the equal +length of time (Δ𝑡lb = 1.2Gyr). We reproduce the log(SFR)-log(𝑀∗) +plane in the right panel of Figure 10 and present the results in Table +2. We find that the 𝑡lb results share the same trends and features +as the previous 𝑧phot version. However, because binning the data in +equal 𝑡lb width removes the unequal-length effect when measuring +the intrinsic MS scatter, we will adopt this for our main analysis. +The left panel of Figure 11 shows the relationship between intrinsic +MS scatter and 𝑡lb for both the redshift and look-back time binning. +In this study, we adopt the weighted linear regression to the intrinsic +MS scatter versus 𝑡lb: +log(𝜎int/𝑀⊙yr−1) = (−0.012 ± 0.002)𝑡lb + (0.432 ± 0.015), +(6) +where 𝜎int is the intrinsic scatter of the MS, and these parameters +are calculated with mass-complete sample. We observe a trend of +MNRAS 000, 1–15 (2022) + +Intrinsic MS Scatter at 0.5 < 𝑧 < 3.0 +7 +Figure 6. Upper left panel: Comparison of measurement uncertainties between default MAGPHYS high-𝑧 (i.e, fixed to 𝑧spec) and MAGPHYS+photoz runs for +the subsample of 3,724 𝜒2-selection galaxies with spectroscopic redshifts. Upper right panel: redshift accuracy ((𝑧phot − 𝑧spec)/(1 + 𝑧spec)) as a function of +𝑧spec. The redshift scatter (𝜎NMAD), catastrophic failure rate (𝜂), and redshift bias (median((𝑧phot − 𝑧spec)/(1 + 𝑧spec))) values are shown at the upper right +corner. Lower panels: Difference in 𝑀∗ and SFR derived by 𝑧phot and 𝑧spec as a function of the 𝑧spec-derived values. The 2D histogram/scatterplot colors range +from blue to yellow with increasing number density. The black line in each sub-diagram is the one-to-one relation as reference; red, green and blue curves +enclose 68%, 95% and 99% populations of sample galaxies within 2𝜎-𝜒2 cut. +decreasing intrinsic MS scatter up to 9.7Gyr (𝑧 ∼ 1.7) as 𝑡lb increases. +The error bars are derived by bootstrap resampling the data in each +bin 100 times. Bins with smaller sample sizes have larger bootstrap +errors. Although the descending rate of intrinsic MS scatter over +look-back time is shallow, the Spearman and Pearson correlation +coefficients (𝑟𝑠 = −0.943 and 𝑟 𝑝 = −0.956, respectively) indicate +a strong monotonic decreasing correlation. Conversely, with 𝑟𝑠 = +−0.486 and 𝑟 𝑝 = −0.837, there is a weaker and tentative correlation +when using redshift binning. This is due to the redshift binning +having a potential upturning feature at 𝑧phot ≥ 2. We suggest this +is a consequence of unequal-length binning for redshift, which will +be discussed in the next paragraph. Furthermore, there is a ‘upturn’ +feature, and the intrinsic MS scatter tends to increase after 𝑡lb ∼ +10Gyr. Given the uncertainty in our intrinsic MS scatter and our +limited sample size at high-𝑧, it is difficult to assess the significance +of this upturn with our current data. +The intrinsic MS scatter may vary with the adopted Δ𝑡lb of each +bin. The right panel of Figure 11 presents the effect of binning the +MNRAS 000, 1–15 (2022) + +4.0 +68% enclosed curve +3.5 +95% enclosed curve +99% enclosed curve +3.0 +Zphot(MAGPHYS) +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0 +1 +2 +3 +4 +ZspecONMAD = 0.086, +0.4 +n = 4.2%, +Zbias = - 0.002 +0.2 +0.0 +0 +-0.2 +-0.4 +Zphot = Zspec ± 0.1(1 + Zspec) +0 +1 +2 +3 +4 +Zspec2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +9 +10 +11 +12 +log(M* /Mo)zspec2.0 +1.5 ++-log(SFR/Moyr- +1.0 +0.5 +0.0 +-0.5 +log(SFR/Moyr- +-1.0 +-1.5 +-2.0 +-1 +0 +1 +2 +3 +log(SFR/Moyr +speo8 +R. Huang et al. +Figure 7. Distribution of the values and uncertainties for the key parameters of our study for the 13,418 galaxies in our sample. Red, green and blue curves +enclose 68%, 95% and 99% populations of sample galaxies within 2𝜎-𝜒2 cut. The median values of 𝑧phot, log(𝑀∗/𝑀⊙) and log(SFR/𝑀⊙yr−1) are 1.20, +10.64 and 1.42, while the median values of uncertainties are 0.05, 0.06 and 0.08 dex, respectively. +Figure 8. Upper panels: Comparison between the uncertainties on 𝑧phot, 𝑀∗ and SFR for MAGPHYS+photo-z runs including IR bands (labeled ‘phot,IR’) and +without IR or radio bands (‘phot,nonIR’). Differing from the cases for 𝑧phot and 𝑀∗, the SFR’s uncertainty increases dramatically for SED fits without the IR +data. The median uncertainty values derived from MAGPHYS+photo-z including the IR bands (distributed along x-axis) are 0.05, 0.06 and 0.08 dex for 𝑧phot, +𝑀∗ and SFR, while the median uncertainty values derived from non-IR runs (distributed along y-axis) are 0.08, 0.08 and 0.25 dex, respectively. The black line +in each sub-diagram is the one-to-one relation as reference; red, green and blue curves enclose 68%, 95% and 99% populations of sample galaxies within 2𝜎-𝜒2 +cut. Lower panels: Difference in measurements between the SED fits with and without IR bands included as a function of the measurements derived from fitting +including IR bands. For the COSMOS2020 data, we find that the dominant factor in accurately constraining the scatter in the main sequence is whether the IR +bands are included to constrain the dust-obscured SFR. +data in different Δ𝑡lb widths. The data are binned into 3, 6, 12, and +24 groups (no less than 100 galaxies in each bin) in 4 different sets +with equal 𝑡lb widths, where 6 is our fiducial number of bins. It can +be seen that the data in the 12 and 24 bins have a similar distribution +statistically relative to our fiducial binning. We find that as the num- +ber of bins increases, the normalisation (intrinsic MS scatter) slightly +decreases. However, the coefficients of the corresponding equations +tend to converge somewhere closely below the current linear re- +gression equation (i.e. the yellow line). Even though the decreasing +binning time scale causes more severe fluctuation along the linear +regression line, the similarity and high absolute values of 𝑟𝑠 and 𝑟 𝑝 +still demonstrate a strong correlation between intrinsic MS scatter +MNRAS 000, 1–15 (2022) + +0.25 +0.20 +0.15 +0.10 +0.05 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Zphot0.5 +0.4 +(log(M* /Mo)) +0.3 +0.2 +0.1 +0.0 +8 +9 +10 +11 +12 +log(M* /Mo)0.5 +0.4 +0.2 +0.1 +0.0 +0 +2 +3 +log(SFR/Moyr-1)1.0 +0.8 +O(Zphot, nonIR) +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +O(Zphot, IR)0.5 +0.4 +o(log(M * /Mo)phot, nonIR) +0.3 +0.2 +0.1 +0.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +o(log(M * /Mo)phot, IR)1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +o(log(SFR */Moyr-1 +)phot,IR)2.0 +1.5 +1.0 +Zphot, IR +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +0 +1 +2 +3 +4 +Zphot, IR2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +8 +9 +10 +11 +12 +log(M * /Mo)phot, IR2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +-1 +0 +3 +log(SFR*/Moyr +phot,IRIntrinsic MS Scatter at 0.5 < 𝑧 < 3.0 +9 +Figure 9. 𝑀∗-SFR relation of our galaxies in 6 redshift bins. All bins are shown through 13,071 selection galaxies with colours from purple to red to green +coding the range of 𝑧phot at 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0. The horizontal dot-dashed line indicates the IR-selection completeness cut in SFR, and the vertical +dashed line indicates the corresponding IR-selection completeness cut in 𝑀∗. The open symbols indicate the mass-incomplete data. We drop the data points +where the numbers of galaxies within a bin are less than 50. The longer error bars indicate the standard deviation of the SFR distribution in each stellar mass +bin, while the shorter black ones represent the median MAGPHYS measurement uncertainties. The gray error bar in the bottom right of each panel denotes the +median uncertainties on 𝑀∗ (along the x-axis) and SFR (along the y-axis) from MAGPHYS+photo-z for the entire redshift bin. The curves are the MS relations +at each redshift epoch from Leslie et al. (2020). The maximum value of log(sSFR/yr−1) ∼ −8 due to the adopted SFR timescale shows as an upper boundary +of the slope in each SFR-𝑀∗ panel, while the bottom right slope represents the sSFR cut at each redshift. +and 𝑡lb. On the other hand, this phenomenon also partially explains +why redshift binning is not a good approach in this study, especially +at high redshifts: a narrower binning time scale may lead to larger +fluctuations in the intrinsic MS scatter. Since there is no significant +discrepancy in intrinsic MS scatter for 𝑛bin ≥ 6, we expect that these +coefficients in linear regression lines approach some values slightly +smaller than the current binning one. Hence, we conclude that the +binning does not strongly affect the trend of intrinsic MS scatter +evolution and we adopt 𝑛bin ≥ 6 as the current 𝑡lb binning number. +MNRAS 000, 1–15 (2022) + +2.5 +Leslie+20 +z = 0.5 +sSFR cut +2.0 +mass-incomplete +★ +mass-complete +1.5 +11 +1.0- +0.5 +0.0 +O +O +-0.5 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +log(M* /Mo)2.5 +Z= 1.0 +2.0 +1.5 +.111 +1.0 +0.5 +0.0 +-0.5 +O +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +log(M* /Mo)3.0 +z= 1.5 +2.5 +2.0 +王 +1.5 +1.0 +0.5 +O +0.0 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +log(M* /Mo)3.0 +Z= 2.0 +2.5 +2.0 +1.5 +C +1.0 +C +0.5 - +O +0.0 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +log(M* /Mo)3.5 +z = 2.5 +3.0 - +2.5 +2.0 +1.5 +1.0 - +O +0.5 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +log(M*/Mo)3.5 +Z=3.0 +3.0- +2.5 +2.0 +1.5 +1.0- +0.5 +8.0 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +log(M* /Mo)10 +R. Huang et al. +Figure 10. The MS evolution and scatter for our sample with the same colour scheme adopted in Figure 9. The shaded regions shown in the diagrams indicate the +range of the reference MS between the upper and lower redshift boundaries for each bin and do not relate to the intrinsic MS scatter. The thick and thin error bars +indicate the observed standard deviation in SFR and the median SFR uncertainty from MAGPHYS+photo-z for each bin, respectively. The left panel shows the +sample binned in equal redshift bins, with the mass-incomplete bins shown as open symbols. The right panel is similar to the left but binned in equal look-back +time bins. Adopting equal-width redshift bins (left-panel) instead of look-back time (right-panel) may impact the measured intrinsic MS scatter because of the +evolution in the MS relation over the redshift range contained within a single bin (i.e., differing widths in shaded region in left-panel relative to right-panel). +Hence, we adopt look-back time bins for our analysis. +Figure 11. The left panel is intrinsic MS scatter vs 𝑧phot and 𝑡lb binning with a linear regression fit to the look-back time bins. The yellow star-like scatter +points are the 𝑡lb binning data. We take the average of the data points in each mass bin and obtain error bars of intrinsic MS scatter by bootstrap resampling the +distribution 100 times based on varying the individual values by their uncertainties and rebinning them. The solid yellow line is the linear fit of intrinsic MS +scatter versus 𝑡lb. The right panel display the effect of binning number. A total of 12,380 galaxies are regrouped in 3, 6, 12 and 24 bins with red circles, yellow +star, green squares and blue diamonds. In each panel, we show the Spearman and Pearson correlation coefficients (𝑟𝑠 and 𝑟𝑝) as well as the corresponding +p-values (𝑝𝑠, 𝑝𝑝) for different binning data. +𝑀∗ +9.5 − 10.0 Log(M⊙) +10.0 − 10.5 Log(M⊙) +10.5 − 11.0 Log(M⊙) +> 11.0 Log(M⊙) +𝑧phot +N +𝜎tot +𝜎meas +𝜎int +N +𝜎tot +𝜎meas +𝜎int +N +𝜎tot +𝜎meas +𝜎int +N +𝜎tot +𝜎meas +𝜎int +0.5 +428 +0.36 +0.09 +0.34 +897 +0.35 +0.08 +0.34 +922 +0.40 +0.07 +0.40 +239 +0.47 +0.06 +0.47 +1.0 +365 +0.30 +0.11 +0.28 +1036 +0.34 +0.10 +0.32 +1660 +0.35 +0.09 +0.34 +831 +0.41 +0.08 +0.40 +1.5 +247 +0.31 +0.10 +0.29 +776 +0.28 +0.09 +0.26 +1430 +0.29 +0.08 +0.28 +953 +0.34 +0.08 +0.34 +2.0 +102 +0.29 +0.09 +0.28 +300 +0.29 +0.09 +0.27 +663 +0.31 +0.08 +0.30 +616 +0.31 +0.08 +0.29 +2.5 +69 +- +- +- +202 +0.26 +0.09 +0.24 +227 +0.35 +0.09 +0.33 +261 +0.30 +0.09 +0.28 +3.0 +25 +- +- +- +78 +- +- +- +74 +0.37 +0.09 +0.36 +72 +0.28 +0.10 +0.26 +Table 1. "N" is the number of galaxies, "𝜎tot" is the galaxy SFR dispersion (standard deviation), "𝜎meas" is the MAGPHYS uncertainty in SFR and "𝜎int" is +the intrinsic MS scatter in each 𝑀∗ bin. The mass-incomplete data are marked as "-", but the number of galaxies in the binning interval are still shown. Since +all the 𝑀∗ < 9.5 log(𝑀⊙) galaxies lie in the mass-incomplete regime, they are not included in the table. +MNRAS 000, 1–15 (2022) + +3.0 +2.5 +2.0 +log(SFR/Moyr-1) +1.5 +1.0 - +0.5 +0.0 +MS at z = 0.5 +MS at z = 3.0 +z: 1.25 ~ 1.75 +-0.5 +MS at z = 1.0 +mass-incomplete +z: 1.75 ~ 2.25 +MS at z = 1.5 +mass-complete +z: 2.25 ~ 2.75 +-1.0 +MS at z = 2.0 +z: 0.40 ~ 0.75 +z: 2.75 ~ 3.25 +MS at z = 2.5 +z: 0.75 ~ 1.25 +-1.5 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M * /Mo)3.0 +2.5 +2.0 +log(SFR/Moyr-1) +1.5 +1.0: +0.5 +0.0 +tib = 4.9Gyr (z=0.48) +tib : 4.3 ~5.5Gyr +tib=6.1Gyr(z=0.66) +tib : 5.5 ~ 6.7Gyr +-0.5 +tib = 7.3Gyr (z=0.90) +tib : 6.7 ~ 7.9Gyr +tib = 8.5Gyr (z=1.22) +tib : 7.9 ~ 9.1Gyr +-1.0 +tib = 9.7Gyr (z=1.70) +tb :9.1~10.3Gyr +tib = 10.9Gyr (z=2.51) +tib :10.3~11.5Gyr +-1.5 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M* /M)Zphot +0.5 +1.0 +1.5 +2.0 +2.5 3.0 +0.45 +linear fit: = -0.012tb+0.432 +tib binning +0.40 +Zphot binning +0.35 +0.30 +0.25 +rs, ps = -0.943, 0.005 +rs, ps = -0.486, 0.329 +rp, Pp = -0.956, 0.003 +rp, Pp = -0.837, 0.038 +0.20 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Look-back time tb (Gyr)0.45 +linear fit (3 bins): -0.017tib+0.477 +linear fit (6 bins): -0.012tib+0.427 +Intrinsic Scatter (log(oint/Moyr +0.40 +linear fit (12 bins): -0.012tib+0.420 +linear fit (24 bins): -0.012tib+0.417 +0.35 +0.30 +ps = -1.000, 0.005 +rp, Pp = -0.951,0.199 += -0.832,0.001 +0.25 +-0.864.0.000 +rs, ps = -0.770, 0.000 +rp, Pp = -0.761, 0.000 +0.20 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Look-back time tb (Gyr)Intrinsic MS Scatter at 0.5 < 𝑧 < 3.0 +11 +𝑀∗ +9.5 − 10.0 Log(M⊙) +10.0 − 10.5 Log(M⊙) +10.5 − 11.0 Log(M⊙) +> 11.0 Log(M⊙) +𝑡lb(Gyr) +N +𝜎tot +𝜎meas +𝜎int +N +𝜎tot +𝜎meas +𝜎int +N +𝜎tot +𝜎meas +𝜎int +N +𝜎tot +𝜎meas +𝜎int +4.9(z=0.48) +226 +0.36 +0.08 +0.35 +427 +0.32 +0.07 +0.32 +343 +0.39 +0.06 +0.39 +40 +- +- +- +6.1(z=0.66) +222 +0.31 +0.09 +0.29 +532 +0.32 +0.08 +0.31 +674 +0.38 +0.07 +0.37 +195 +0.48 +0.08 +0.47 +7.3(z=0.90) +240 +0.28 +0.11 +0.26 +648 +0.32 +0.09 +0.31 +1082 +0.35 +0.09 +0.33 +504 +0.40 +0.07 +0.39 +8.5(z=1.22) +219 +0.34 +0.11 +0.32 +696 +0.33 +0.10 +0.31 +1128 +0.30 +0.09 +0.28 +758 +0.38 +0.09 +0.37 +9.7(z=1.70) +280 +0.28 +0.09 +0.27 +855 +0.29 +0.09 +0.27 +1607 +0.32 +0.08 +0.31 +1316 +0.32 +0.08 +0.31 +10.9(z=2.51) +49 +- +- +- +131 +0.26 +0.07 +0.25 +140 +0.35 +0.09 +0.34 +157 +0.29 +0.10 +0.27 +Table 2. Look-back time version of Table 1 by collecting and reassigning the 12,380 mass-complete data to 6 𝑡lb bins. We exclude bins in which the number of +galaxies is less than 50 (> 1011𝑀⊙ at 𝑡lb = 4.9Gyr), together with the mass-incomplete data. In general, the intrinsic MS scatter is still significantly larger than +measurement uncertainty in new binning criteria, which avoids the effect of nonlinear 𝑡lb widths in previous 𝑧phot binning. +4 DISCUSSION +4.1 Comparison to the halo mass-stellar mass relation +The stochastic events that occurred throughout an SF galaxy’s history, +including as galaxy mergers and supernova & AGN feedback, are +assumed to be responsible for the inherent MS scatter. The amount +of burstiness in SFH, which is probably related to galactic feedback +mechanisms, is reflected in the distribution and evolution of intrinsic +MS scatter. +In the left panel of Figure 12, we show the distribution of the +intrinsic MS scatter versus 𝑀∗. We see a minimum intrinsic MS +scatter of ∼ 0.35 dex at 𝑀∗ ∼ 1010.25𝑀⊙. We see a higher increase +of ∼ 0.1 dex at higher mass (> 1011𝑀⊙) with decreasing look- +back time for low redshifts (𝑡lb: 4.9 - 8.5 Gyr) and a relatively flat +relation at higher redshifts (𝑡lb ≳ 9.7Gyr). In the low mass end +(∼ 109𝑀⊙), where the galaxies are mass-incomplete, the intrinsic +scatter rises from 0.35 to 0.6 dex. For some of the redshift bins, +this type of trend is qualitatively similar to the turnover that occurs +in the halo mass-stellar mass (HMSM) relation (see Figure 2 of +Wechsler & Tinker (2018)). The shape of the HMSM relation is +thought to be a consequence of feedback, with SF feedback reducing +the SF efficiency in low-mass galaxies and AGN feedback reducing +the SF efficiency in high-mass galaxies, with a turnover at halo mass +𝑀h ∼ 1012𝑀⊙, which is coincidentally corresponding to the upturn +point of 𝑀∗-𝜎int panel at 𝑀∗ ∼ 1010.25𝑀⊙ in this study. The intrinsic +scatter in the MS is also expected to be linked to feedback, and this +may account for similarities in the observed trends with the HMSM +relation. +In the right panel of Figure 12, we present a toy model where we +relate the HMSM relation with the intrinsic MS scatter. We adopt +the best-fit functional form and the parameterised data of the stellar +mass halo mass (SMHM) relation from Equations 21 & 22 and Table +2 of Behroozi et al. (2010). Behroozi et al. (2010) parameterize the +evolution of SMHM relation in terms of 𝑀1, 𝑀∗,0, 𝛽, 𝛿 and 𝛾. All +these variables vary with the scale factor 𝑎. For our model, we convert +the standard SMHM relation into HMSM fraction vs stellar mass (i.e., +log(𝑀h/𝑀∗) vs log(𝑀∗), instead of log(𝑀∗/𝑀h) vs log(𝑀h). This +change results in the HMSM relation having an upturn shape instead +of the usual downturn shape (because we invert the values in the y- +axis ratio). This modified HMSM relation presents a similar turnover +feature as the ‘U-shaped’ distribution shown in the left subplot of +Figure 12. Considering time evolution, we renormalize the HMSM +relation to match our data by multiplying the HMSM fraction by an +arbitrary coefficient 𝑘𝑖 (𝑖 indicates different 𝑡lb), which is computed +by the non-linear regression: +log(𝜎int/𝑀⊙yr−1) = 𝑘𝑖 · 𝑓HMSM(𝑎), +(7) +𝑡lb +𝑘𝑖 +4.9Gyr (z=0.48) +0.20 +6.1Gyr (z=0.66) +0.19 +7.3Gyr (z=0.90) +0.17 +8.5Gyr (z=1.22) +0.15 +9.7Gyr (z=1.70) +0.12 +10.9Gyr (z=2.51) +0.10 +Table 3. Best-fit values for the constant factor 𝑘𝑖 at different bins of look-back +time for our toy model given by Equation 7. +where 𝜎int is the intrinsic MS scatter, 𝑘𝑖 is a constant factor (see +Table 3) and 𝑓HMSM(𝑎) = 𝑀h/𝑀∗ is the HMSM fraction that varies +with the scale factor, 𝑎 (Behroozi et al. 2010). The Equation 7 does a +reasonable job of matching the trends for the first four time bins (4.9- +8.5Gyr), but the final two bins (9.7 and 10.9Gyr) favor a flatter shape +than our toy model at high 𝑀∗. At the lower and higher mass ends, +the increasing intrinsic MS scatter may be driven by the feedback +of supernovas and AGNs, respectively. In contrast, in the mid-range +of stellar mass (10.0 < log(𝑀∗/𝑀⊙) < 10.5), the intrinsic MS +scatter relation reaches a minimum (maximum in HMSM relation), +reflecting the maximum conversion efficiency of gas to baryon and +is thought to be linked to a minimum in the influence of starburst +and galaxy feedback. We notice a large discrepancy between our data +with shifted HMSM fraction at higher redshifts, which may be due to +the low quality of observational data at high redshifts or differences +in feedback mechanisms in the earlier universe. For example, high-𝑧 +observational limitation can lead to larger uncertainties on SFR and +make it more difficult to constrain the intrinsic MS scatter. On the +other hand, weaker AGN feedback for high-𝑀∗ galaxies at high-𝑧 +may also account for the almost constant intrinsic MS scatter. +4.2 Comparison to observational studies +We compare our measurements of the intrinsic MS scatter with some +previous studies in Figure 13. Traditionally, a redshift-independent +width of MS at either 0.2 or 0.25 dex is founded in observations +(Daddi et al. 2007; Noeske et al. 2007; Whitaker et al. 2012; Ciesla +et al. 2014; Speagle et al. 2014; Kurczynski et al. 2016), which is +lower than our results. In contrast, Kurczynski et al. (2016); Santini +et al. (2017); Davies et al. (2022) report a redshift evolution of +intrinsic MS scatter, which spans a wider range from 0.2 to 0.9 dex. +Our data show a similar trend to the ‘U-shaped’ distribution de- +scribed in Davies et al. (2022). They used the DEVILS survey, con- +taining ∼60,000 galaxies with spectroscopic redshifts ranging from +0.1 to 0.85. In the case of 𝑧phot = 0.5, where our samples have red- +shift overlap, the intrinsic MS scatter trend along the 𝑀∗ is highly +consistent with the ‘U-shaped’ distribution, except we have signifi- +cantly smaller intrinsic MS scatter (0.24 - 0.40 dex in this study, while +∼0.4 - 0.8 dex in Davies et al. (2022)). At low stellar mass, Davies +MNRAS 000, 1–15 (2022) + +12 +R. Huang et al. +Figure 12. Left panel: the intrinsic MS scatter 𝜎(log(SFR)) vs 𝑀∗ for our 6 look-back time bins. Solid lines connect the mass-complete data and dashed lines +indicate the mass-incomplete regions. Right panel: we show a fit of our toy model (relation indicated in the panel), based on the halo mass to stellar mass fraction +vs stellar mass relation Behroozi et al. (2010) normalized by a constant factor, normalized by a constant factor, relative to our intrinsic MS scatter. Similar to the +left, we indicate the mass-complete and mass-incomplete regions with solid and dashed lines, respectively. +Figure 13. A comparison of our intrinsic MS scatter to previous studies. The first three panels display the distribution of intrinsic MS scatter over 𝑀∗ at +𝑧phot < 1, 1 < 𝑧phot < 2 and 𝑧phot > 2, respectively. The star symbols are the results in this study. The pentagon symbols show the results from Davies et al. +(2022), while the curves in the top left panel are polynomial fitting curves to their data. The filled pentagons are the fitting range of Davies et al. (2022). The +circles are the results from Kurczynski et al. (2016), while the squares represent the data from Santini et al. (2017). The horizontal dashed line in the bottom +indicates a constant intrinsic MS scatter at 0.25 dex (Daddi et al. 2007; Ciesla et al. 2014; Speagle et al. 2014), while the solid line represents a non-evolving +scatter at 0.20 dex (Noeske et al. 2007; Whitaker et al. 2012; Pessa et al. 2021). Our data are quantitatively similar to Kurczynski et al. (2016); Davies et al. +(2022). +MNRAS 000, 1–15 (2022) + +0.7 +tib = 4.9Gyr (z=0.48) +tib = 6.1Gyr (z=0.66) +0.6 +tib = 7.3Gyr (z=0.90) +tib = 8.5Gyr (z=1.22) +0.5 +tib = 9.7Gyr (z=1.70) +log(oint/Moyr-1 +tib = 10.9Gyr (z=2.51) +0.4 +0.3 +0.2 +mass-incomplete +mass-complete +0.1 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M* /Mo)0.7 +0.6 +log(oint/Moyr-1) = kj - fHMSM +log(oint/Moyr-1) +0.5 +0.4 +0.3 +0.2 +0.1 +8.5 +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +log(M* /Mo)0.8 +z<1 +0.7 +0.6 +log(oint/Moyr +0.5 +0.4 +0.3 +0.2 +7 +8 +9 +10 +11 +12 +log(M* /Mo)0.8 +1<<2 +0.7 +log(int/Moyr-1) +0.6 +0.5 +0.4 +0.3 +0.2 +8 +9 +10 +12 +11 +log(M*/Mo)0.8 +Z>2 +0.7 +(t-/w/ul0)60l +0.6 +0.5 +0.4 +0.3 +0.2 +8 +9 +10 +12 +7 +11 +log(M*/Mo)constant scatter = 0.2 dex +Davies+22, z:0.7-0.85 +constant scatter = 0.25 dex +Kurczynski+16, z:0.5-1.0 +★★★★★★ +tib = 4.9Gyr (z=0.48) +Kurczynski+16, z:1.0-1.5 +tib = 6.1Gyr (z=0.66) +Kurczynski+16, z:1.5-2.0 +tib = 7.3Gyr (z=0.90) +Kurczynski+16, z:2.0-2.5 +tib = 8.5Gyr (z=1.22) +Kurczynski+16, z:2.5-3.0 +tib = 9.7Gyr (z=1.70) +Santini+17, z:1.3-2 +tib = 10.9Gyr (z=2.51) +Santini+17, z:2-3 +Davies+22, z:0.1-0.25 +Santini+17, z:3-4 +Davies+22, z:0.25-0.4 +Santini+17, z:4-5 +Davies+22, z:0.4-0.55 +Santini+17, z:5-6 +Davies+22, z:0.55-0.7Intrinsic MS Scatter at 0.5 < 𝑧 < 3.0 +13 +et al. (2022) suggest that the intrinsic MS scatter is driven by stochas- +tic starbursts and stellar feedback events; while the galaxies become +more massive and reach intermediate stellar mass (around 1010𝑀⊙), +the galaxies are too massive so that the effect of star formation and +galaxy feedback is less significant. In this study, intrinsic MS scatter +increases at high stellar mass (log(𝑀∗/𝑀⊙) ≳ 10.3), consistent with +the ‘U-shaped’ distribution. Davies et al. (2022) conclude that AGN +feedback leads to a large scatter at the high mass end. We note that in +our selection, we removed sources with current AGN signatures (see +Section 2.1), but that the feedback from previous AGN will still affect +the MS scatter over longer timescales than the AGN duty cycle. As the +redshift increases, more data in the low stellar mass end are identified +as mass-incomplete due to IR selection. However, we can still rec- +ognize that the intrinsic MS scatter tends to increase when galaxies +become more massive for 𝑀∗ > 1010𝑀⊙. With increasing redshift, +we notice that the right half of the ‘U-shaped’ distribution becomes +flattered and even decreases at 𝑡lb = 10.9Gyr (𝑧phot = 2.51). This +suggests that the star-forming and feedback activity or efficiency for +high-mass galaxies in the early universe may differ relative to lower +redshifts. +Regarding the higher intrinsic MS scatter in Davies et al. (2022) +relative to our results, we think this is due to the following reasons: +(1) They do not include photo-z uncertainties in the SED modeling, +which results in an underestimate of the measurement uncertainty +(on 𝑀∗ and SFR), and hence, an overestimation of the intrinsic +MS scatter. (2) They derive galaxy properties for the D10 field of +DEVILS (Davies et al. 2018) by using different SED fitting code, +PROSPECT (Robotham et al. 2020). Large differences in obtained +properties are produced by different derivation techniques applied +to various photometric data (see the comparison between PROSPECT +and MAGPHYS3 in Thorne et al. (2021)). (3) They adopt the U-V-J +selection rather than sSFR cut so that the samples in Davies et al. +(2022) contain low-sSFR galaxies (see the discussion of these two +selection criteria in Appendix A). However, in our selection criteria, +these galaxies are identified as quenched and removed. The addition +of quenched galaxies severely enlarges the MS scatter. As will be +shown in Section 4.3 when comparing to simulations, the amplitude +of the intrinsic MS scatter is very sensitive to the choice of sSFR +cuts. For example, Leja et al. (2022) demonstrate that fixed-sSFR +cuts may reduce the inferred MS scatter, particularly at the highest +stellar mass. (4) They do not strictly require detection in the IR +bands. As discussed in Section 3.1.3, the SED fitting in the absence +of IR information results in considerable uncertainty of derived SFR, +which may increase the SFR standard deviation and inferred intrinsic +MS scatter. +4.3 Comparison to theoretical studies +We first compare to results from the SHARK semi-analytic models (left +panel of Figure 14; Lagos et al. 2018). For SHARK, we select galaxies +with stellar masses between 108.25 −1011.75𝑀⊙ within ±1 dex along +the MS for each redshifts. We calculate the standard deviations of +the median SFRs for selected galaxies and present the results as +the dot-dashed lines in Figure 14. A notable difference from the +observational data is that the overall scatter in SHARK is larger than +observational data in the mass-complete range, particularly for the +highest redshift bin. We find a common trend that the SHARK results +3 We pick MAGPHYS+photo-z in this study because this code can treat red- +shift as a free parameter and derive the 𝑧phot and the corresponding measure- +ment uncertainty. +follow a similar ‘U-shaped’ distribution at each redshift, though the +minimum and maximum points occur at a stellar mass < 1010𝑀⊙ and +> 1010.75𝑀⊙, respectively. We suggest that the consistent ‘upturn’ +feature in SHARK also indicates the effect of past AGNs for 𝑀∗ ≳ +1010𝑀⊙. We also observe a flat or decreasing scatter in SHARK for +galaxies in the range of 𝑀∗ ≥ 1010.75𝑀⊙. This occurrence indicates +that galaxies in this mass range are shifted below the chosen sSFR +limit because AGNs have a more significant impact on them. We also +investigate the effect of the sSFR cut on intrinsic MS scatter and find +that the stricter sSFR cut leads to a smaller amplitude of the intrinsic +MS scatter. For instance, the intrinsic MS scatter will be reduced by +∼ 1 dex overall when we pick a selection with a narrower sSFR cut, +such as ±0.75 dex along the MS, rather than ±1 dex. +Next, we compare the results with EAGLE hydrodynamical sim- +ulations (right panel of Figure 14; Matthee & Schaye 2019). We +redivide the data in redshift binning at 𝑧phot = 0.5, 1.0, 2.0 and 3.0 +for a proper comparison with Figure 3 in Matthee & Schaye (2019). +Matthee & Schaye (2019) adopt SF galaxies with evolved sSFR se- +lection (i.e., log(sSFR/yr−1) = −10.4 at 𝑧 = 0.5 and increases to +−9.4 at 𝑧 = 3) and measure the scatter from the residuals by us- +ing the non-parametric local polynomial regression method. Then +they obtain the intrinsic MS scatter by subtracting the observational +errors derived by median uncertainties of the observational sample +from Chang et al. (2015). The EAGLE results suggest lower intrinsic +MS scatter at higher redshift for 𝑀∗ < 109.8𝑀⊙ and similar intrinsic +MS scatter for 𝑀∗ > 109.8𝑀⊙, with a downward ‘U-shaped’ feature +appearing at z = 2 and 3. This shape differs substantially from our +findings. Unlike the decreasing trend with stellar mass from simu- +lation, the intrinsic MS scatter in this study decreases initially but +increases at the high mass end. Matthee & Schaye (2019) consider +that supermassive black hole growth accounts for the increasing scat- +ter at 𝑀∗ ≈ 109.8𝑀⊙ at high redshift. However, our results suggest +the influence of the feedback mechanism from previous AGNs might +be more significant at higher stellar masses (𝑀∗ ≳ 1010.3𝑀⊙) at low +redshift. The intrinsic MS scatter at each redshift bin is also larger +than Matthee & Schaye (2019). We suspect that different physics +(e.g., feedback, SF model) adopted in EAGLE give rise to the quanti- +tative difference to our results and from SHARK. +5 CONCLUSIONS +In this paper, using the selection of 12,380 SF galaxies from the +COSMOS2020 database and adopting the MAGPHYS+photo-z SED +fitting code, we characterise the intrinsic scatter of galaxy MS over +the redshift range 0.5 < 𝑧 < 3.0. +• We find that the intrinsic MS scatter is larger than the measure- +ment uncertainty by a factor of 1.4-2.6 when IR data is available for +accurately constraining the dust-obscured star formation (Section 3), +with measured MS scatter in the range of 0.26-0.47 dex. +• For the COSMOS2020 sample, the inclusion of IR data is the +dominant factor (over 𝑧phot uncertainty), affecting the accuracy of +measuring the scatter on the MS. +• Binning the data according to either redshift or look-back time, +we find a slightly negative correlation between intrinsic MS scatter +and look-back time (Equation 6) but with an upturn at 𝑡lb ≳ 10Gyr. +• To connect the intrinsic MS scatter with the feedback mecha- +nism, we present a toy model that uses the Behroozi et al. (2010) +SMHM relation (Equation 7), which does a reasonable job of match- +ing the distribution of intrinsic MS scatter over 𝑀∗ and redshift +(Figure 12), although with less agreement at the highest redshifts. +MNRAS 000, 1–15 (2022) + +14 +R. Huang et al. +Figure 14. Left panel: we compare our results with SHARK models and show the sensitivity of the intrinsic MS scatter to the chosen boundary of sSFR (i.e., ±1 +and ±0.75 dex). The trends in SHARK show rough qualitative agreement with our observational results but with slight differences in the normalization, which +is sensitive to the adopted sSFR cut. Right panel: we overlap the results from the SHARK (dot-dashed lines) and EAGLE (solid bands) simulations (Lagos et al. +2018; Matthee & Schaye 2019) with our data. For both panels, we have modified our bins to be consistent with the bins used in Matthee & Schaye (2019). +At 𝑀∗ > 109.8𝑀⊙, where we are mostly mass-complete, we find differing trends between the observational results and the EAGLE simulations, and also note +there are large differences in the trends inferred from SHARK and EAGLE. We obtain error bars of intrinsic MS scatter by bootstrap resampling the data by their +uncertainties 100 times and remeasuring the intrinsic scatter. +• We compare our results to other observational studies of the +MS scatter. Differing from a non-evolving, mass-independent scatter, +our results are qualitatively similar to the ‘U-shaped’ intrinsic MS +scatter distribution with stellar mass and redshifts found in Davies +et al. (2022). +• We compare the intrinsic MS scatter to some theoretical studies. +The consistent upturn trend in SHARK models suggests the agreement +of the feedback mechanism from past AGN activity for galaxies with +𝑀∗ > 1010𝑀⊙. These comparisons highlight the significant influ- +ence that sSFR cuts can have on the measured value of the ‘intrinsic’ +MS scatter and that particular care needs to be taken with such com- +parisons. We also find that the behaviour of intrinsic MS scatter +diverges significantly between our study and EAGLE Simulation. +In the future, the most significant gain in our understanding of the +evolution in the MS scatter will come from deeper surveys in rest- +frame IR to enable accurate characterization of the MS scatter at both +low stellar masses and higher redshifts, where our current sample is +severely limited. There is a weak agreement between observation +data and theory in Equation 7 for high-𝑧 and low mass cases. In +particular, better sampling in these regimes will provide a clear test +of whether our toy model linking the MS scatter to the HMSM +relation is reasonable. Alternatively, we also plan to explore less- +restrictive selection criteria in the IR bands to push our sample to +include more low-𝑀∗ and/or high-𝑧 sources from existing surveys. +ACKNOWLEDGEMENTS +The authors appreciate the referee, Antonios Katsianis, who provided +valuable and insightful comments on the manuscript. Parts of this +research were supported by the Australian Research Council Centre +of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO +3D), through project number CE170100013. We thank the COSMOS +team for making the data products that we used in this project. K.G. is +supported by the Australian Research Council through the Discovery +Early Career Researcher Award (DECRA) Fellowship DE220100766 +funded by the Australian Government. We also thank Jorryt Matthee +for correspondence regarding his study using the EAGLE survey. This +research was conducted on Ngunnawal Indigenous land. +DATA AVAILABILITY +The observational data used in this paper are publicly available +through catalog and imaging data releases from the COSMOS survey +team (see Section 2.1). Other data products can be made available +upon reasonable request to the first author. +REFERENCES +Arnouts S., et al., 2002, MNRAS, 329, 355 +Barro G., et al., 2017, ApJ, 840, 47 +Battisti A. J., et al., 2019, ApJ, 882, 61 +Battisti A. J., et al., 2022, MNRAS, 513, 4431 +Behroozi P. S., Conroy C., Wechsler R. H., 2010, ApJ, 717, 379 +Bisigello L., Caputi K. 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Since the sSFR is computed by multi- +band SED fitting, we expect this technique will be more accurate in +excluding quenched galaxies than a color-color cut based on U, V, +and J bands. We find that 75% of galaxies below our sSFR cut are +excluded by the U-V-J cut, so the majority of ‘quenched’ galaxies in +our samples are also designated as ‘passive’ in the U-V-J diagram. +However, only 5.17% galaxies that were eliminated by U-V-J selec- +tion lie below our sSFR cut. Hence, the bulk of galaxies excluded +by the colour-colour cut for our sample are SF galaxies; presumably, +they are dusty SF galaxies. Therefore, we adopt an sSFR cut rather +than U-V-J cut to remove quenched galaxies for our analysis. +MNRAS 000, 1–15 (2022) + +16 +R. Huang et al. +Figure A1. Left panel: U-V-J diagram lot for the 13,071 𝜒2 selection galaxies. Right panel: in addition to Figure 4, we show the galaxies removed by U-V-J cut +from Whitaker et al. (2012). 64 galaxies are excluded by the sSFR cut, and 929 galaxies are excluded by the U-V-J cut, whereas only 48 of them are marked as +quenched galaxies jointly by both selections. +MNRAS 000, 1–15 (2022) + +3 +2 +1 +U +U-V-J cut +Encloses 68% +Encloses 95% +removed by sSFR cut +Encloses 99% +0 +1 +2 +3 +468% enclosed curve +95% enclosed curve +8 +sSFR cut: 0.57Zphot - 11.60 +removed by UVl cut +log(sSFR/yr-1) +-10 +-12 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Zphot \ No newline at end of file diff --git a/KtFQT4oBgHgl3EQfTjYq/content/tmp_files/2301.13293v1.pdf.txt b/KtFQT4oBgHgl3EQfTjYq/content/tmp_files/2301.13293v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..93bb0086873cc32084adb275d28a4dd3b504cba6 --- /dev/null +++ b/KtFQT4oBgHgl3EQfTjYq/content/tmp_files/2301.13293v1.pdf.txt @@ -0,0 +1,909 @@ +OVERCOMING SIMPLICITY BIAS IN DEEP NETWORKS USING A +FEATURE SIEVE +Rishabh Tiwari, Pradeep Shenoy +Google Research +{rishabhtiwari,shenoypradeep}@google.com +February 1, 2023 +ABSTRACT +Simplicity bias is the concerning tendency of deep networks to over-depend on simple, weakly +predictive features, to the exclusion of stronger, more complex features. This causes biased, incorrect +model predictions in many real-world applications, exacerbated by incomplete training data containing +spurious feature-label correlations. We propose a direct, interventional method for addressing +simplicity bias in DNNs, which we call the feature sieve. We aim to automatically identify and +suppress easily-computable spurious features in lower layers of the network, thereby allowing the +higher network levels to extract and utilize richer, more meaningful representations. We provide +concrete evidence of this differential suppression & enhancement of relevant features on both +controlled datasets and real-world images, and report substantial gains on many real-world debiasing +benchmarks (11.4% relative gain on Imagenet-A; 3.2% on BAR, etc). Crucially, we outperform +many baselines that incorporate knowledge about known spurious or biased attributes, despite our +method not using any such information. We believe that our feature sieve work opens up exciting new +research directions in automated adversarial feature extraction & representation learning for deep +networks. +1 +Introduction +Deep networks are known to be vulnerable to a number of failure modes; in particular, simplicity bias is the tendency +of DNNs to prioritize weak predictive features over stronger, more difficult-to-extract features [1]. This bias has +been studied analytically [2] as well as empirically using natural images (texture bias [3]) and carefully controlled +synthetic datasets [4] that independently manipulate feature complexity and predictive power. Such learning biases have +significant real-world consequences too, resulting for instance in biased decision-making in AI-assisted workflows for +face recognition, healthcare, credit rating, etc. Figure 1 illustrates the idea behind simplicity bias, and some real-world +consequences. As a result, much recent work aims to debias neural network models via a variety of approaches to +achieve more equitable outcomes [5, 6, 7, 8]. +Previous approaches towards debiasing neural networks include data manipulation (augmentation & adversarial +training [9, 10], data reweighting [11], multiple training environments [12, 13], and loss function design (robust +learning [2], fairness objectives [14]). Other approaches include diversity-enhanced ensembles [10, 15, 16] and +architecture optimization [17]. +We propose a novel, direct approach towards addressing simplicity bias in neural networks: an adversarial learning +challenge that forces the network to learn sophisticated feature representations. We refer to this learning challenge +as a feature sieve enforced by an auxiliary network (Figure 2a;b). Our primary intuition is that simple features are +computable early in the neural network, and proliferate throughout the deeper layers, thereby hindering the learning +of complex features. We therefore propose to use the auxiliary network to alternately predict labels using available +features at some intermediate level (i.e., identify simple predictive features), and erase those features from the early +layers of the network, using a “forgetting loss” (Figure 2b; details in Section 3). Critically, our proposal does not depend +arXiv:2301.13293v1 [cs.LG] 30 Jan 2023 + +Overcoming simplicity bias in deep networks using a feature sieve +Figure 1: Simplicity bias and spurious features. a) DNNs focus on color to the exclusion of shape when both are +predictive. b) Image misclassified as elephant due to overdependence on texture features (adapted from [3]). c) +Classifiers mislabel blond-haired male faces as female. +Figure 2: SIFER workflow and results. a) We use an auxiliary network to alternately identify predictive features and +erase them only at lower network layers. By positioning the auxiliary network at different depths, we control the +complexity of erased features. See Section 3 for details. b) Our approach successfully suppresses digit and enhances +CIFAR decodability at higher layers for CIFAR_MNIST dataset. c) We show significant gains over other approaches on +many real-world debiasing benchmarks. +on any specific definition or complexity class of “simple features”, and instead automatically customizes to the data +characteristics using generalization error estimates. +We explicate our approach & its inner workings using experiments on controlled datasets (CMNIST, CIFAR_MINST), +and demonstrate its practical value on real-world debiasing benchmarks (BAR [11], CelebA [18], NICO [19], ImageNet- +9 [20] & ImageNet-A [21]); in nearly all experiments we establish substantial gains over other competitive approaches +for the relevant datasets. Figure 2c provides a quick visual summary of our findings. +Summing up, we propose SIFER: Sieving Features for Robust learning, a novel approach towards mitigating simplicity +bias, thereby debiasing neural networks from spurious correlations in data. Our contributions are listed below. +2 + +VS +a. Shape suppression (ColoredMNlsT) +b. Texture bias +c. Attribute bias +(ImageNet) +(CelebA)training +forgetting +9 +y +y +L4 +L4 +Aux +L3 +Aux +L3 +个 +L2 +L2 +x: +L1 +L1 +x→ +a) +Feature Sieve workflowERM +Feature Sieve +1.0 +L4 decodability +ComplexFeatures +SimpleFeatures +ComplexFeatures +0.8 +SimpleFeatures +0.6 +0.4 +10 15 20 25 30 35 40 0 510 15 20 25 30 35 40 +5 +Epochs +b) Suppression of simple featuresFEATURE SEVEVS +BASELINES +12 +11.14 +10 +Rel Acc Gain +8 +6 +4 +4 +3.15 +2 +1.14 +0.11 +0 +Imagenet-A +BAR +CelebA-Hair +Nico-Animal +(c) Real-world impactOvercoming simplicity bias in deep networks using a feature sieve +• We propose and formalize the idea of a feature sieve for mitigating simplicity bias, and provide an automated +learning recipe to control feature complexity based on validation set. +• We show, using controlled datasets, the effectiveness of our approach in enhancing the decodability of complex +features. We also demonstrate the customizability of our approach based on the desired generalization goals +on a given dataset–in short, our work is not restricted only to suppressing simple features, but is more broadly +a controllable feature tradeoff tool. +• We show significant gains in debiasing classifiers on real-world datasets: 3.2%, 4%, 11.1% relative gains over +baselines on BAR, ImageNet-9, ImageNet-A (Figure 2c). Crucially, we do not use foreknowledge of biased +features / input dimensions in obtaining these results, unlike many of the baselines we outperform. +• Finally, we show using feature importance visualizations that SIFER is able to correctly identify important +visual features of a scene, while suppressing irrelevant but spuriously-label-correlated background features +(Figure 4); this underscores the relevance of SIFER to real-world feature understanding. +We hope that our work with SIFER encourages further work in designing interesting computational barriers for neural +networks; by automating the extraction and combination of diverse features ordered by complexity and predictive power, +we could make significant progress towards the debiasing of machine learning models. +2 +Related Work +2.1 +Simplicity bias +Shah et al. [1] showed that neural networks trained with SGD are biased to learn the simplest predictive features in the +data while ignoring others. Numerous studies have attempted to investigate the correlation and impact of such shortcuts, +yielding a wealth of intriguing findings [4, 22]. +2.2 +Debiasing Spurious Correlations +Unlike our work, the majority of previous work on mitigating simplicity bias uses explicit biased-attribute labels +[23, 24, 25, 26, 27, 28] in their debiasing recipes. This reduces their practicality since both identifying, and manually +labeling biased instances and dimensions in real-life data are significant barriers. Only recently, the focus has shifted +towards debiasing without using explicit attribute labels [10, 11, 15, 16, 29]. Here we discuss different technical +approaches used by previous work in both of the above directions: +Alternate Networks: LfF [11] and LWBC [15] initially train a prejudiced network and try to debias the second network +by focusing on samples that go against the bias. +Ensemble: LWBC [15] and ESB [16] both create a classifier ensemble; the former work enforces debiasing via +reweighting of training instances, while the latter incorporates a diversity constraint in the ensemble. +Architecture Design: NAS-OoD [17] adds an OOD generalization criterion to network architecture search training to +construct inherently more robust network architectures. OccamNet [29] adds a few inductive biases in the network–for +instance, explaining the dataset with simple hypothesis using minimum depth of the network, and applying spatial +localization assumptions about unbiased (visual) features in order to filter spurious features. +Multiple Environments: IRM [12] uses the theory of causal bayesian networks to find an invariant feature representation +using multiple training environments with different bias correlations. REx [27] tries to improve on the worst linear +combinations of risks from different training environments. CaaM [30] learns causal attention by partitioning the data +on-the-go to break correlation with bias. +Augmentations: DecAug [28] proposed a semantic augmentation and feature decomposition approach to disentangle +context features from category related features. Niu el al. [10] adds adversarial augmentations to the image while +training to avoid over-reliance on spurious visual cues. This work is conceptually closest to our work, in that it builds +an ensemble where previous components compete with a new classifier to encourage it to learn diverse hypotheses. Our +approach directly addresses the competitive development of features within a network (the “heart” of the simplicity bias +challenge); we outperform them on the BAR dataset [11] (Table 2), while being more computationally parsimonious. +3 + +Overcoming simplicity bias in deep networks using a feature sieve +3 +SIFER: a feature sieve for bias mitigation +3.1 +Preliminaries & intuition +We start from the assumption that simple features are (by definition) quickly learned, available early in the neural +network stack (i.e., in layers closer to the input), and are made easily available throughout the subsequent layers (see +e.g., Hermann et al. [4] for substantial supportive evidence for these assumptions). Further, the ubiquitous presence +of simple features prevents acquisition of more complex hypotheses by subsequent NN layers, due to the so-called +simplicity bias inherent in NN training methods (see e.g., [1, 2] for theoretical results supporting these claims). +Thus, our primary goal is to identify and actively suppress simple / spurious predictive features, so as to create room for +the learning of complex predictive features at higher layers of the NN–an approach we refer to as a “feature sieve”. +We include another key consideration in the design of our approach: do not leverage any a priori information of simple +features, or even the function class / degree of complexity of simple features. To support this design goal, we a) build +into our design the knobs that control tradeoffs between simpler- and more complex-to-compute features, and b) focus +on reducing generalization error as the objective in setting these knobs. This allows us to not only automatically discover +useful tradeoffs, but also to ensure that our trained classifiers are overall more accurate than standard baselines. +3.2 +The alternating identify-and-erase workflow +Figure 2(a) provides an overview of SIFER. Briefly, we use an auxiliary network, working at an intermediate level +of representation in the neural network, to identify predictive features (simple / spurious) in the representation, and +subsequently to erase them at the lower layers of the primary network. This is a direct operationalization of our primary +goal stated above. +Identifying simple features: The training of the primary and auxiliary networks proceed in conventional fashion via +forward- and back-propagation (Figure 2(a), left panel, black & blue arrows respectively). By controlling the auxiliary +network’s capacity and the layer of the primary network to which it is attached, we can control the complexity of the +predictive features it can identify. +Applying the feature sieve: We aim to erase the identified features in the early layers of the neural network, by the +combination of the following steps: a) The parameters of auxiliary layer (A) are frozen and only that portion of the +main network (Md) which is before the auxiliary layer is kept trainable– this is the region where we wish to “forget” +the simple features, and b) We apply a forgetting loss (Lf) at the output layer of the auxiliary network. +ˆyaux += +A(Md(x)) +(1) +yep += +[ 1 +n, 1 +n, ...] (n entries) +(2) +Lf += +CE(ˆyaux, yep) +(3) +where x, yep, ˆyaux and n represent input images, a pseudo-label with uniform probability across classes, the prediction +from auxiliary layer, and number of classes respectively. +Iterative optimization: A challenge is that this process of identification and sieving is dynamic in nature; in particular, +the two steps may interfere with each other. In order to handle this challenge, we interleave the two steps such that each +forgetting step happens after regular intervals of some minibatch iterations (F) which is treated as a hyperparameter +selected using the validation set. +The entire learning recipe is summed up in Algorithm 1 +4 + +Overcoming simplicity bias in deep networks using a feature sieve +Algorithm 1: SIFER: Mitigating simplicity bias +Input +:Pretrained Model Weights W; +training data D; training iters N +Hparams :Aux Depth AD; Aux Position AP +main_lr_weight α1; aux_lr_weight α2 +aux_forget_weight α3; forget_after_iters F +Output +:robust model weights W +for k = 1 . . . N do +(x, y) ← sample(D) +ˆy, ˆyaux ← Forward_with_aux(x, AD, AP , W) +L1 ← CE(ˆy, y) +L2 ← CE(ˆyaux, y) +Lf ← CE(ˆyaux, U) +L ← α1L1 + α2L2 +if k % F == 0 then +L ← L + α3Lf +end +∇W ← Backward(L) +W ← OptimizeStep(∇W) +end +3.3 +Controllability of the feature sieve +As remarked earlier, we aim to autodiscover notions of and tradeoffs between so-called simple and complex features, as +relevant for the specific dataset on hand. The feature sieve approach described here allows for many mechanisms to +control this discovery & tradeoff. The primary parameters are the position & depth of the auxiliary network (AP , AD) +which implicitly control the function complexity of the features available for discovery by the auxiliary network; and +the auxiliary forgetting weight α3, which controls the degree to which the discovered features are suppressed. The +interleaving of the feature identifying & feature sieving steps is controlled by the parameter Fn–again, based on the +specific dataset and the nature of the features contained, this controls the dynamics of the training procedure. +Finally, we set these hyperparameters based on the goal of minimizing validation error–this ensures not only that the +parameters are chosen using unbiased estimates of generalization, but also that at a minimum, we perform better than +the standard training baseline (which, as the trivial solution of not-forgetting, is included in the search space for the +feature sieve). +4 +Experiment setup +4.1 +Datasets for studying simplicity bias +CMNIST: Colored-MNIST is a 2-class synthetic dataset used to study simplicity bias, we use labels 0/1 for digits 0 & +1 respectively from the MNIST dataset. A color channel (red, green) is artificially added to each example with perfect +correlation between color and digit. +CIFAR-MNIST: This is a binary classification dataset consisting of paired-composite images–Class 0 pairing MNIST +0s with CIFAR automobiles, and Class 1 pairing MNIST 1s with CIFAR’s truck images. +For CMNIST and CIFAR-MNIST, the training set contains perfectly predictive simple and complex features; by training +a classifier and then manipulating the test set to break one of these correlations, one can examine which features are +being used by the trained classifier. +4.2 +Real-world debiasing benchmarks +BAR: Biased Activity Recognition [11] is a real-world image benchmark for classifying human actions (images) into 6 +classes; each training image contains spurious correlations with background features (e.g., rocks with climbing). The +test set contains the same set of actions but with different backgrounds (e.g., ice with climbing). The training data has +no bias-conflicting examples, which makes this a challenging benchmark. +NICO: NICO [19] is a real-world benchmark for out-of-distribution robustness. Following [28], we used its Animal +subset containing 10 object classes and 10 context labels. The training set only contains 7 contexts for each object class +5 + +Overcoming simplicity bias in deep networks using a feature sieve +while the validation and test set contains 3 extra unseen contexts (total 10). Unlike the majority of the baselines, we +don’t use context label attributes in train, validation, or test. +ImageNet-9: ImageNet-9 [20] is a subset of ImageNet [31] containing 9 super classes. It has been established that this +subset has a spurious correlation between object labels and image texture. We followed the setting used by [15] and +[32] for creating train and val split. We report the average accuracy on the validation split. +ImageNet-A: ImageNet-A [21] contains handpicked real-world images misclassified by models trained on ImageNet. +Since these misclassifications are due to over-reliance on spurious features like color&texture, we use this dataset for +evaluating models trained on ImageNet-9 as a robustness challenge (i.e., OOD test set). +4.3 +Training procedure & metrics +For all our synthetic as well as real-world experiments we consistently used ResNet-18, an auxiliary layer that uses +the same layer structure as of BasicBlock of ResNet with varying depth, optimized using SGD optimizer with a fixed +learning rate of 0.001. For real-world experiments the model is loaded with ImageNet pre-trained weights. We repeat +the experiments with 5 different random seeds and report the mean and std deviation of results. Refer Appendix A for +more details. +Choice of Validation Set: For BAR, since there is no val split given, we study it under two settings, in first, we use +20% of images from the test set and call it as OOD valset. In the second, which is more harder and realistic, we use +20% images from the train set calling it In-Domain (ID) valset. For NICO-Animal, CelebA Hair and ImageNet-9, we +use the already given validation split. +Evaluation Metrics: Accuracy means average accuracy on all examples. Unbiased means accuracy averaged over +each label-context group. This metric is more fair when there is a huge imbalance between the groups. Conflicting +means accuracy only on the bias conflicting examples. +We used Accuracy for BAR, NICO and Imagenet-9 / ImageNet-A and Unbiased for CelebA-Hair dataset as the +performance metric on validation data for hyperparameter search and early stopping. +5 +Results +Figure 3: Decodability of Simple (MNIST) and Complex (CIFAR) features across layers of ResNet-50 with a) Normal +ERM training b) with SIFER +5.1 +Suppressing simple features +We first studied the effectiveness of SIFER on targeted suppression of specific features. To do this, we experimented with +the CIFAR_MNIST dataset, which consists of composite pairings of MNIST & CIFAR images, each fully predictive +of the assigned label (see Section 4 for more details). DNNs are known to entirely ignore the CIFAR feature on this +training dataset–when the CIFAR component is randomized at test-time, accuracy is unaffected, but when the MNIST +6 + +Layer 1 +Layer 2 +Layer 3 +Layer 4 +1.0 +1.0 +1.0 +0.8 +0.8 +0.8 +0.8 +oComplexFeatures +RM +·Simple Features +0.6+ +0.6 +0.6 +0.6 +(Acc) +Decodability +0.4 +0.4 +0.4 +0.4 +10 15 20 25 3035 40 +510 15 20 25 3035 40 +510 15 20 25 30 35 40 +510 1520 25 30 35 40 +0 + Sieve +1.0 +1.0 +1.0 +1.0 +Feature +0.8 +0.8 +0.8 +0.8 +ComplexFeatures +· Simple Features +0.6+ +0.6 +0.6 +0.6 +0.4 +0.4 +0.4 +0.4 +10 15 202530 35 40 +5 +101520 25 30 35 +5 +10′152025 +40 +30 35 40 +1015 +20 25 +30 35 40 +0 +0 +0 +5 +EpochsOvercoming simplicity bias in deep networks using a feature sieve +Table 1: Feature Controllability. +DataSet +Target Feature +SIFER (Ours) +ERM +SR +CR +SR +CR +CMNIST +Complex (Digit) +99.54±0.19 +58.14±10.69 +56.96±6.59 +92.21±3.92 +Simple (Color) +52.44±1.22 +99.64±1.30 +49.20±2.60 +96.27±0.99 +CIFAR_MNIST +Complex (CIFAR) +62.37±4.62 +48.93±1.92 +58.14±1.60 +100 +Simple (Digit) +47.17±0.14 +99.83±0.29 +49.20±2.60 +100 +component is randomized, accuracy drops to chance. We refer to the MNIST component as the simple feature, and +CIFAR as the complex feature. +Figure 3 shows the layerwise decodability1 of simple and complex features at different layers of the neural network, +as tracked across epochs in the learning process. We contrast standard training (top row) against SIFER (bottom +row). Standard training overemphasizes the simple feature at higher layers, and does not pay attention to the complex +feature. We note, particularly, that the complex CIFAR feature is in fact decodable to some extent in earlier layers of +the ERM classifier, but is suppressed in later layers, due to the preponderance of the simple feature. In contrast, the +auxiliary forgetting loss in SIFER effectively suppresses the simple feature in the earlier layers, and thereby enhances +the decodability of the complex feature in the higher layers. This shows that removing the availability of spurious +simple features is a direct method of overcoming simplicity bias. +What makes these findings more interesting is the fact that no knowledge of which features in particular are considered +simple or complex was used in any way whatsoever in this experiment. In other words, SIFER organically discovers +and suppresses the by-design simple feature purely through the use of the strategically placed auxiliary network, and its +configuration via the training recipe. +5.2 +Feature controllability using SIFER +We described in Section 3.3 the various degrees of freedom in SIFER for identifying and suppressing complex features. +Since these DOFs are optimized on a validation set (i.e., driven by generalization error), this gives us a simple method +of customizing SIFER to the characteristics of the dataset, and in particular, the needs of the downstream task–e.g., +generalization to related domain data. We demonstrate this capability by conducting studies on controlled datasets +CMNIST & CIFAR_MNIST. In each, the training data were designed to have pairings of simple and complex features +((color,digit) and (image,digit) respectively) where both simple and complex features were fully predictive. We then +trained a SIFER classifier for different choices of validation set, representing which feature we actually wanted the +classifier to focus on. This was achieved by randomizing the “spurious” feature in the validation dataset, and choosing +all our hyperparameters based on that validation dataset. This represents a real-world scenario where small amounts of +vetted data are available for optimization of a model, but (re)-labeling or manipulating all the training data is infeasible. +Table 1 shows the results, comparing SIFER against ERM. Our approach shows higher accuracy for chosen features +(higher diagonal terms) than for spurious features (off-diagonal terms), driven by the choice of the validation data. In +contrast, ERM primarily focuses on the simple features, irrespective of the choice of validation set (higher second +column numbers). Thus, our method is in fact able to focus on the relevant feature–be it simple or complex–in an easily +controllable manner. +5.3 +Debiasing real-world datasets +Our method outperforms baselines on four different real world datasets–BAR [11], CelebA Hair [18], NICO [19] and +Imagenet-A [21], by large margins (upto 11%, see Figure 2c for a quick summary). Critically, we chose in all our +experiments to not use any knowledge of which attribute labels are considered spurious in each dataset–this is because +in real-world scenarios, it is difficult to know in advance which attributes may end up containing biased information, or +to label data according to those attributes in order to do targeted debiasing of models. Nevertheless, we outperform the +other baselines, including many that do use attribute labels as part of their training procedure. +1Here, “decodability” is the accuracy of a linear decoder trained on the learned representations of that layer, to predict the specific +feature of interest. +2Architecture design optimization based method, hence unfair to compare directly against other methods. +7 + +Overcoming simplicity bias in deep networks using a feature sieve +Figure 4: Examples of SIFER’s focus on relevant features while suppressing irrelevant background information. Top +row: Input images; Middle row: GRAD-CAM-derived feature importance visualizations for the ERM classifier; Bottom +row: feature importance for SIFER. First 3 columns from BAR [11], and last 3 columns from NICO-Animal [19]. +Table 2: Classification Accuracy (%) on test set of BAR Dataset. +Method +Used OOD Val +Accuracy +ERM + +51.85± 5.92 +BiaSwap [33] + +52.44 +LfF [11] + +62.98± 2.76 +PGI [34] + +65.19± 1.32 +EIIL [35] + +65.44± 1.17 +ESB [16] + +67.10± 0.30 +Roadblock [10] + +69.51± 2.43 +Debian [14] + +69.88± 2.92 +SIFER (Ours) + +72.08± 0.38 +ERM + +35.32± 0.46 +ReBias [32] + +37.02± 0.26 +LfF [11] + +48.15± 0.93 +SSL+ERM [15] + +60.88± 0.80 +LWBC [15] + +62.03± 0.74 +ESB [16] + +64.40± 0.20 +SIFER (Ours) + +65.75± 1.84 +Mitigating Spurious Correlations: Biased Activity Recognition (BAR) and CelebA Hair Dataset represent back- +ground and gender bias in real life. In the BAR training set, human activity (image categories) is spuriously correlated +with the background in which those activities are performed; in CelebA Hair, hair color is strongly correlated with +gender. Both BAR and CelebA are heavily biased and contain no or very few conflicting examples–eg. CelebA Hair +has only 1% of men with blond hair in the train set. +For BAR, since no validation set is provided, we show results both using in-distribution and out-distribution validation +sets to compare against all the baselines that do and don’t require conflicting examples in the validation set. Note +that methods that work on the principle of reweighting conflicting samples in the trainset (eg LWBC [15]) typically +8 + +Overcoming simplicity bias in deep networks using a feature sieve +Table 3: Unbiased and Conflicting Accuracy metrics (%) on Test set of CelebA Hair Dataset +Method +Spurious +Attribs +Unbiased +Conflict +DRO [25] + +85.43± 0.53 +83.40± 0.67 +EnD [36] + +91.21± 0.22 +87.45± 1.06 +CSAD [37] + +89.36 +87.53 +ERM + +70.25± 0.35 +52.52± 0.19 +LfF [11] + +84.24± 0.37 +81.24± 1.38 +SSL+ERM [15] + +80.48± 0.91 +66.79± 2.20 +LWBC [15] + +88.90± 1.55 +87.22± 1.14 +SIFER (Ours) + +89.00± 0.92 +88.04± 1.25 +Table 4: Classification Accuracy (%) on test set of NICO Dataset. Most of the competititive baselines eg. DecAug, +DRO, etc use spurious attribute labels for training, still we outperform all of them. +Method +Accuracy +ERM +75.87 +IRM [12] +59.17 +REx [27] +74.31 +JiGen [38] +84.95 +Mixup [39] +80.27 +Cumix [40] +76.78 +MTL [41] +78.89 +DANN [42] +75.59 +CORAL [43] +80.27 +MMD [44] +70.91 +DRO [25] +77.61 +CNBB [19] +78.16 +DecAug [28] +85.23 +SIFER (Ours) +86.20± 0.85 +NAS-OoD2 [17] +88.72 +adds 1% of conflicting samples from the test set to the training data. We do not make these changes to the training +data. We outperform baselines in both settings by more than 1-2% absolute accuracy, refer Table 2. Interestingly, our +method with only in-distribution validation data outperforms most baselines that leverage an additional OOD validation +set, showing the superior generalization and robust feature learning capacity of our method. Table 3 shows results on +CelebA dataset, we get almost the same unbiased accuracy as LWBC and improve upon conflicting accuracy. +Domain-shift Generalization: NICO introduces three new contexts in which object classes appear in the validation +and test set, that are absent in the training set. Table 4 shows classification accuracy on test set. Our method outperforms +all the baselines. Unlike a majority of the baselines, we do not use spurious attribute labels (context labels) while +training our model. This shows the value of SIFER in domain shift generalization without knowledge of domain IDs. +Robustness to Texture bias: Table 5 shows results on ImageNet-9, which is known to be biased towards texture, and +ImageNet-A, which consists of natural images that have bias-conflicting features. This setting is closest to real-world +scenarios for texture bias. We improve the previous best baseline by 3% absolute on ImageNet-9 validation set and +by 4% absolute on ImageNet-A test set. Thus, SIFER encourages learning features robust to texture bias, improving +performance on both the in-distribution validation set as well as bias-conflicting test set. Two critical findings here +are a) that SIFER did not sacrifice in-distribution accuracy through the process of sieving simple features, and b) the +learned classifier robustly transfers over to a novel test set, where it provides even larger gains. +9 + +Overcoming simplicity bias in deep networks using a feature sieve +Table 5: Classification Accuracy (%) on Validation set of ImageNet-9 and test set of ImageNet-A. +Method +Spurious +Attribs +ImageNet-9 +ImageNet-A +Accuracy +Accuracy +StylisedIN [3] + +88.4± 0.5 +24.6± 1.4 +LearnedMixin [45] + +64.1± 4.0 +15.0± 1.6 +RUBi [46] + +90.5± 0.3 +27.7± 2.1 +ERM + +90.8± 0.6 +24.9± 1.1 +BagNet18 [47] + +67.7± 0.3 +18.8± 1.15 +ReBias [32] + +91.9± 1.7 +29.6± 1.6 +LfF [11] + +86.00 +24.60 +CaaM [30] + +95.70 +32.80 +SSL+ERM [15] + +94.18± 0.07 +34.21± 0.49 +LWBC [15] + +94.03± 0.23 +35.97± 0.49 +SIFER + +97.78± 0.12 +39.98± 0.81 +5.4 +SIFER focuses on relevant information +We visualize the information in an image that is relevant to a given classifier [48], in order to verify whether our feature +sieving results in semantically relevant modifications to learned classifiers. Figure 4 shows this evaluation, contrasting +ERM classifier’s regions of focus (middle row) and SIFER’s regions of focus (bottom row) on a range of input images +(top row, drawn from BAR & NICO). Interestingly, not only does SIFER correctly focus on the central object of interest, +but also it is able to effectively suppress the (spuriously label-correlated) background information, which is highly +valued by the ERM classifier. This undercores SIFER’s ability to carefully differentiate between relevant and irrelevant +features, rather than some notion of simple vs complex features alone. +6 +Discussion & conclusion +We proposed SIFER–a novel feature sieve approach towards addressing simplicity bias and spurious correlations in +deep neural networks. Our proposal introduces an auxiliary network attached to the deep network which alternately +identifies and suppresses predictive features. The approach is controllable through the use of configuration parameters +optimized using validation data; thus, it requires no foreknowledge or hand-coding of the notion of “simple features”. +We demonstrated on controlled datasets the ability of SIFER to automatically identify and suppress features; further, +we showed that, strictly speaking, SIFER rebalances the role of various features in a controllable manner driven by +the needs of generalization. We showed using extensive experiments on real-world data that our approach provides +significant gains–3-11% relative accuracy improvements on BAR, NICO, and Imagenet-A. We believe our work is a +small, important first step in a fruitful new direction of research. We hope that follow-on work will build on the notion +of the feature sieve, developing effective computational barriers that encourage deep networks to discover and utilize +richer, more powerful featural representations. Our current approach strikes a balance between various competing +features, guided by generalization error estimates (validation error). One could potentially extract even more value if +different feature classes could be isolated into (relatively) independent predictors, then combined effectively. This is, +for instance, the approach taken by Niu et al. [10]. Thus, a straightforward next step we aim to explore is the study of +ensembling approaches to combine a range of features of varying complexity & predictive power, and methods for +efficiently learning them. We also hope to develop a systematic theoretical understanding of feature sieve approaches +and their role in supervised learning using DNNs. +References +[1] Harshay Shah, Kaustav Tamuly, Aditi Raghunathan, Prateek Jain, and Praneeth Netrapalli. The pitfalls of +simplicity bias in neural networks. 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Table 6 shows +the hyperparamters search space for all the hyperparameters that we tune on the basis of validation set. +Table 6: Range for hyperparameters search. +Hparam +Range +AD +[1, 9] +AP +[1, 3] +α1 +loguniform(10−2, 101) +α2 +loguniform(1, 103) +α3 +loguniform(1, 103) +F +[1, 9] ∗ 10 +B +Baselines +Here we list and briefly explain all the baselines that we compare against on real world datasets: +BiaSwap [33] proposes a bias-tailored augmentation-based approach for learning debiased representation without +requiring supervision on the bias type. they divide the data into bias-guiding and bias-conflicting groups and then swaps +the bias in bias guiding group. +LfF [11] uses generalized cross-entropy initially trains a prejudiced net-work and tries to debias the second network by +focusing weighing on samples that go against the bias. +IRM [12] uses theory of causal bayesian networks to find an invariant feature representation using multiple training +environments with different bias correlations. +REx [27] proposed a min-max algorithm to optimize for the worst linear combination of risks on different environments. +EIIL [35] optimizes for bias group assignment to automatically identify the bias groups to maximize IRM. +PGI [34] follows EIIL to identify bias groups by training a small neural network. +Evading Simplicity Bias (ESB) [16] creates a ensemble of diverse classifiers by incorporating a diversity regularizer +between the gradients while training. +Roadblock [10] adds adversarial augmentations to the image while training to avoid over-reliance on spurious visual +cues. +Debian [14] trains two networks in alternate manner namely discoverer and classifier, the discoverer tries to find +multiple unknown biases of the classifier without any annotations of biases, and the classifier aims at unlearning the +biases identified by the discoverer. +ReBias [32] propose a novel framework to train a de-biased representation by encouraging it to be different from a set +of representations that are biased by design. +LWBC [15] employs a committee of classifiers as an auxiliary module that identifies bias-conflicting data and assigns +large weights to them when training the main classifier. +Group-DRO [25] minimizes for worst-case training loss over a set of pre-defined groups. +EnD [36] proposes a regularization technique that uses the bias attributes to prevent deep models from learning +spurious biases by inserting an information bottleneck. +CSAD [37], given the bias attributes, explicitly extracts target and bias features disentangled from the latent +representation generated by a feature extractor and then learns to discover and remove the correlation between the +target and bias features. +JiGen [38] jointly classifies objects and solves unsupervised jigsaw tasks. +Cumix [40] mixes up data and labels from different domains to be able to recognize unseen categories in unseen +domains. +MTL [41] argue that problem of Domain Generalization can be viewed as a kind of supervised learning problem by +14 + +Overcoming simplicity bias in deep networks using a feature sieve +augmenting the original feature space with the marginal distribution of feature vectors. +DANN [42] proposes a representation learning approach such that features are not predictive of the domain from which +the model is being trained on. +CORAL [43] proposes an unsupervised domain adaptation method that aligns the second-order statistics of the source +and target distributions with a linear transformation. +MMD [44] extend adversarial autoencoders by imposing the Maximum Mean Discrepancy measure to align the +distributions among different domains, and matching the aligned distribution to an arbitrary prior distribution via +adversarial feature learning. +CNBB [19] is an OoD learning method that based on sample reweighting inspired by causal inference. +DecAug [28] proposed a semantic augmentation and feature decomposition approach to distangle context features from +category related features. +NAS-OoD [17] adds an OOD generalization criterion to network architecture search training to construct inherently +more robust network architectures. +StylisedIN [3] showed that ImageNet is texture biased and works on improving shape bias. +LearnedMixin [45] trains a robust model as part of an ensemble with the naive one in order to encourage it to focus on +other patterns in the data that are more likely to generalize. +CaaM [30] learns causal attention by partitioning the data on-the-go to break correlation with bias. +15 + diff --git a/KtFQT4oBgHgl3EQfTjYq/content/tmp_files/load_file.txt b/KtFQT4oBgHgl3EQfTjYq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c48a6d0d9da7f73965bbe83ae5328108862ed4c --- /dev/null +++ b/KtFQT4oBgHgl3EQfTjYq/content/tmp_files/load_file.txt @@ -0,0 +1,645 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf,len=644 +page_content='OVERCOMING SIMPLICITY BIAS IN DEEP NETWORKS USING A FEATURE SIEVE Rishabh Tiwari, Pradeep Shenoy Google Research {rishabhtiwari,shenoypradeep}@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='com February 1, 2023 ABSTRACT Simplicity bias is the concerning tendency of deep networks to over-depend on simple, weakly predictive features, to the exclusion of stronger, more complex features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This causes biased, incorrect model predictions in many real-world applications, exacerbated by incomplete training data containing spurious feature-label correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We propose a direct, interventional method for addressing simplicity bias in DNNs, which we call the feature sieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We aim to automatically identify and suppress easily-computable spurious features in lower layers of the network, thereby allowing the higher network levels to extract and utilize richer, more meaningful representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We provide concrete evidence of this differential suppression & enhancement of relevant features on both controlled datasets and real-world images, and report substantial gains on many real-world debiasing benchmarks (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4% relative gain on Imagenet-A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='2% on BAR, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Crucially, we outperform many baselines that incorporate knowledge about known spurious or biased attributes, despite our method not using any such information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We believe that our feature sieve work opens up exciting new research directions in automated adversarial feature extraction & representation learning for deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 1 Introduction Deep networks are known to be vulnerable to a number of failure modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' in particular, simplicity bias is the tendency of DNNs to prioritize weak predictive features over stronger, more difficult-to-extract features [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This bias has been studied analytically [2] as well as empirically using natural images (texture bias [3]) and carefully controlled synthetic datasets [4] that independently manipulate feature complexity and predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Such learning biases have significant real-world consequences too, resulting for instance in biased decision-making in AI-assisted workflows for face recognition, healthcare, credit rating, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Figure 1 illustrates the idea behind simplicity bias, and some real-world consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' As a result, much recent work aims to debias neural network models via a variety of approaches to achieve more equitable outcomes [5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Previous approaches towards debiasing neural networks include data manipulation (augmentation & adversarial training [9, 10], data reweighting [11], multiple training environments [12, 13], and loss function design (robust learning [2], fairness objectives [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Other approaches include diversity-enhanced ensembles [10, 15, 16] and architecture optimization [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We propose a novel, direct approach towards addressing simplicity bias in neural networks: an adversarial learning challenge that forces the network to learn sophisticated feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We refer to this learning challenge as a feature sieve enforced by an auxiliary network (Figure 2a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Our primary intuition is that simple features are computable early in the neural network, and proliferate throughout the deeper layers, thereby hindering the learning of complex features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We therefore propose to use the auxiliary network to alternately predict labels using available features at some intermediate level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=', identify simple predictive features), and erase those features from the early layers of the network, using a “forgetting loss” (Figure 2b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' details in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Critically, our proposal does not depend arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='13293v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='LG] 30 Jan 2023 Overcoming simplicity bias in deep networks using a feature sieve Figure 1: Simplicity bias and spurious features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' a) DNNs focus on color to the exclusion of shape when both are predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' b) Image misclassified as elephant due to overdependence on texture features (adapted from [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' c) Classifiers mislabel blond-haired male faces as female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Figure 2: SIFER workflow and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' a) We use an auxiliary network to alternately identify predictive features and erase them only at lower network layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' By positioning the auxiliary network at different depths, we control the complexity of erased features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' See Section 3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' b) Our approach successfully suppresses digit and enhances CIFAR decodability at higher layers for CIFAR_MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' c) We show significant gains over other approaches on many real-world debiasing benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' on any specific definition or complexity class of “simple features”, and instead automatically customizes to the data characteristics using generalization error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We explicate our approach & its inner workings using experiments on controlled datasets (CMNIST, CIFAR_MINST), and demonstrate its practical value on real-world debiasing benchmarks (BAR [11], CelebA [18], NICO [19], ImageNet- 9 [20] & ImageNet-A [21]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' in nearly all experiments we establish substantial gains over other competitive approaches for the relevant datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Figure 2c provides a quick visual summary of our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Summing up, we propose SIFER: Sieving Features for Robust learning, a novel approach towards mitigating simplicity bias, thereby debiasing neural networks from spurious correlations in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Our contributions are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 2 VS a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Shape suppression (ColoredMNlsT) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Texture bias c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Attribute bias (ImageNet) (CelebA)training forgetting 9 y y L4 L4 Aux L3 Aux L3 个 L2 L2 x: L1 L1 x→ a) Feature Sieve workflowERM Feature Sieve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0 L4 decodability ComplexFeatures SimpleFeatures ComplexFeatures 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8 SimpleFeatures 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 10 15 20 25 30 35 40 0 510 15 20 25 30 35 40 5 Epochs b) Suppression of simple featuresFEATURE SEVEVS BASELINES 12 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='14 10 Rel Acc Gain 8 6 4 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='15 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='11 0 Imagenet-A BAR CelebA-Hair Nico-Animal (c) Real-world impactOvercoming simplicity bias in deep networks using a feature sieve We propose and formalize the idea of a feature sieve for mitigating simplicity bias, and provide an automated learning recipe to control feature complexity based on validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We show, using controlled datasets, the effectiveness of our approach in enhancing the decodability of complex features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We also demonstrate the customizability of our approach based on the desired generalization goals on a given dataset–in short, our work is not restricted only to suppressing simple features, but is more broadly a controllable feature tradeoff tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We show significant gains in debiasing classifiers on real-world datasets: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='2%, 4%, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='1% relative gains over baselines on BAR, ImageNet-9, ImageNet-A (Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Crucially, we do not use foreknowledge of biased features / input dimensions in obtaining these results, unlike many of the baselines we outperform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Finally, we show using feature importance visualizations that SIFER is able to correctly identify important visual features of a scene, while suppressing irrelevant but spuriously-label-correlated background features (Figure 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' this underscores the relevance of SIFER to real-world feature understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We hope that our work with SIFER encourages further work in designing interesting computational barriers for neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' by automating the extraction and combination of diverse features ordered by complexity and predictive power, we could make significant progress towards the debiasing of machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='1 Simplicity bias Shah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' [1] showed that neural networks trained with SGD are biased to learn the simplest predictive features in the data while ignoring others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Numerous studies have attempted to investigate the correlation and impact of such shortcuts, yielding a wealth of intriguing findings [4, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='2 Debiasing Spurious Correlations Unlike our work, the majority of previous work on mitigating simplicity bias uses explicit biased-attribute labels [23, 24, 25, 26, 27, 28] in their debiasing recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This reduces their practicality since both identifying, and manually labeling biased instances and dimensions in real-life data are significant barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Only recently, the focus has shifted towards debiasing without using explicit attribute labels [10, 11, 15, 16, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Here we discuss different technical approaches used by previous work in both of the above directions: Alternate Networks: LfF [11] and LWBC [15] initially train a prejudiced network and try to debias the second network by focusing on samples that go against the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Ensemble: LWBC [15] and ESB [16] both create a classifier ensemble;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' the former work enforces debiasing via reweighting of training instances, while the latter incorporates a diversity constraint in the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Architecture Design: NAS-OoD [17] adds an OOD generalization criterion to network architecture search training to construct inherently more robust network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' OccamNet [29] adds a few inductive biases in the network–for instance, explaining the dataset with simple hypothesis using minimum depth of the network, and applying spatial localization assumptions about unbiased (visual) features in order to filter spurious features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Multiple Environments: IRM [12] uses the theory of causal bayesian networks to find an invariant feature representation using multiple training environments with different bias correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' REx [27] tries to improve on the worst linear combinations of risks from different training environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' CaaM [30] learns causal attention by partitioning the data on-the-go to break correlation with bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Augmentations: DecAug [28] proposed a semantic augmentation and feature decomposition approach to disentangle context features from category related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Niu el al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' [10] adds adversarial augmentations to the image while training to avoid over-reliance on spurious visual cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This work is conceptually closest to our work, in that it builds an ensemble where previous components compete with a new classifier to encourage it to learn diverse hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Our approach directly addresses the competitive development of features within a network (the “heart” of the simplicity bias challenge);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' we outperform them on the BAR dataset [11] (Table 2), while being more computationally parsimonious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 3 Overcoming simplicity bias in deep networks using a feature sieve 3 SIFER: a feature sieve for bias mitigation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='1 Preliminaries & intuition We start from the assumption that simple features are (by definition) quickly learned, available early in the neural network stack (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=', in layers closer to the input), and are made easily available throughout the subsequent layers (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=', Hermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' [4] for substantial supportive evidence for these assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Further, the ubiquitous presence of simple features prevents acquisition of more complex hypotheses by subsequent NN layers, due to the so-called simplicity bias inherent in NN training methods (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=', [1, 2] for theoretical results supporting these claims).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Thus, our primary goal is to identify and actively suppress simple / spurious predictive features, so as to create room for the learning of complex predictive features at higher layers of the NN–an approach we refer to as a “feature sieve”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We include another key consideration in the design of our approach: do not leverage any a priori information of simple features, or even the function class / degree of complexity of simple features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' To support this design goal, we a) build into our design the knobs that control tradeoffs between simpler- and more complex-to-compute features, and b) focus on reducing generalization error as the objective in setting these knobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This allows us to not only automatically discover useful tradeoffs, but also to ensure that our trained classifiers are overall more accurate than standard baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='2 The alternating identify-and-erase workflow Figure 2(a) provides an overview of SIFER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Briefly, we use an auxiliary network, working at an intermediate level of representation in the neural network, to identify predictive features (simple / spurious) in the representation, and subsequently to erase them at the lower layers of the primary network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This is a direct operationalization of our primary goal stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Identifying simple features: The training of the primary and auxiliary networks proceed in conventional fashion via forward- and back-propagation (Figure 2(a), left panel, black & blue arrows respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' By controlling the auxiliary network’s capacity and the layer of the primary network to which it is attached, we can control the complexity of the predictive features it can identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Applying the feature sieve: We aim to erase the identified features in the early layers of the neural network, by the combination of the following steps: a) The parameters of auxiliary layer (A) are frozen and only that portion of the main network (Md) which is before the auxiliary layer is kept trainable– this is the region where we wish to “forget” the simple features, and b) We apply a forgetting loss (Lf) at the output layer of the auxiliary network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' ˆyaux = A(Md(x)) (1) yep = [ 1 n, 1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='] (n entries) (2) Lf = CE(ˆyaux, yep) (3) where x, yep, ˆyaux and n represent input images, a pseudo-label with uniform probability across classes, the prediction from auxiliary layer, and number of classes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Iterative optimization: A challenge is that this process of identification and sieving is dynamic in nature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' in particular, the two steps may interfere with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' In order to handle this challenge, we interleave the two steps such that each forgetting step happens after regular intervals of some minibatch iterations (F) which is treated as a hyperparameter selected using the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The entire learning recipe is summed up in Algorithm 1 4 Overcoming simplicity bias in deep networks using a feature sieve Algorithm 1: SIFER: Mitigating simplicity bias Input :Pretrained Model Weights W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' training data D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' training iters N Hparams :Aux Depth AD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Aux Position AP main_lr_weight α1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' aux_lr_weight α2 aux_forget_weight α3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' forget_after_iters F Output :robust model weights W for k = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' N do (x, y) ← sample(D) ˆy, ˆyaux ← Forward_with_aux(x, AD, AP , W) L1 ← CE(ˆy, y) L2 ← CE(ˆyaux, y) Lf ← CE(ˆyaux, U) L ← α1L1 + α2L2 if k % F == 0 then L ← L + α3Lf end ∇W ← Backward(L) W ← OptimizeStep(∇W) end 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='3 Controllability of the feature sieve As remarked earlier, we aim to autodiscover notions of and tradeoffs between so-called simple and complex features, as relevant for the specific dataset on hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The feature sieve approach described here allows for many mechanisms to control this discovery & tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The primary parameters are the position & depth of the auxiliary network (AP , AD) which implicitly control the function complexity of the features available for discovery by the auxiliary network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' and the auxiliary forgetting weight α3, which controls the degree to which the discovered features are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The interleaving of the feature identifying & feature sieving steps is controlled by the parameter Fn–again, based on the specific dataset and the nature of the features contained, this controls the dynamics of the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Finally, we set these hyperparameters based on the goal of minimizing validation error–this ensures not only that the parameters are chosen using unbiased estimates of generalization, but also that at a minimum, we perform better than the standard training baseline (which, as the trivial solution of not-forgetting, is included in the search space for the feature sieve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 4 Experiment setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='1 Datasets for studying simplicity bias CMNIST: Colored-MNIST is a 2-class synthetic dataset used to study simplicity bias, we use labels 0/1 for digits 0 & 1 respectively from the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' A color channel (red, green) is artificially added to each example with perfect correlation between color and digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' CIFAR-MNIST: This is a binary classification dataset consisting of paired-composite images–Class 0 pairing MNIST 0s with CIFAR automobiles, and Class 1 pairing MNIST 1s with CIFAR’s truck images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' For CMNIST and CIFAR-MNIST, the training set contains perfectly predictive simple and complex features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' by training a classifier and then manipulating the test set to break one of these correlations, one can examine which features are being used by the trained classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='2 Real-world debiasing benchmarks BAR: Biased Activity Recognition [11] is a real-world image benchmark for classifying human actions (images) into 6 classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' each training image contains spurious correlations with background features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=', rocks with climbing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The test set contains the same set of actions but with different backgrounds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=', ice with climbing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The training data has no bias-conflicting examples, which makes this a challenging benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' NICO: NICO [19] is a real-world benchmark for out-of-distribution robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Following [28], we used its Animal subset containing 10 object classes and 10 context labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The training set only contains 7 contexts for each object class 5 Overcoming simplicity bias in deep networks using a feature sieve while the validation and test set contains 3 extra unseen contexts (total 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Unlike the majority of the baselines, we don’t use context label attributes in train, validation, or test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' ImageNet-9: ImageNet-9 [20] is a subset of ImageNet [31] containing 9 super classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' It has been established that this subset has a spurious correlation between object labels and image texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We followed the setting used by [15] and [32] for creating train and val split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We report the average accuracy on the validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' ImageNet-A: ImageNet-A [21] contains handpicked real-world images misclassified by models trained on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Since these misclassifications are due to over-reliance on spurious features like color&texture, we use this dataset for evaluating models trained on ImageNet-9 as a robustness challenge (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=', OOD test set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='3 Training procedure & metrics For all our synthetic as well as real-world experiments we consistently used ResNet-18, an auxiliary layer that uses the same layer structure as of BasicBlock of ResNet with varying depth, optimized using SGD optimizer with a fixed learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' For real-world experiments the model is loaded with ImageNet pre-trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We repeat the experiments with 5 different random seeds and report the mean and std deviation of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Refer Appendix A for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Choice of Validation Set: For BAR, since there is no val split given, we study it under two settings, in first, we use 20% of images from the test set and call it as OOD valset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' In the second, which is more harder and realistic, we use 20% images from the train set calling it In-Domain (ID) valset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' For NICO-Animal, CelebA Hair and ImageNet-9, we use the already given validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Evaluation Metrics: Accuracy means average accuracy on all examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Unbiased means accuracy averaged over each label-context group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This metric is more fair when there is a huge imbalance between the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Conflicting means accuracy only on the bias conflicting examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We used Accuracy for BAR, NICO and Imagenet-9 / ImageNet-A and Unbiased for CelebA-Hair dataset as the performance metric on validation data for hyperparameter search and early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 5 Results Figure 3: Decodability of Simple (MNIST) and Complex (CIFAR) features across layers of ResNet-50 with a) Normal ERM training b) with SIFER 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='1 Suppressing simple features We first studied the effectiveness of SIFER on targeted suppression of specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' To do this, we experimented with the CIFAR_MNIST dataset, which consists of composite pairings of MNIST & CIFAR images, each fully predictive of the assigned label (see Section 4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' DNNs are known to entirely ignore the CIFAR feature on this training dataset–when the CIFAR component is randomized at test-time, accuracy is unaffected, but when the MNIST 6 Layer 1 Layer 2 Layer 3 Layer 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8 oComplexFeatures RM Simple Features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 (Acc) Decodability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 10 15 20 25 3035 40 510 15 20 25 3035 40 510 15 20 25 30 35 40 510 1520 25 30 35 40 0 Sieve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0 Feature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8 ComplexFeatures Simple Features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 10 15 202530 35 40 5 101520 25 30 35 5 10′152025 40 30 35 40 1015 20 25 30 35 40 0 0 0 5 EpochsOvercoming simplicity bias in deep networks using a feature sieve Table 1: Feature Controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' DataSet Target Feature SIFER (Ours) ERM SR CR SR CR CMNIST Complex (Digit) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='19 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='14±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='69 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='96±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='59 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='21±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='92 Simple (Color) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='44±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='22 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='64±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='30 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='20±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='60 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='99 CIFAR_MNIST Complex (CIFAR) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='37±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='62 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='93±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='92 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='14±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='60 100 Simple (Digit) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='14 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='29 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='20±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='60 100 component is randomized, accuracy drops to chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We refer to the MNIST component as the simple feature, and CIFAR as the complex feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Figure 3 shows the layerwise decodability1 of simple and complex features at different layers of the neural network, as tracked across epochs in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We contrast standard training (top row) against SIFER (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Standard training overemphasizes the simple feature at higher layers, and does not pay attention to the complex feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We note, particularly, that the complex CIFAR feature is in fact decodable to some extent in earlier layers of the ERM classifier, but is suppressed in later layers, due to the preponderance of the simple feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' In contrast, the auxiliary forgetting loss in SIFER effectively suppresses the simple feature in the earlier layers, and thereby enhances the decodability of the complex feature in the higher layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This shows that removing the availability of spurious simple features is a direct method of overcoming simplicity bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' What makes these findings more interesting is the fact that no knowledge of which features in particular are considered simple or complex was used in any way whatsoever in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' In other words, SIFER organically discovers and suppresses the by-design simple feature purely through the use of the strategically placed auxiliary network, and its configuration via the training recipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='2 Feature controllability using SIFER We described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='3 the various degrees of freedom in SIFER for identifying and suppressing complex features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Since these DOFs are optimized on a validation set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=', driven by generalization error), this gives us a simple method of customizing SIFER to the characteristics of the dataset, and in particular, the needs of the downstream task–e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=', generalization to related domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We demonstrate this capability by conducting studies on controlled datasets CMNIST & CIFAR_MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' In each, the training data were designed to have pairings of simple and complex features ((color,digit) and (image,digit) respectively) where both simple and complex features were fully predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We then trained a SIFER classifier for different choices of validation set, representing which feature we actually wanted the classifier to focus on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This was achieved by randomizing the “spurious” feature in the validation dataset, and choosing all our hyperparameters based on that validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This represents a real-world scenario where small amounts of vetted data are available for optimization of a model, but (re)-labeling or manipulating all the training data is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Table 1 shows the results, comparing SIFER against ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Our approach shows higher accuracy for chosen features (higher diagonal terms) than for spurious features (off-diagonal terms), driven by the choice of the validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' In contrast, ERM primarily focuses on the simple features, irrespective of the choice of validation set (higher second column numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Thus, our method is in fact able to focus on the relevant feature–be it simple or complex–in an easily controllable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='3 Debiasing real-world datasets Our method outperforms baselines on four different real world datasets–BAR [11], CelebA Hair [18], NICO [19] and Imagenet-A [21], by large margins (upto 11%, see Figure 2c for a quick summary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Critically, we chose in all our experiments to not use any knowledge of which attribute labels are considered spurious in each dataset–this is because in real-world scenarios, it is difficult to know in advance which attributes may end up containing biased information, or to label data according to those attributes in order to do targeted debiasing of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Nevertheless, we outperform the other baselines, including many that do use attribute labels as part of their training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 1Here, “decodability” is the accuracy of a linear decoder trained on the learned representations of that layer, to predict the specific feature of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 2Architecture design optimization based method, hence unfair to compare directly against other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 7 Overcoming simplicity bias in deep networks using a feature sieve Figure 4: Examples of SIFER’s focus on relevant features while suppressing irrelevant background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Top row: Input images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Middle row: GRAD-CAM-derived feature importance visualizations for the ERM classifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Bottom row: feature importance for SIFER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' First 3 columns from BAR [11], and last 3 columns from NICO-Animal [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Table 2: Classification Accuracy (%) on test set of BAR Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Method Used OOD Val Accuracy ERM \x13 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='85± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='92 BiaSwap [33] \x13 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='44 LfF [11] \x13 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='98± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='76 PGI [34] \x13 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='19± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='32 EIIL [35] \x13 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='44± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='17 ESB [16] \x13 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='10± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='30 Roadblock [10] \x13 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='51± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='43 Debian [14] \x13 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='88± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='92 SIFER (Ours) \x13 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='08± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='38 ERM \x17 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='32± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='46 ReBias [32] \x17 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='02± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='26 LfF [11] \x17 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='15± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='93 SSL+ERM [15] \x17 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='88± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='80 LWBC [15] \x17 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='03± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='74 ESB [16] \x17 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='40± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='20 SIFER (Ours) \x17 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='75± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='84 Mitigating Spurious Correlations: Biased Activity Recognition (BAR) and CelebA Hair Dataset represent back- ground and gender bias in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' In the BAR training set, human activity (image categories) is spuriously correlated with the background in which those activities are performed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' in CelebA Hair, hair color is strongly correlated with gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Both BAR and CelebA are heavily biased and contain no or very few conflicting examples–eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' CelebA Hair has only 1% of men with blond hair in the train set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' For BAR, since no validation set is provided, we show results both using in-distribution and out-distribution validation sets to compare against all the baselines that do and don’t require conflicting examples in the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Note that methods that work on the principle of reweighting conflicting samples in the trainset (eg LWBC [15]) typically 8 Overcoming simplicity bias in deep networks using a feature sieve Table 3: Unbiased and Conflicting Accuracy metrics (%) on Test set of CelebA Hair Dataset Method Spurious Attribs Unbiased Conflict DRO [25] \x13 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='43± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='53 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='40± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='67 EnD [36] \x13 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='21± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='22 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='45± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='06 CSAD [37] \x13 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='36 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='53 ERM \x17 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='25± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='35 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='52± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='19 LfF [11] \x17 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='24± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='37 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='24± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='38 SSL+ERM [15] \x17 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='48± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='91 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='79± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='20 LWBC [15] \x17 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='90± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='55 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='22± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='14 SIFER (Ours) \x17 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='00± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='92 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='04± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='25 Table 4: Classification Accuracy (%) on test set of NICO Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Most of the competititive baselines eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' DecAug, DRO, etc use spurious attribute labels for training, still we outperform all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Method Accuracy ERM 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='87 IRM [12] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='17 REx [27] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='31 JiGen [38] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='95 Mixup [39] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='27 Cumix [40] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='78 MTL [41] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='89 DANN [42] 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='59 CORAL [43] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='27 MMD [44] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='91 DRO [25] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='61 CNBB [19] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='16 DecAug [28] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='23 SIFER (Ours) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='20± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='85 NAS-OoD2 [17] 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='72 adds 1% of conflicting samples from the test set to the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We do not make these changes to the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We outperform baselines in both settings by more than 1-2% absolute accuracy, refer Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Interestingly, our method with only in-distribution validation data outperforms most baselines that leverage an additional OOD validation set, showing the superior generalization and robust feature learning capacity of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Table 3 shows results on CelebA dataset, we get almost the same unbiased accuracy as LWBC and improve upon conflicting accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Domain-shift Generalization: NICO introduces three new contexts in which object classes appear in the validation and test set, that are absent in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Table 4 shows classification accuracy on test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Our method outperforms all the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Unlike a majority of the baselines, we do not use spurious attribute labels (context labels) while training our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This shows the value of SIFER in domain shift generalization without knowledge of domain IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Robustness to Texture bias: Table 5 shows results on ImageNet-9, which is known to be biased towards texture, and ImageNet-A, which consists of natural images that have bias-conflicting features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This setting is closest to real-world scenarios for texture bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We improve the previous best baseline by 3% absolute on ImageNet-9 validation set and by 4% absolute on ImageNet-A test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Thus, SIFER encourages learning features robust to texture bias, improving performance on both the in-distribution validation set as well as bias-conflicting test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Two critical findings here are a) that SIFER did not sacrifice in-distribution accuracy through the process of sieving simple features, and b) the learned classifier robustly transfers over to a novel test set, where it provides even larger gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 9 Overcoming simplicity bias in deep networks using a feature sieve Table 5: Classification Accuracy (%) on Validation set of ImageNet-9 and test set of ImageNet-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Method Spurious Attribs ImageNet-9 ImageNet-A Accuracy Accuracy StylisedIN [3] \x13 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 LearnedMixin [45] \x13 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='1± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='0± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 RUBi [46] \x13 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='5± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='7± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='1 ERM \x17 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='9± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='1 BagNet18 [47] \x17 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='7± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='8± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='15 ReBias [32] \x17 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='9± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='6 LfF [11] \x17 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='00 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='60 CaaM [30] \x17 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='70 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='80 SSL+ERM [15] \x17 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='18± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='07 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='21± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='49 LWBC [15] \x17 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='03± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='23 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='97± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='49 SIFER \x17 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='78± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='12 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='98± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='81 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='4 SIFER focuses on relevant information We visualize the information in an image that is relevant to a given classifier [48], in order to verify whether our feature sieving results in semantically relevant modifications to learned classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Figure 4 shows this evaluation, contrasting ERM classifier’s regions of focus (middle row) and SIFER’s regions of focus (bottom row) on a range of input images (top row, drawn from BAR & NICO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Interestingly, not only does SIFER correctly focus on the central object of interest, but also it is able to effectively suppress the (spuriously label-correlated) background information, which is highly valued by the ERM classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This undercores SIFER’s ability to carefully differentiate between relevant and irrelevant features, rather than some notion of simple vs complex features alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 6 Discussion & conclusion We proposed SIFER–a novel feature sieve approach towards addressing simplicity bias and spurious correlations in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Our proposal introduces an auxiliary network attached to the deep network which alternately identifies and suppresses predictive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The approach is controllable through the use of configuration parameters optimized using validation data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' thus, it requires no foreknowledge or hand-coding of the notion of “simple features”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We demonstrated on controlled datasets the ability of SIFER to automatically identify and suppress features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' further, we showed that, strictly speaking, SIFER rebalances the role of various features in a controllable manner driven by the needs of generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We showed using extensive experiments on real-world data that our approach provides significant gains–3-11% relative accuracy improvements on BAR, NICO, and Imagenet-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We believe our work is a small, important first step in a fruitful new direction of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We hope that follow-on work will build on the notion of the feature sieve, developing effective computational barriers that encourage deep networks to discover and utilize richer, more powerful featural representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Our current approach strikes a balance between various competing features, guided by generalization error estimates (validation error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' One could potentially extract even more value if different feature classes could be isolated into (relatively) independent predictors, then combined effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' This is, for instance, the approach taken by Niu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Thus, a straightforward next step we aim to explore is the study of ensembling approaches to combine a range of features of varying complexity & predictive power, and methods for efficiently learning them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We also hope to develop a systematic theoretical understanding of feature sieve approaches and their role in supervised learning using DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' References [1] Harshay Shah, Kaustav Tamuly, Aditi Raghunathan, Prateek Jain, and Praneeth Netrapalli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The pitfalls of simplicity bias in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:9573–9585, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' [2] Mohammad Pezeshki, Oumar Kaba, Yoshua Bengio, Aaron C Courville, Doina Precup, and Guillaume Lajoie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Gradient starvation: A learning proclivity in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:1256–1272, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 10 Overcoming simplicity bias in deep networks using a feature sieve [3] Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A Wichmann, and Wieland Brendel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Imagenet-trained cnns are biased towards texture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' increasing shape bias improves accuracy and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' [4] Katherine Hermann and Andrew Lampinen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' What shapes feature representations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' exploring datasets, architectures, and training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:9995–10006, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' [5] Ninareh Mehrabi, Fred Morstatter, Nripsuta Saxena, Kristina Lerman, and Aram Galstyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' A survey on bias and fairness in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' ACM Computing Surveys (CSUR), 54(6):1–35, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' [6] Muhammad Bilal Zafar, Isabel Valera, Manuel Gomez Rodriguez, and Krishna P Gummadi.' 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using a feature sieve Appendix A Additional Details For all experiments we consistently used ResNet-18, an auxiliary layer that uses the same layer structure as of BasicBlock of ResNet with varying depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The ResNet network is composed of 4 layers modules (each itself is made up of 2 BasicBlocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We apply auxiliary layer only at the end of the of layers except layer 4, this gives us 3 different choice for aux position(AP ) which we treat as hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' The network is optimized using SGD optimizer with a fixed learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' For real-world experiments the model is loaded with ImageNet pre-trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' We repeat the experiments with 5 different random seeds and report the mean and std deviation of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Table 6 shows the hyperparamters search space for all the hyperparameters that we tune on the basis of validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Table 6: Range for hyperparameters search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Hparam Range AD [1, 9] AP [1, 3] α1 loguniform(10−2, 101) α2 loguniform(1, 103) α3 loguniform(1, 103) F [1, 9] ∗ 10 B Baselines Here we list and briefly explain all the baselines that we compare against on real world datasets: BiaSwap [33] proposes a bias-tailored augmentation-based approach for learning debiased representation without requiring supervision on the bias type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' they divide the data into bias-guiding and bias-conflicting groups and then swaps the bias in bias guiding group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' LfF [11] uses generalized cross-entropy initially trains a prejudiced net-work and tries to debias the second network by focusing weighing on samples that go against the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' IRM [12] uses theory of causal bayesian networks to find an invariant feature representation using multiple training environments with different bias correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' REx [27] proposed a min-max algorithm to optimize for the worst linear combination of risks on different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' EIIL [35] optimizes for bias group assignment to automatically identify the bias groups to maximize IRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' PGI [34] follows EIIL to identify bias groups by training a small neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Evading Simplicity Bias (ESB) [16] creates a ensemble of diverse classifiers by incorporating a diversity regularizer between the gradients while training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Roadblock [10] adds adversarial augmentations to the image while training to avoid over-reliance on spurious visual cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Debian [14] trains two networks in alternate manner namely discoverer and classifier, the discoverer tries to find multiple unknown biases of the classifier without any annotations of biases, and the classifier aims at unlearning the biases identified by the discoverer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' ReBias [32] propose a novel framework to train a de-biased representation by encouraging it to be different from a set of representations that are biased by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' LWBC [15] employs a committee of classifiers as an auxiliary module that identifies bias-conflicting data and assigns large weights to them when training the main classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Group-DRO [25] minimizes for worst-case training loss over a set of pre-defined groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' EnD [36] proposes a regularization technique that uses the bias attributes to prevent deep models from learning spurious biases by inserting an information bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' CSAD [37], given the bias attributes, explicitly extracts target and bias features disentangled from the latent representation generated by a feature extractor and then learns to discover and remove the correlation between the target and bias features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' JiGen [38] jointly classifies objects and solves unsupervised jigsaw tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' Cumix [40] mixes up data and labels from different domains to be able to recognize unseen categories in unseen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' MTL [41] argue that problem of Domain Generalization can be viewed as a kind of supervised learning problem by 14 Overcoming simplicity bias in deep networks using a feature sieve augmenting the original feature space with the marginal distribution of feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' DANN [42] proposes a representation learning approach such that features are not predictive of the domain from which the model is being trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' CORAL [43] proposes an unsupervised domain adaptation method that aligns the second-order statistics of the source and target distributions with a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' MMD [44] extend adversarial autoencoders by imposing the Maximum Mean Discrepancy measure to align the distributions among different domains, and matching the aligned distribution to an arbitrary prior distribution via adversarial feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' CNBB [19] is an OoD learning method that based on sample reweighting inspired by causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' DecAug [28] proposed a semantic augmentation and feature decomposition approach to distangle context features from category related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' NAS-OoD [17] adds an OOD generalization criterion to network architecture search training to construct inherently more robust network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' StylisedIN [3] showed that ImageNet is texture biased and works on improving shape bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' LearnedMixin [45] trains a robust model as part of an ensemble with the naive one in order to encourage it to focus on other patterns in the data that are more likely to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' CaaM [30] learns causal attention by partitioning the data on-the-go to break correlation with bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFQT4oBgHgl3EQfTjYq/content/2301.13293v1.pdf'} diff --git a/L9E4T4oBgHgl3EQf8g5c/content/tmp_files/2301.05348v1.pdf.txt b/L9E4T4oBgHgl3EQf8g5c/content/tmp_files/2301.05348v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..771a810fd7649e76fdcfbfa8715eb0e757a3ad03 --- /dev/null +++ b/L9E4T4oBgHgl3EQf8g5c/content/tmp_files/2301.05348v1.pdf.txt @@ -0,0 +1,3827 @@ +arXiv:2301.05348v1 [math.NT] 13 Jan 2023 +REPRESENTATIONS OF THE p-ADIC GSpin4 AND GSpin6 AND THE ADJOINT +L-FUNCTION +MAHDI ASGARI AND KWANGHO CHOIY +Abstract. We prove a conjecture of Gross-Prasad and Rallis about determination of generic L-packets in +terms of the analytic properties of the adjoint L-function for p-adic general even spin groups of semi-simple +ranks 2 and 3. +We also explicitly write the adjoint L-function for each L-packet in terms of the local +Langlands L-functions for the general linear groups. +1. Introduction +In this article, we provide further details on the local L-packets for the non-Archimedean split general spin +groups GSpin4 and GSpin6, following our earlier work [AC17]. We then use our explicit description of these +L-packets to prove a conjecture of Gross-Prasad and Rallis, determining which of the L-packets are “generic” +(i.e., contain an irreducible representation with a Whittaker model) in terms of the analytic properties of +the adjoint L-function of the packet. We also write the adjoint L-function for each L-packet in terms of the +local Langlands L-functions of the general linear groups. In addition to details about the representations +that our results provide, given that the adjoint L-functions have a significant role in the Gan-Gross-Prasad +conjectures, we expect that our results in this paper would be helpful in that direction as well. +Let F be a p-adic field of characteristic zero. Denote by WF the Weil group of F and let W ′ +F = WF ×SL2(C) +be the Weil-Deligne group of F. Let G be a connected, reductive, linear algebraic group over F. The local +Langlands Conjecture (LLC) predicts a surjective, finite-to-one map L from the set Irr(G) of equivalence +classes of irreducible, smooth, complex representations of G(F) to the set Φ(G) of �G-conjugacy classes of +L-parameters of G(F), i.e., admissible homomorphisms φ : W ′ +F −→ LG. Here, LG denotes the L-group of G +with �G = LG0 its connected component, i.e., the complex dual of G [Bor79]. Among other properties, the +map L is supposed to preserve the local L-, ǫ-, and γ-factors. Moreover, the (finite) fibers Πφ, for φ ∈ Φ(G), +of the map L are called the L-packets of G and their structures are expected to be controlled by certain +finite subgroups of �G. +Consider the split general spin groups G = GSpin4 and G = GSpin6, of type D2 = A1 × A2 and D3 = A3 +respectively, whose algebraic structure we review in Section 2.3. +We constructed most of the L-packets +for these two groups in [AC17] and proved that they satisfy the expected properties of preservation of the +local factors and their internal structure. We review and complete the construction of these L-packets. In +particular, using the classification of representations of GLn, we give more explicit descriptions of the L- +packets for GSpin4 and GSpin6 in terms of given representations of GL2 ×GL2 and GL4 ×GL1, respectively. +As a byproduct, we are able to give the criteria for determining the size of the L-packets for GSpin4 and +GSpin6 (see Sections 4 and 5). +The known cases of the LLC for the p-adic groups include GLn[HT01, Hen00, Sch13]; SLn [GK82]; +non-quasi-split F-inner forms of GLn and SLn [HS12, ABPS16]; GSp4 and Sp4 [GT11, GT10]; non-quasi- +split F-inner form GSp1,1 of GSp4 [GT14]; Sp2n, SOn, and quasi-split SO∗ +2n [Art13]; Un [Rog90, Mok15]; +non quasi-split F-inner forms of Un [Rog90, KMSW14]; non-quasi-split F-inner form Sp1,1 of Sp4 [Cho17]; +GSpin4, GSpin6 and their inner forms [AC17]; GSp2n and GO2n [Xu18]. +Going back to the case of general G, assume that ρ is a finite-dimensional complex representation of LG. +When LLC is known, one can define the local Langlands L-functions +L(s, π, ρ) = L(s, ρ ◦ φ) +for each π ∈ Πφ. Here, the L-factors on the right hand side are the Artin local factors associated to the +given representation of W ′ +F . +1 + +2 +MAHDI ASGARI AND KWANGHO CHOIY +B. Gross and D. Prasad, following a remark of S. Rallis, conjectured (in the generality of quasi-split +groups) that the local L-packet Πφ(G) is generic if and only if the adjoint L-function L(s, Ad ◦ φ) is regular +at s = 1 [GP92, Conj. 2.6]. Here, Ad denotes the adjoint representation of LG on the dual Lie algebra �g of +�G. (Note that in the body of this paper we use Ad exclusively for the restriction of the adjoint representation +to the derived group of �g to distinguish it from the full adjoint L-function, which would have an extra factor +of the L-function for the trivial character when �g has a one-dimensional center.) +We prove the above conjecture for the groups GSpin4 and GSpin6 as a consequence of our construction +of the L-packets for these groups. In fact, we prove the conjecture for a larger class of groups G = Gr,s +m,n, +which are given as subgroups of GLm × GLn satisfying a certain determinant equality (2.6). We are able to +work in the slightly larger generality because, as in the construction of the L-packets, we use the approach +of restricting representations from GLm(F) × GLn(F) to the subgroup G. +Moreover, we also give the adjoint L-function in all cases for G explicitly in terms of local Langlands L- +functions of the general linear groups. While we are able to prove the Gross-Prasad-Rallis conjecture already +without the explicit knowledge of the adjoint L-function, the explicit description of the adjoint L-function +certainly also verifies the conjecture and we include it here since it may lead to other number theoretic or +representation theoretic results. +Finally, we take this opportunity to correct a few inaccuracies in [AC17]. They do not affect the main +results in that paper and fix some errors in our description of the L-packets. The details are given in Section +6. +Acknowledgements. We are grateful to Behrang Noohi and Ralf Schmidt for helpful discussions. K. Choiy +was supported by a gift from the Simons Foundation (#840755). +2. Preliminaries +2.1. Local Langlands Correspondence (LLC). Let p be a prime number and let F be a p-adic field +of characteristic zero, i.e., a finite extension of Qp. We fix an algebraic closure ¯F of F. Denote the ring of +integers of F by OF and its unique maximal ideal by PF . Moreover, let q denote the cardinality of the +residue field OF /PF and fix a uniformizer ̟ with |̟|F = q−1. Also, let WF denote the Weil group of F, +W ′ +F the Weil-Deligne group of F, and Γ the absolute Galois group Gal( ¯F/F). Throughout the paper, we +will use the notation ν(·) = | · |F . +Let G be a connected, reductive, linear algebraic group over F. Fixing Γ-invariant splitting data we define +the L-group of G as a semi-direct product LG := �G ⋊ Γ, where �G = LG0 denotes the connected component +of the L-group of G, i.e., the complex dual of G (see [Bor79, §2]). +LLC (still conjectural in this generality) asserts that there is a surjective, finite-to-one map from the set +Irr(G) of isomorphism classes of irreducible smooth complex representations of G(F) to the set Φ(G) of +�G-conjugacy classes of L-parameters, i.e., admissible homomorphisms ϕ : W ′ +F −→ LG. +Given ϕ ∈ Φ(G), its fiber Πϕ(G), which is called an L-packet for G, is expected to be controlled by +a certain finite group living in the complex dual group �G. Furthermore, for π ∈ Πϕ(G) and ρ a finite +dimensional algebraic representation of LG one defines the local factors +L(s, π, ρ) += +L(s, ρ ◦ φ), +(2.1) +ǫ(s, π, ρ, ψ) += +ǫ(s, ρ ◦ φ, ψ), +(2.2) +γ(s, π, ρ, ψ) += +γ(s, ρ ◦ φ, ψ). +(2.3) +provided that LLC is known for the case in question. Here, the factors on the right are Artin factors. +2.2. The Adjoint L-Function. What we recall in this subsection holds for G quasi-split ([GP92, §2]). +However, for simplicity we will take G to be split over F since the groups we are working with in this +article are split. When G is split over F, we may replace the L-group LG by its connected component +�G = LG0. Take ρ to be the adjoint action of �G on its Lie algebra. Then we obtain the adjoint L-function +L(s, π, Ad � +G) = L(s, Ad � +G ◦ φ) for all π ∈ Πϕ(G). The following is a conjecture of D. Gross and D. Prasad, +suggested by a remark of S. Rallis (see [GP92, Conj. 2.6]). + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +3 +Conjecture 2.1. Πϕ(G) contains a generic member if and only if L(s, Ad � +G ◦ φ) is regular at s = 1. +(Equivalently, π is generic if and only if L(s, π, Ad � +G) is regular at s = 1.) +The conjecture is known in many cases in which the LLC is known. To mention a few, it was verified for +GLn by B. Gross and D. Prasad [GP92], for GSp4 in [GT11] and, for non-supercuspidals, in [AS08], and for +SO and Sp groups, it follows from the work of Arthur on endoscopic classification [Art13]. We will verify +this conjecture for the small rank split groups GSpin4 and GSpin6. +2.3. The Groups GSpin4 and GSpin6. We gave detailed information about the structure of these two +groups (as well as their inner forms) in [AC17, §2.2]. For now we just recall the incidental isomorphisms +GSpin4 +∼= +{(g1, g2) ∈ GL2 × GL2 : det g1 = det g2} +(2.4) +GSpin6 +∼= +� +(g1, g2) ∈ GL1 × GL4 : g2 +1 = det g2 +� +. +(2.5) +While our main interests in this article are the split general spin groups GSpin4 and GSpin6, for the +purposes of Conjecture 2.1 it is no more difficult, and perhaps also more natural, to consider a slightly more +general setup as follows. +Fix integers m, n ≥ 1 and r, s ≥ 1 and assume that gcd(r, s) = 1. Define +G = Gr,s +m,n := {(g, h) ∈ GLm × GLn | (det g)r = (det h)s} +(2.6) +Proposition 2.2. The group Gr,s +m,n is a split, connected, reductive, linear algebraic group over F. +Proof. Let X = (Xij) and Y = (Ykl) be m×m and n×n matrices, respectively. It is clear that Gr,s +m,n, being an +almost direct product of SLm×SLn and a torus, is reductive. The only issue that requires justification is that +the polynomial f(X, Y ) = (det X)r − (det Y )s is irreducible in F[Xij, Ykl] if and only if d = gcd(r, s) = 1. It +is clear that if d > 1, then f is reducible since it would be divisible by (det X)(r/d) − (det Y )(s/d). It remains +to show that if d = 1, then f(X, Y ) is irreducible. This assertion should be easy to see via elementary +arguments considering the polynomials in a possible factorization of f. However, we prove it below as a +special case of a more general fact. +Assume that f(x, y) is an (arbitrary) irreducible polynomial in F[x, y]. Let +p(x1, x2, . . . , xa) ∈ F[x1, x2, . . . , xa] +and +p(y1, y2, . . . , yb) ∈ F[y1, y2, . . . , yb] +be two polynomials such that p − α and q − α are irreducible for all constants α. Then, f(p, q) is irreducible +in F[x1, x2, . . . , xa, y1, y2, . . . , yb]. +Our Proposition would clearly follow from the above assertion since (det −α) is always an irreducible +polynomial and it is well-known that the two-variable polynomial xr − ys is irreducible in F[x, y] provided +that d = gcd(r, s) = 1. +To prove the assertion above, we proceed as follows. By base extension to an algebraic closure we may +assume, without loss of generality, that F is algebraically closed. +Let A be the subscheme of Spec F[x1, x2, . . . , xa, y1, y2, . . . , yb] defined by f(p, q), and let B be the sub- +scheme of Spec F[x, y] defined by xr − ys. The latter is irreducible since xr − ys is an irreducible polynomial +by our assumption that d = 1. There is a natural map A → B which has irreducible (geometric) fibers. The +result now follows from the following claim. +Claim: Let g : A → B be an open morphism of schemes of finite type over an algebraically closed field F +such that the (geometric) fibers of g are irreducible and B is irreducible. Then A is irreducible. +To see the claim let U be an open in A. We want to show that for any other open V , we have that U ∩ V +is nonempty. Since B is irreducible and g is open, we have that g(U) ∩ g(V ) is nonempty so there is a fiber +F0 of g such that F0 ∩ U and F0 ∩ V are nonempty. Hence, by irreducibility of F0, they have a nonempty +intersection in F0. In particular, U ∩ V is nonempty, which gives the claim. +It only remains to check that the map A → B above is open. In fact, it is flat since it is a base extension +of the cartesian product of two flat morphisms p : Spec F[x1, ..., xa] → Spec F[x] and q : Spec F[y1, ..., yb] → +Spec F[y]. (Here, we are using the fact that Spec F[x] is a curve.) This finishes the proof. +□ +Of particular interest to us in this paper are the cases +• m = n = 2 and r = s = 1, when G = GSpin4, and + +4 +MAHDI ASGARI AND KWANGHO CHOIY +• m = 1, n = 4 and r = 2, s = 1, when G = GSpin6. +The (connected) L-group of G is +LGr,s 0 +m,n = �G ∼= (GLm(C) × GLn(C))/{(z−rIm, zsIn) : z ∈ C×} +(2.7) +and we have the exact sequence +1 −→ {(z−rIm, zsIn) : z ∈ C×} ∼= C× −→ GLm(C) × GLn(C) +prr,s +m,n +−−−−→ � +Gr,s +m,n −→ 1. +(2.8) +2.4. Computation of the Adjoint L-Function for G. Let π be an irreducible admissible representation +of G(F). There exist irreducible admissible representations πm and πn of GLm(F) and GLn(F), respectively, +such that +π ֒→ ResGLm(F )×GLn(F ) +G(F ) +(πm ⊗ πn) . +(2.9) +Let Ad � +G denote the adjoint action of �G on its Lie algebra +�g = {(X, Y ) ∈ glm(C) × gln(C) | r tr(X) = s tr(Y )} . +(2.10) +In what follows, let us write +Ad � +G = triv ⊕Ad +(2.11) +and for i ∈ {m, n} we similarly write Adi = Ad� +GLi = triv ⊕Ad, where Ad here denotes the action of GLi(C) +on the space of traceless i × i complex matrices sli(C). +Let φπ : WF × SL2(C) → �G be the L-parameter of π and let φi : WF × SL2(C) → GLi(C), i = m, n, be +the L-parameter of πi. Recall by (2.8) that we have a natural map +pr = prr,s +m,n : GLm(C) × GLn(C) −→ �G. +(2.12) +Then we have +φπ = pr ◦ (φm ⊗ φn). +(2.13) +Since the subgroup {(z−rIm, zsIn) : z ∈ C×} is central in GLm(C)×GLn(C) the following diagram commutes. +GLm(C) × GLn(C) +AutC (glm(C) × gln(C)) +WF × SL2(C) +�G +AutC (�g) +Adm⊗Adn +pr +φm⊗φn +φπ +Ad � +G +Note that the adjoint action Adm of GLm(C) on glm(C) preserves the trace, and similarly for n, so we +obtain a right downward arrow by simply restricting any automorphism to the set of those pairs satisfying +the trace equality in (2.10). We have +L(s, 1F ×)L(s, π, Ad) · L(s, 1F ×) += +L(s, π, Ad � +G) · L(s, 1F ×) += +L(s, Ad � +G ◦ φπ) · L(s, 1F ×) += +L (s, (Adm ⊗ Adn) ◦ (φm ⊗ φn)) += +L(s, Adm ◦ φm)L(s, Adn ◦ φn) += +L(s, πm, Adm)L(s, πn, Adn) += +L(s, 1F ×)2L(s, πm, Ad)L(s, πn, Ad). +(2.14) +Therefore, we obtain the more convenient equality +L(s, π, Ad) = L(s, πm, Ad)L(s, πn, Ad), +(2.15) + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +5 +which holds thanks to our choice of the notation Ad. In Section 3.2 this relation helps verify Conjecture 2.1 +for the groups of interest to us. +3. Genericity and The Conjecture of Gross-Prasad and Rallis +3.1. Restriction of Generic Representations. Let us write □D for the group Hom(□, C×) of all contin- +uous characters on a topological group □. Dente by □der the derived group of □. Let G and �G be connected, +reductive, linear, algebraic groups over F satisfying the property that +Gder = �Gder ⊆ G ⊆ �G. +(3.1) +For any connected, reductive, linear, algebraic group □ over F, we write Irrsc(□) and Irresq(□) for the set of +equivalence classes of supercuspidal and essentially square-integrable representations of □(F), respectively. +Assume �G and G to be F-split. Let �B be a Borel subgroup of �G with Levi decomposition �B = �T �U. Then +B = �B ∩ G is a Borel subgroup of G with B = T U. Note that T = �T ∩ G and �U = U. Let ψ be a generic +character of U(F). From [Tad92, Proposition 2.8] we know that given a ψ-generic irreducible representation +�σ of �G(F) we have a unique ψ-generic σ of G(F) such that +σ ֒→ Res +� +G +G(�σ). +The generic character associated with σ is not unique though. +Proposition 3.1. Each generic character associated with σ is determined up to the action of �T(F)/T (F). +Proof. We let �σ ∈ Irr( �G) be ψ-generic. Then there is a unique ψ-generic σψ ∈ Π�σ(G). On the other hand, +for each σ ∈ Π�σ(G) there exists t ∈ �T(F)/T (F) ∼= �G/G(F) such that σ = tσψ, where tσψ(g) = σ(t−1gt). +This implies that σ is tψ-generic. Here tψ is defined as tψ(u) = ψ(t−1ut). +□ +Remark 3.2. We say σ ∈ Irr(G), resp. �σ ∈ Irr( �G), is generic if it is ψ-generic with respect to some generic +character ψ. With this notation, σ ∈ Irr(G) is generic if and only if is �σ ∈ Irr( �G). +3.2. Criterion for Genericity. In this section we verify Conjecture 2.1 for the small rank general spin +groups we are considering in this article. +Theorem 3.3. Let G = Gr,s +m,n be the group defined in (2.6). Let π be an irreducible admissible representation +of G(F). Then π is generic if and only if L(s, π, Ad) is regular at s = 1. +Proof. Given π there exist irreducible admissible representations πm of GLm(F) and πn of GLn(F) such +that π is a subrepresentation of the restriction to G(F) of πm ⊗ πn as in (2.9). Now, π is generic if and only +if both πm and πn are generic. By the truth of Conjecture 2.1 for the general linear groups, the latter is +equivalent to both L(s, πm, Ad) and L(s, πn, Ad) being regular at s = 1. Hence, by (2.15) and the fact that +neither of the L-functions can have a zero at s = 1, we have that π is generic if and only if L(s, π, Ad) is +regular at s = 1. This proves the theorem. +□ +As we observed in Section 2.3, the split groups GSpin4 and GSpin6 are special cases of Gr,s +m,n. Therefore, +we have the following. +Corollary 3.4. Conjecture 2.1 holds for the groups GSpin4 and GSpin6. +4. Representations of GSpin4 +In this section we list all the irreducible representations of GSpin4(F) and then calculate their associated +adjoint L-function explicitly. To this end, we give the nilpotent matrix associated to their parameter in each +case. +4.1. The Reprsentations. + +6 +MAHDI ASGARI AND KWANGHO CHOIY +4.1.1. Classification of representations of GSpin4. Following [AC17], we have +1 −→ GSpin4(F) −→ GL2(F) × GL2(F) −→ F × −→ 1. +(4.1) +Recall that +GSpin4(F) ∼= {(g1, g2) ∈ GL2(F) × GL2(F) : det g1 = det g2}, +(4.2) +LGSpin4 = � +GSpin4 = GSO4(C) ∼= (GL2(C) × GL2(C))/{(z−1, z) : z ∈ C×}, +(4.3) +and +1 −→ C× −→ GL2(C) × GL2(C) +pr4 +−→ � +GSpin4 −→ 1. +(4.4) +When convenient, we view GSO4 as the group similitude orthogonal 4 × 4 matrices with respect to the +anti-diagonal matrix +J = J4 = + + +0 +0 +0 +1 +0 +0 +1 +0 +0 +1 +0 +0 +1 +0 +0 +0 + + . +(4.5) +The Lie algebra of this group is also defined with respect to J and an element X in this Lie algebra satisfies +tXJ + JX = 0. +4.1.2. Construction of the L-packets of GSpin4 (recalled from [AC17]). Given σ ∈ Irr(GSpin4) we have a lift +�σ ∈ Irr(GL2 × GL2) such that +σ ֒→ ResGL2×GL2 +GSpin4 +(�σ). +It follows form the LLC for GLn [HT01, Hen00, Sch13] that there is a unique �ϕ�σ ∈ Φ(GL2 × GL2) corre- +sponding to the representation �σ. We now have a surjective, finite-to-one map +L4 : Irr(GSpin4) +−→ +Φ(GSpin4) +(4.6) +σ +�−→ +pr4 ◦ �ϕ�σ, +which does not depend on the choice of the lifting �σ. Then, for each ϕ ∈ Φ(GSpin4), all inequivalent +irreducible constituents of �σ constitutes the L-packet +Πϕ(GSpin4) := Π�σ(GSpin4) = +� +σ +��� σ ֒→ ResGL2×GL2 +GSpin4 +(�σ) +� � +∼= . +(4.7) +Here, �σ is the member in the singleton Π�ϕ(GL2 × GL2) and �ϕ ∈ Φ(GL2 × GL2) is such that pr4 ◦ �ϕ = ϕ. We +note that the construction does not depends on the choice of �ϕ, due to the LLC for GL2, [GK82, Lemma +2.4], [Tad92, Corollary 2.5], and [HS12, Lemma 2.2]. Further details may be found in [AC17, Section 5.1]. +4.1.3. The L-parameters of GL2. We recall the generic representations of GL2(F) in this paragraph. We +refer to [Wed08, Kud94, GR10] for details. Let χ : F × → C× denote a continuous quasi-character of F ×. +By Zelevinski ([Zel80, Theorem 9.7] or [Kud94, Theorem 2.3.1]) we know that the generic representations +of GL2 are: the supercuspidals, St ⊗ (χ ◦ det) where St denotes the Steinberg representation, and normally +induced representations iGL2 +GL1×GL1(χ1 ⊗ χ2) with χ1 ̸= χ2ν±1. The only non-generic representation is χ◦ det . +4.2. Generic Representations of GSpin4. Following [AC17, Section 5.3], given ϕ ∈ Φ(GSpin4), fix the +lift +�ϕ = �ϕ1 ⊗ �ϕ2 ∈ Φ(GL2 × GL2) +with �ϕi ∈ Φ(GL2) such that ϕ = pr4 ◦ �ϕ. Let +�σ = �σ1 ⊠ �σ2 ∈ Π�ϕ(GL2 × GL2) +be the unique member such that {�σi} = Π�ϕi(GL2). +Recall the notation +IGSpin4(�σ) := +� +χ ∈ (GL2(F) × GL2(F)/GSpin4(F))D ��� �σ ⊗ χ ∼= �σ +� +. +Then we have +Πϕ(GSpin4) +1−1 +←→ IGSpin4(�σ), +(4.8) + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +7 +and we recall that, by [AC17, Proposition 5.7], we have +IGSpin4(�σ) = +� ISL2(�σ1), +if �σ2 ∼= �σ1�η for some �η ∈ (F ×)D; +ISL2(�σ1) ∩ ISL2(�σ2), +if �σ2 ̸∼= �σ1�η for any �η ∈ (F ×)D. +(4.9) +4.2.1. Irreducible Parameters. Let ϕ ∈ Φ(GSpin4) be irreducible. Then �ϕ, �ϕ1, and �ϕ2 are all irreducible. +By Section 3.1, we have the following. +Proposition 4.1. Let ϕ ∈ Φ(GSpin6) be irreducible. Then every member in Πϕ(GSpin4) is supercuspidal +and generic. +To study the internal structure of Πϕ(GSpin4), by (4.8), we need to know the structure of IGSpin4(�σ), as +we now recall from [AC17]. +gnr-(a) When �σ2 ∼= �σ1�η for some �η ∈ (F ×)D, we have +IGSpin4(�σ) ∼= + + + +{1}, +if �ϕ1 (and hence also �ϕ2) is primitive or non-trivial on SL2(C); +Z/2Z, +if �ϕ1 (and hence also �ϕ2) is dihedral w.r.t. one quadratic extension; +(Z/2Z)2, +if �ϕ1 (and hence also �ϕ2) is dihedral w.r.t. three quadratic extensions. +gnr-(b) When �σ2 ̸∼= �σ1�η for any �η ∈ (F ×)D, then by (4.9) we have +IGSpin4(�σ) ∼= {1} or Z/2Z. +Since �σ2 ̸∼= �σ1�η for any �η ∈ (F ×)D, the case of both �ϕ1 and �ϕ2 being diredral w.r.t. three quadratic +extensions is excluded. Thus, we have the following list: +• If at least one of �ϕi is primitive, then IGSpin4(�σ) ∼= {1}. +• If both are dihedral, then IGSpin4(�σ) ∼= Z/2Z. +From [AC17, Proposition 2.1], we recall the identification +∆∨ = {β∨ +1 = f ∗ +11 − f ∗ +12, β∨ +2 = f ∗ +21 − f ∗ +22} , +(4.10) +using the notation fij and f ∗ +ij, 1 ≤ i, j ≤ 2, for the usual Z-basis of characters and cocharacters of GL2 ×GL2 +and β1, β2 denote the simple roots of GSpin4. We can use this identification to relate the nilpotent matrices +associated to the parameters of GL2 × GL2 and GSpin4, respectively. +For both (a) and (b) above, we have +NGL2(C)×GL2(C) = +��0 +0 +0 +0 +� +, +�0 +0 +0 +0 +�� +(4.10) +⇐⇒ NGSO4(C) = 04×4. +Remark 4.2. We note that case (b) above was mentioned, less precisely, in [AC17, Remark 5.10]. +4.2.2. Reducible Parameters. If ϕ ∈ Φ(GSpin4) is reducible, then at least one �ϕi must be reducible. Since +the number of irreducible constituents in ResGL2 +SL2 (�σi) is at most 2, we have ISL2(�σi) ∼= {1}, or Z/2Z. This +implies that +IGSpin4(�σ) ∼= {1}, or Z/2Z. +If �ϕi is reducible and generic, then �σi is either the Steinberg representation twisted by a character or +an irreducibly induced representation from the Borel subgroup of GL2. We make case-by-case arguments as +follows. +gnr-(i) Note that the Steinberg representation of GL2 × GL2 is of the form StGL2 ⊠ StGL2. We have +ResGL2×GL2 +GSpin4 +(StGL2 ⊠ StGL2) = StGSpin4 +(4.11) +and +ResGL2×GL2 +GSpin4 +(StGL2 ⊗ χ1 ⊠ StGL2 ⊗ χ2) = StGSpin4 ⊗ χ +for some χ. We have IGSpin4(�σ) ∼= {1} as IG(StG) ∼= {1}. Thus, by (4.9), the L-packet remains a +singleton and the restriction is irreducible. +• To determine χ, we use the required properties of χ1, χ2. Using +T = +���a +0 +0 +b +� +, +�c +0 +0 +d +�� ���� ab = cd +� +, +(4.12) +we have χ1(ab) = χ2(cd) ⇔ χ1 = χ2. Denote χ1 = χ2 by χ. + +8 +MAHDI ASGARI AND KWANGHO CHOIY +For (4.11), we have +NGL2(C)×GL2(C) = +��0 +1 +0 +0 +� +, +�0 +1 +0 +0 +�� +(4.10) +⇐⇒ NGSO4(C) = + + +0 +1 +1 +0 +0 +0 +0 +−1 +0 +0 +0 +−1 +0 +0 +0 +0 + + +gnr-(ii) Next we consider +ResGL2×GL2 +GSpin4 +� +iGL2 +GL1×GL1(χ1 ⊗ χ2) ⊠ StGL2 ⊗ χ +� +. +(4.13) +By (4.9), the fact that �σ2 ̸∼= �σ1�η for any �η ∈ (F ×)D, and since IG(StG) ∼= {1}, it follows that +IGSpin4(�σ) ∼= {1}. +Thus, the L-packet remains a singleton and the restriction (4.13) is irreducible. +• To describe the restriction (4.13), we proceed similarly as above. We have +χ1(a)χ2(b) = χ(cd) = χ(ab) ⇔ χ1χ−1(a) = χ−1 +2 χ(b) +Specializing to a = b and c = d in the center, we have +χ1χ2χ−2 = 1 +For (4.13) , we have +NGL2(C)×GL2(C) = +�� +0 +0 +0 +0 +� +, +� +0 +1 +0 +0 +�� +(4.10) +⇐⇒ NGSO4(C) = + + +0 +0 +1 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 + + . +gnr-(iii) We consider +ResGL2×GL2 +GSpin4 +� +iGL2 +GL1×GL1(χ1 ⊗ χ2) ⊠ iGL2 +GL1×GL1(χ3 ⊗ χ4) +� += iGSpin4 +T +� +χ1 ⊗ χ2, χ3 ⊗ χ1χ2χ−1 +3 +� +. +Here, χ1 ̸= χ2ν±1 and χ3 ̸= χ4ν±1. Note that by (4.9) this induced representation may be irre- +ducible or consist of two irreducible inequivalent constituents. We have +NGL2(C)×GL2(C) = +�� +0 +0 +0 +0 +� +, +� +0 +0 +0 +0 +�� +(4.10) +⇐⇒ NGSO4(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + . +gnr-(iv) Given a supercuspidal �σ ∈ Irr(GL2), we consider +ResGL2×GL2 +GSpin4 +(�σ ⊠ StGL2 ⊗ χ) . +(4.14) +Since IG(StG) ∼= {1}, due to (4.9), the restriction (4.14) is irreducible. We then have +NGL2(C)×GL2(C) = +��0 +0 +0 +0 +� +, +�0 +1 +0 +0 +�� +(4.10) +⇐⇒ NGSO4(C) = + + +0 +0 +1 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 + + . +gnr-(v) Given supercuspidal �σ ∈ Irr(GL2), we next consider +ResGL2×GL2 +GSpin4 +� +�σ ⊠ iGL2 +GL1×GL1(χ1 ⊗ χ2) +� +. +Note from (4.9) that this may be irreducible or consist of two irreducible inequivalent constituents. +We have +NGL2(C)×GL2(C) = +�� +0 +0 +0 +0 +� +, +� +0 +0 +0 +0 +�� +(4.10) +⇐⇒ NGSO4(C) = 04×4. + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +9 +4.3. Non-Generic Representations of GSpin4. If σ ∈ Irr(GSpin4) is non-generic, then σ is of the form +ResGL2×GL2 +GSpin4 +((χ ◦ det) ⊠ �σ) , +(4.15) +with �σ ∈ Irr(GL2). Note this restriction is irreducible due to (4.9), and that as χ ◦ det is non-generic, so is +the restriction σ for any �σ ∈ Irr(GL2). +For �σ = St ∈ Irr(GL2), we have +NGL2(C)×GL2(C) = +��0 +0 +0 +0 +� +, +�0 +1 +0 +0 +�� +(4.10) +⇐⇒ NGSO4(C) = + + +0 +0 +1 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 + + , +and otherwise we have +NGL2(C)×GL2(C) = +��0 +0 +0 +0 +� +, +�0 +0 +0 +0 +�� +(4.10) +⇐⇒ NGSO4(C) = 04×4. +We summarize the above information about the representations of GSpin4 in Table 1. +4.4. Computation of the Adjoint L-function for GSpin4. We now give explicit expressions for the +adjoint L-function for each of the representations of GSpin4(F). +We start by recalling that the adjoint +L-functions of the representations �σ ∈ Irr(GL2) are as follows. +L(s, �σ, Ad2) = + + + + + + + + + +L(s)2L(s, χ1χ−1 +2 )L(s, χ−1 +1 χ2), +if �σ = iGL2 +GL1×GL1(χ1 ⊠ χ2) with χ1χ−1 +2 +̸= ν±1; +L(s)L(s + 1), +if �σ = StGL2 ⊗ χ; +L(s)L(s, �σ, Sym2 ⊗ω−1 +�σ ), +if �σ is supercuspidal; +L(s)2L(s − 1)L(s + 1), +if �σ = χ ◦ det . +Here, L(s) = L(s, 1F ×). Recall our choice of notation +L(s, �σ, Ad2) = L(s)L(s, �σ, Ad). +Combining with (2.14), Sections 4.2.1 and 4.2.2, we have the following. +gnr-(a)&(b) Given a supercuspidal σ ∈ Irr(GSpin4), we recall that +σ ⊂ ResGL2×GL2 +GSpin4 +(�σ1 ⊠ �σ2) +for some supercuspidal �σ1 ⊠ �σ2 ∈ Irr(GL2 × GL2). By (2.15) we have +L(s, σ, Ad) = L(s, �σ1, Sym2 ⊗ω−1 +�σ1 )L(s, �σ2, Sym2 ⊗ω−1 +�σ2 ). +gnr-(i) Given +σ = StGSpin4 ⊗ χ ∈ Irr(GSpin4), +by (2.15) we have +L(s, σ, Ad) = L(s + 1)2. +gnr-(ii) Given σ ∈ Irr(GSpin4) such that +σ = ResGL2×GL2 +GSpin4 +� +iGL2 +GL1×GL1(χ1 ⊗ χ2) ⊠ StGL2 ⊗ χ +� +, +by (2.15) we have +L(s, σ, Ad) = L(s)L(s, χ1χ−1 +2 )L(s, χ−1 +1 χ2)L(s + 1). +gnr-(iii) Given σ ∈ Irr(GSpin4) such that +σ ⊂ ResGL2×GL2 +GSpin4 +� +iGL2 +GL1×GL1(χ1 ⊗ χ2) ⊠ iGL2 +GL1×GL1(χ3 ⊗ χ4) +� +by (2.15) we have +L(s, σ, Ad) = L(s)2L(s, χ1χ−1 +2 )L(s, χ−1 +1 χ2)L(s, χ3χ−1 +4 )L(s, χ−1 +3 χ4). + +10 +MAHDI ASGARI AND KWANGHO CHOIY +gnr-(iv) Given σ ∈ Irr(GSpin4) such that +σ = ResGL2×GL2 +GSpin4 +(�σ ⊠ StGL2 ⊗ χ) +by (2.15) we have +L(s, σ, Ad) = L(s, �σ2, Sym2 ⊗ω−1 +�σ2 )L(s + 1). +gnr-(v) Given σ ∈ Irr(GSpin4) such that +σ ⊂ ResGL2×GL2 +GSpin4 +� +�σ ⊠ iGL2 +GL1×GL1(χ1 ⊗ χ2) +� +by (2.15) we have +L(s, σ, Ad) = L(s)L(s, �σ2, Sym2 ⊗ω−1 +�σ2 )L(s, χ1χ−1 +2 )L(s, χ−1 +1 χ2). +nongnr Given a non-generic σ ∈ Irr(GSpin4), from (4.15), we recall that +σ = ResGL2×GL2 +GSpin4 +(χ ◦ det ⊠ �σ) +and by (2.15) we have +L(s, σ, Ad) = L(s)L(s − 1)L(s + 1)L(s, �σ, Ad). +We summarize the explicit computations above in Table 2. +5. Representations of GSpin6 +We now list all the representations of GSpin6(F) and then calculate their associated adjoint L-function +explicitly. Again, we do this explicit calculation by finding the 6 × 6 nilpotent matrix in the complex dual +group GSO6(C) in each case that is associated with the parameter of the representation. +5.1. The Represenations. +5.1.1. Classification of representations of GSpin6. Again, following [AC17], we have +1 −→ GSpin6(F) −→ GL1(F) × GL4(F) −→ F × −→ 1. +(5.1) +Recall that +GSpin6(F) ∼= +� +(g1, g2) ∈ GL1(F) × GL4(F) : g2 +1 = det g2 +� +, +(5.2) +LGSpin6 = � +GSpin6 = GSO6(C) ∼= (GL1(C) × GL4(C))/{(z−2, z) : z ∈ C×}, +(5.3) +and +1 −→ C× −→ GL1(C) × GL4(C) +pr6 +−→ � +GSpin6 −→ 1. +(5.4) +Just as the rank two case, here too we view GSO6 as the group similitude orthogonal 6 × 6 matrices with +respect to the analogous 6 × 6, anti-diagonal, matrix J = J6 as in (4.5), and similarly define its Lie algebra +with respect to J. +5.1.2. Construction of the L-packets of GSpin6 (recalled from [AC17]). Given σ ∈ Irr(GSpin6) we have a lift +�σ ∈ Irr(GL1 × GL4) such that +σ ֒→ ResGL1×GL4 +GSpin6 +(�σ). +It follows from the LLC for GLn [HT01, Hen00, Sch13] that there is a unique �ϕ�σ ∈ Φ(GL1 × GL4) corre- +sponding to the representation �σ. We now have a surjective, finite-to-one map +L6 : Irr(GSpin6) +−→ +Φ(GSpin6) +(5.5) +σ +�−→ +pr6 ◦ �ϕ�σ, +which does not depend on the choice of the lifting �σ. Then, for each ϕ ∈ Φ(GSpin6), all inequivalent +irreducible constituents of �σ constitutes the L-packet +Πϕ(GSpin6) := Π�σ(GSpin6) = +� +σ : σ ֒→ ResGL1×GL4 +GSpin6 +(�σ) +� � +∼=, +(5.6) +where �σ is the unique member of Π�ϕ(GL1 × GL4) and �ϕ ∈ Φ(GL1 × GL4) is such that pr6 ◦ �ϕ = ϕ. We note +that the construction does not depends on the choice of �ϕ. Further details may be found in [AC17, Section +6.1]. + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +11 +Following [AC17, Section 6.3], given ϕ ∈ Φ(GSpin6), fix the lift +�ϕ = �η ⊗ �ϕ0 ∈ Φ(GL1 × GL4) +with �ϕ0 ∈ Φ(GL4) such that ϕ = pr6 ◦ �ϕ. Let +�σ = �η ⊠ �σ0 ∈ Π�ϕ(GL1 × GL4) +be the unique member such that {�σ0} = Π�ϕ0(GL4). +Recall that +IGSpin6(�σ) := +� +�χ ∈ +� +GL1(F) × GL4(F)/GSpin6(F) +�D +: �σ ⊗ �χ ∼= �σ +� +. +Then we have +Πϕ(GSpin6) +1−1 +←→ IGSpin6(�σ), +(5.7) +and by [AC17, Lemma 6.5 and Proposition 6.6] we have +IGSpin6(�σ) ∼= {�χ ∈ ISL4(�σ0) : �χ2 = 1F ×} +(5.8) +and any �χ ∈ IGSpin6(�σ) is of the form +�χ = (�χ′)−2 ⊠ �χ′, +for some �χ′ ∈ (F ×)D. +5.2. Generic Representations of GSpin6. Thanks to the group structure (5.2) and the relation of generic +representations in Section 3.1, in order to classify the generic representations of GSpin6, it suffices to classify +the generic representations of GL4. +Here are two key facts from the GL theory. +• Recall from [Zel80, Theorem 9.7] and [Kud94, Theorem 2.3.1] that a generic representation of GL4 +is of the form +iGL4 +M♭ (σ♭) +where M♭ runs through any F-Levi subgroup of GL4 (including GL4 itself) and σ♭ is any essentially +square-integrable representation of M♭. +• For their L-parameters, we note from [Kud94, §5.2] that the generic representations of GL4 have +Langlands parameters (i.e., 4-dimensional Weil-Deligne representations (ρ, N)) of the form +(ρ1 ⊗ sp(r1)) ⊗ .. ⊗ (ρt ⊗ sp(rt)) +with t ≤ 4, where ρi’s are irreducible and no two segments are linked. +5.2.1. Irreducible Parameters. Let ϕ ∈ Φ(GSpin6) be irreducible. Then �ϕ and �ϕ0 are also irreducible. By +Section 3.1, we have the following. +Proposition 5.1. Let ϕ ∈ Φ(GSpin6) be irreducible. Every member in Πϕ(GSpin6) is supercuspidal and +generic. +To see the internal structure of Πϕ(GSpin6), we need, by (5.7), to know the detailed structure of IGSpin6(�σ) +as follows. +gnr-(a) Given σ ∈ Irrsc(GSpin6), we have +�σ = �σ0 ⊠ �η ∈ Irrsc(GL4 × GL1). +(5.9) +From [AC17, Proposition 2.1], we recall the identification: +∆∨ = {β∨ +1 = f ∗ +2 − f ∗ +3 , β∨ +2 = f ∗ +1 − f ∗ +2 , β∨ +3 = f ∗ +3 − f ∗ +4 } . +(5.10) +using the notation fij and f ∗ +ij, 1 ≤ i, j ≤ 4, for the usual Z-basis of characters and cocharacters of +GL4. Also, {β1, β2, β3} are the simple roots of GSpin6. +We have +NGL4(C)×GL1(C) = (04×4, 0) +(5.10) +⇐⇒ NGSO6(C) = 06×6. + +12 +MAHDI ASGARI AND KWANGHO CHOIY +5.2.2. Reducible Parameters. When �ϕ0 is not irreducible, we have proper parabolic inductions. An exhaus- +tive list of F-Levi subgroups M of GSpin6 (up to isomorphism) is as follows. +• M ∼= GL1 × GL1 × GL1 × GL1 = � +M ∩ GSpin6, where � +M = (GL1 × GL1 × GL1 × GL1) × GL1. +• M ∼= GL2 × GL1 × GL1 = � +M ∩ GSpin6, where � +M = (GL2 × GL1 × GL1) × GL1. +• M ∼= GL3 × GL1 = � +M ∩ GSpin6, where � +M = (GL3 × GL1) × GL1. (Note: The factor GL1 of M is +GSpin0 by convention.) +• M ∼= GL1 × GSpin4 = � +M ∩ GSpin6, where � +M = (GL2 × GL2) × GL1. +• M ∼= GSpin6 = � +M ∩ GSpin6, where � +M = GL4 × GL1. +(Note that M ∼= GL2 × GL2 does not occur on this list.) We now consider each case and, by abuse of +notation, conflate algebraic groups and their F-points. +gnr-(I) M ∼= GL1 × GL1 × GL1 × GL1 and � +M = (GL1 × GL1 × GL1 × GL1) × GL1. +Given χi ∈ (F ×)D we consider +iGSpin6 +M +(χ1 ⊠ χ2 ⊠ χ3 ⊠ χ4). +(5.11) +Write χ1 ⊠ χ2 ⊠ χ3 ⊠ χ4 = (�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η)|M with �χi, �η ∈ (F ×)D so that +�χ1�χ2�χ3�χ4 = �η2. +Then we have the following relations +χ1 = �χ1, χ2 = �χ2, χ3 = �χ3, χ4 = �η2(�χ2�χ3�χ4)−1. +(5.12) +By Section 3.1, we know that the representation (5.11) is generic if and only if its lift +iGL4×GL1 +� +M +(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η) +(5.13) +is generic if and only if +iGL4 +GL1×GL1×GL1×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4) +(5.14) +is generic. By the classification of the generic representations of GLn ([Zel80, Theorem 9.7] and +[Kud94, Theorem 2.3.1]), this amounts to (5.14) being irreducible. By [Kud94, Theorem 2.1.1] +and [BZ77, Zel80], the necessary and sufficient condition for this to occur is that there is no pair +i, j with i ̸= j such that +�χi = ν �χj. +We have +NGL4(C)×GL1(C) = (04×4, 0) +(5.10) +⇐⇒ NGSO6(C) = 06×6 +gnr-(II) M ∼= GL2 × GL1 × GL1 and � +M = (GL2 × GL1 × GL1) × GL1. +Given σ0 ∈ Irresq(GL2) and χ1, χ2 ∈ (F ×)D, we consider +iGSpin6 +M +(σ0 ⊠ χ1 ⊠ χ2). +(5.15) +Write σ0 ⊠ χ1 ⊠ χ2 = (�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η)|M with �σ0 ∈ Irresq(GL2), �χi, �η ∈ (F ×)D. +Given (g, h1, h2, h3) ∈ � +M with det(gh1h2) = h2 +3, +• if we set (g, h1, h3) ∈ M, we have +�σ0(g)�χ1(h1)�χ2(h2)�η(h3) += +�σ0(g)�χ1(h1)�χ2(det g−1h−1 +1 h2 +3)�η(h3) += +(�σ0 �χ−1 +2 +◦ det)(g)(�χ1 �χ−1 +2 )(h1)(�χ2 +2�η)(h3) += +σ(g)χ1(h1)χ2(h3). +Then we have +�σ0 = σ0�χ2, �χ1 = χ1�χ2, �η = χ2�χ−2 +2 . + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +13 +• If we set (g, h2, h3) ∈ M, we have +�σ0(g)�χ1(h1)�χ2(h2)�η(h3) += +�σ0(g)�χ1(det g−1h−1 +2 h2 +3)�χ2(h2)�η(h3) += +(�σ0 �χ−1 +1 +◦ det)(g)(�χ2 �χ−1 +1 )(h2)(�χ2 +1�η)(h3) += +σ(g)χ1(h2)χ2(h3). +Then we have +�σ0 = σ0�χ1, �χ2 = χ2�χ1, �η = χ1�χ−2 +1 . +(5.16) +As before, the representation (5.15) is generic if and only if its lift +iGL4×GL1 +� +M +(�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η) +(5.17) +is generic if and only if +iGL4 +GL2×GL1×GL1(�σ0 ⊠ �χ1 ⊠ �χ2) +(5.18) +is generic. Again by the classification of the generic representations of GLn this amounts to (5.18) +being irreducible. Hence, we must have +�χ1 ̸= ν±1�χ2. +In other words, given (g, h1, h2, h3) ∈ � +M with det(gh1h2) = h2 +3, +• if we set (g, h1, h3) ∈ M, then +χ1 ̸= ν±1; +• if we set (g, h2, h3) ∈ M, then +χ2 ̸= ν±1. +We have the following two cases. If σ0 is supercuspidal, then +NGL4(C)×GL1(C) = (04×4, 0) +(5.10) +⇐⇒ NGSO6(C) = 06×6. +If σ0 is non-supercuspidal, then +NGL4(C)×GL1(C) = + + + + + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +. +gnr-(III) M ∼= GL3 × GL1 and � +M = (GL3 × GL1) × GL1. +Given σ0 ∈ Irresq(GL3) and χ ∈ (F ×)D, we consider +iGSpin6 +M +(σ0 ⊠ χ). +(5.19) +Write σ0 ⊠ χ = (�σ0 ⊠ �χ ⊠ �η)|M with �σ0 ∈ Irresq(GL3), �χ, �η ∈ (F ×)D. +Given (g, h1, h2) ∈ � +M with det(gh1) = h2 +2, if we set (g, h2) ∈ M, then we have +�σ0(g)�χ(h1)�η(h2) += +�σ0(g)�χ(det g−1h2 +2)�η(h2) +(5.20) += +(�σ0�χ−1 ◦ det)(g)(�χ2�η)(h2) += +σ(g)χ(h2). +Then, we have +�σ0 = σ0�χ +and +�η = χ2�χ−2. +As before, (5.19) is generic if and only if its lift +iGL4×GL1 +� +M +(�σ0 ⊠ �χ ⊠ �η) +(5.21) +is generic if and only if +iGL4 +GL3×GL1(�σ0 ⊠ �χ) +(5.22) + +14 +MAHDI ASGARI AND KWANGHO CHOIY +is generic. This amounts to (5.22) being irreducible as before, which is always true since �σ0 is +an essentially square integrable representation of GL3. Note that by the classification of essen- +tially square-integrable representations of GL3 ([Kud94, Proposition 1.1.2]), �σ0 must be either +supercuspidal or the unique subrepresentation of +iGL3 +GL1×GL1×GL1 +� +νχ ⊠ χ ⊠ ν−1χ +� +(5.23) +with any χ ∈ (F ×)D. +We have the following two cases. If σ0 is supercuspidal, then +NGL4(C)×GL1(C) = (04×4, 0) +(5.10) +⇐⇒ NGSO6(C) = 06×6. +If σ0 is the non-supercuspidal, unique, subrepresentation of (5.23), then +NGL4(C)×GL1(C) = + + + + + + +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +1 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 + + +. +gnr-(IV) M ∼= GL1 × GSpin4 and � +M = (GL2 × GL2) × GL1. +Given σ0 ∈ Irresq(GSpin4) and χ ∈ (F ×)D we consider +iGSpin6 +M +(χ ⊠ σ0). +(5.24) +Write χ ⊠ σ0 ⊂ (�σ1 ⊠ �σ2 ⊠ �η)|M with �σi ∈ Irresq(GL2), �η ∈ (F ×)D. +As before, (5.24) is generic if and only if its lift +iGL4×GL1 +� +M +(�σ1 ⊠ �σ2 ⊠ �η) +(5.25) +is generic if and only if +iGL4 +GL2×GL2(�σ1 ⊠ �σ2) +(5.26) +is generic. This amounts to (5.26) being irreducible. Thus, we must have +�σ1 ̸= ν±1�σ2. +We have several cases to consider. If σ0 is supercuspidal (so are �σ1 and �σ2), then +NGL4(C)×GL1(C) = (04×40) +(5.10) +⇐⇒ NGSO6(C) = 06×6. +If σ0 is non-supercuspidal, then for supercuspidal �σ1 and non-supercuspidal �σ2 we have +NGL4(C)×GL1(C) = + + + + + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +; +for non-supercuspidal �σ1 and supercuspidal �σ2 we have +NGL4(C)×GL1(C) = + + + + + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +; + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +15 +and for non-supercuspidal �σ1 and �σ2 we have +NGL4(C)×GL1(C) = + + + + + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +. +gnr-(V) M ∼= GSpin6 and � +M = GL4 × GL1. +Given σ ∈ Irresq(GSpin6) \ Irrsc(GSpin6), we consider +σ ⊂ (�σ ⊠ �η)|M +with �σ ∈ Irresq(GL4)\Irrsc(GL4), �η ∈ (F ×)D. Here, we note that ϕ ∈ Φ(GSpin6) is not irreducible +and neither �σ nor σ is supercuspidal. It is clear that σ is generic as �σ ⊠ �η is. By the classification +of essentially square-integrable representations of GL4 ([Kud94, Proposition 1.1.2]), �σ must be the +unique subrepresentation of either +iGL4 +GL1×GL1×GL1×GL1 +� +ν3/2 �χ ⊠ ν1/2�χ ⊠ ν−1/2�χ ⊠ ν−3/2�χ +� +(5.27) +with any �χ ∈ (F ×)D (i.e., �σ = StGL4 ⊗ �χ), or of +iGL4 +GL2×GL2 +� +ν1/2�τ ⊠ ν−1/2�τ +� +(5.28) +with any �τ ∈ Irrsc(GL2). +Now, for (5.27) we have +NGL4(C)×GL1(C) = + + + + + + +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +1 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 + + +; +and for (5.28) we have +NGL4(C)×GL1(C) = + + + + + + +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +. +(We note, cf. [Tat79, (4.1.5)], that NGL4(C) is of the form O2×2 ⊗ I2×2 + +� +0 +1 +0 +0 +� +⊗ I2×2.) +5.3. Non-Generic Representaions of GSpin6. Using the transitivity of the parabolic induction and the +classification of generic representations of GLn, ([Zel80, Theorem 9.7] and [Kud94, Theorem 2.3.1]), the +non-generic representations of GSpin6 are as follows. +nongnr-(A) M ∼= GL1 × GL1 × GL1 × GL1 and � +M = (GL1 × GL1 × GL1 × GL1) × GL1. +Given χi ∈ (F ×)D, by Section 3.1 and using (5.12), the representation (5.11) contains a +non-generic constituent if and only if the same is true for +iGL4×GL1 +� +M +(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η) +(5.29) +if and only if +iGL4 +GL1×GL1×GL1×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4) +(5.30) + +16 +MAHDI ASGARI AND KWANGHO CHOIY +contains a non-generic constituent. +This amounts to (5.30) being reducible. +As before, the +necessary and sufficient condition for this to occur is that there is some pair i, j with i ̸= j such +that �χi = ν �χj. +By the Langlands classification and the description of constituents of the parabolic induction +(see [Zel80, Theorem 7.1], [Rod82, Theorem 7.1], and [Kud94, Theorems 2.1.1 §5.1.1]), each +constituent can be described as a Langlands quotient, denoted by Q(...), as follows. +The first case is when there is only one pair, say �χ1 = ν1/2�χ and �χ2 = ν−1/2�χ for some +�χ ∈ (F ×)D while �χ3 ̸= ν±1�χj for j ̸= 3 and �χ4 ̸= ν±1�χj for j ̸= 4. Then we have the non-generic +constituent +Q +� +[ν1/2�χ], [ν−1/2�χ], [�χ3], [�χ4] +� +, +(5.31) +which is the Langlands quotient of +iGL4 +GL2×GL1×GL1 +� +Q +� +[ν1/2�χ], [ν−1/2�χ] +� +⊠ �χ3 ⊠ �χ4 +� += iGL4 +GL2×GL1×GL1 ((�χ ◦ det) ⊠ �χ3 ⊠ �χ4) . +We have +NGL4(C)×GL1(C) = (04×4, 0) +(5.10) +⇐⇒ NGSO6(C) = 06×6. +Note that the other constituent of this induced representation, which is generic, is +Q +� +[ν−1/2�χ, ν1/2�χ], [�χ3], [�χ4] +� += +iGL4 +GL2×GL1×GL1 +� +Q +� +[ν−1/2 �χ, ν1/2�χ] +� +⊠ �χ3 ⊠ �χ4 +� += +iGL4 +GL2×GL1×GL1 ((St ⊗ �χ) ⊠ �χ3 ⊠ �χ4) . +The next case is when there are two pairs, say �χ1 = ν �χ, �χ2 = �χ, and �χ3 = ν−1�χ for some +�χ ∈ (F ×)D and �χ4 ̸= ν±1�χi for i = 1, 2, 3. +Then we have the following three non-generic +constituents: +Q +� +[ν �χ], [�χ], [ν−1�χ], [�χ4] +� += +iGL4 +GL3×GL1((�χ ◦ det) ⊠ �χ3 ⊠ �χ4); +(5.32) +Q +� +[�χ, ν �χ], [ν−1�χ], [�χ4] +� +; +(5.33) +Q +� +[ν �χ], [�χ, ν−1�χ], [�χ4] +� +. +(5.34) +For (5.32) we have +NGL4(C)×GL1(C) = (04×4, 0) +(5.10) +⇐⇒ NGSO6(C) = 06×6, +for (5.33) we have +NGL4(C)×GL1(C) = + + + + + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +, +and for (5.34) we have +NGL4(C)×GL1(C) = + + + + + + +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 + + +. +Finally, in the case where we have three pairs we are in the situation of (5.27). Then we have +the following seven non-generic constituents: +Q +� +[ν3/2�χ], [ν1/2�χ], [ν−1/2�χ], [ν−3/2�χ] +� += �χ ◦ det; +(5.35) +Q +� +[ν1/2�χ, ν3/2�χ], [ν−1/2�χ], [ν−3/2�χ] +� +; +(5.36) + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +17 +Q +� +[ν3/2�χ], [ν−1/2�χ, ν1/2�χ], [ν−3/2�χ] +� +; +(5.37) +Q +� +[ν3/2�χ], [ν1/2�χ], [ν−3/2 �χ, ν−1/2�χ] +� +; +(5.38) +Q +� +[ν1/2 �χ, ν3/2�χ], [ν−3/2 �χ, ν−1/2�χ] +� +; +(5.39) +Q +� +[ν−1/2�χ, ν1/2�χ, ν3/2�χ], [ν−3/2�χ] +� +; +(5.40) +Q +� +[ν3/2 �χ], [ν−3/2�χ, ν−1/2�χ, ν1/2�χ] +� +. +(5.41) +For (5.35) we have +NGL4(C)×GL1(C) = (04×4, 0) +(5.10) +⇐⇒ NGSO6(C) = 06×6, +for (5.36) we have +NGL4(C)×GL1(C) = + + + + + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +, +for (5.37) we have +NGL4(C)×GL1(C) = + + + + + + +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 + + +, +for (5.38) we have +NGL4(C)×GL1(C) = + + + + + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +, +for (5.39) we have +NGL4(C)×GL1(C) = + + + + + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +, +for (5.40) we have +NGL4(C)×GL1(C) = + + + + + + +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +1 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 + + +, + +18 +MAHDI ASGARI AND KWANGHO CHOIY +and for (5.41) we have +NGL4(C)×GL1(C) = + + + + + + +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 + + +. +nongnr-(B) M ∼= GL2 × GL1 × GL1 and � +M = (GL2 × GL1 × GL1) × GL1. +Given σ0 ∈ Irr(GL2) and χ1, χ2 ∈ (F ×)D, we consider +iGSpin6 +M +(σ0 ⊠ χ1 ⊠ χ2). +(5.42) +Write +σ0 ⊠ χ1 ⊠ χ2 = (�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η)|M +with �σ0 ∈ Irr(GL2) and �χi, �η ∈ (F ×)D. By (5.16), it follows that (5.42) contains a non-generic +constituent if and only if its lift +iGL4×GL1 +� +M +(�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η) +(5.43) +contains a non-generic constituent if and only if +iGL4 +GL2×GL1×GL1(�σ0 ⊠ �χ1 ⊠ �χ2) +(5.44) +does. Recalling nongnr-(A), it is sufficient to consider the case of �σ0 ∈ Irr(GL2), �χ1 = ν1/2 �χ, +and �χ2 = ν−1/2�χ for �χ ∈ (F ×)D, where the segment ∆�σ0 of �σ0 does not precede either �χ1 or +�χ2. We then have the following sole non-generic constituent: +Q([∆�σ0], [ν1/2�χ], [ν−1/2 �χ]). +(5.45) +We have +NGL4(C)×GL1(C) = (04×4, 0) +(5.10) +⇐⇒ NGSO6(C) = 06×6. +nongnr-(C) M ∼= GL3 × GL1 and � +M = (GL3 × GL1) × GL1. +Given a non-generic σ0 ∈ Irr(GL3) and any χ ∈ (F ×)D, we consider +iGSpin6 +M +(σ0 ⊠ χ). +(5.46) +Write +σ0 ⊠ χ = (�σ0 ⊠ �χ ⊠ �η)|M +with non-generic �σ0 ∈ Irr(GL3) and �χ, �η ∈ (F ×)D. As in (5.20) we have +�σ0 = σ0�χ, +and +�η = χ2�χ−2. +As before, (5.46) contains a non-generic constituent if and only if its lift +iGL4×GL1 +� +M +(�σ0 ⊠ �χ ⊠ �η) +(5.47) +also contains one if and only if +iGL4 +GL3×GL1(�σ0 ⊠ �χ) +(5.48) +does. To have a non-generic �σ0 of GL3(F), the irreducible representation �σ0 must be some +constituent in a reducible induction. This case has been covered in nongnr-(A) and (B) above. +nongnr-(D) M ∼= GL1 × GSpin4 and � +M = (GL2 × GL2) × GL1. +Given a non-generic σ0 ∈ Irr(GSpin4), by Section 4.3, we know that it must be of the form +ResGL2×GL2 +GSpin4 +((χ ◦ det) ⊠ �σ) +for �σ ∈ Irr(GL2). For η ∈ (F ×)D, the induced representation +iGSpin6 +M +((χ ◦ det) ⊠ �σ ⊠ η) +(5.49) + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +19 +contains a non-generic constituent if and only if so does +iGL4 +GL2×GL2((χ ◦ det) ⊠ �σ), +which is always the case. Therefore, if �σ is supercuspidal, then +NGL4(C)×GL1(C) = (04×4, 0) +(5.10) +⇐⇒ NGSO6(C) = 06×6. +If �σ is non-supercuspidal, then it suffices to consider the case �σ = StGL2 ⊗ η with η ∈ (F ×)D +since the other case has been covered in nongnr-(A). Thus, we have +NGL4(C)×GL1(C) = + + + + + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + , 0 + + + + +(5.10) +⇐⇒ NGSO6(C) = + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + +. +nongnr-(E) M ∼= GSpin6 and � +M = GL4 × GL1. +Given a non-generic σ ∈ Irr(GSpin6), it must be of the form +ResGL4×GL1 +GSpin6 +(�χ ◦ det ⊠�η) = χ ◦ det, +(5.50) +for some �χ, �η ∈ (F ×)D. This is the case Q([ν3/2�χ], [ν1/2�χ], [ν−1/2�χ], [ν−3/2�χ]) in nongnr-(A). +5.4. Computation of the Adjoint L-function for GSpin6. We now give explicit expressions for the +adjoint L-function of each of the representations of GSpin6(F). Recall that if we have a parameter (φ, N) +with N a nilpotent matrix on the vector space V , then its adjoint L-function is +L(s, φ, Ad) = det +� +1 − q−sAd(φ)|V I +N +�−1 , +where VN = ker(N), V I the vectors fixed by the inertia group, and V I +N = V I ∩ VN. Below for the cases +where N is non-zero, we write ker(Ad(N)) and we use Lα to denote the root group associated with the root +α. +We now consider each case. Using (2.14) and Sections 5.2, and 5.3, we have the following. +gnr-(a) Given σ ∈ Irrsc(GSpin6), we have �σ = �σ0 ⊠ �η ∈ Irrsc(GL4 × GL1). Then +L(s, 1F ×)L(s, σ, Ad) = L(s, �σ0, Ad� +GL4) +or +L(s, σ, Ad) = L(s, �σ0, Ad). +gnr-(I) Given M ∼= GL1 × GL1 × GL1 × GL1 and � +M = (GL1 × GL1 × GL1 × GL1) × GL1, we recall +iGL4 +GL1×GL1×GL1×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4) +must be irreducible. Thus, given σ ∈ Irr(GSpin6) such that +σ = iGSpin6 +M +(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4), +we have +L(s, σ, Ad) = L(s)3 � +i̸=j +L(s, �χi�χ−1 +j ). +gnr-(II) Given M ∼= GL2 × GL1 × GL1 and � +M = (GL2 × GL1 × GL1) × GL1, for σ0 ∈ Irresq(GL2) and +χ1, χ2 ∈ (F ×)D, we have an irreducible induced representation +σ = iGSpin6 +M +(σ0 ⊠ χ1 ⊠ χ2) = ResGL4×GL1 +GSpin6 +� +iGL4 +GL2×GL1×GL1(�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η) +� +, +for some �σ0 ∈ Irresq(GL2), and �χi, �η ∈ (F ×)D. For supercuspidal �σ0 we have +L(s, σ, Ad) += +L(s)2L(s, �σ0, Ad)L(s, �σ0 × �χ−1 +1 )L(s, �σ∨ +0 × �χ1) +L(s, �σ0 × �χ−1 +2 )L(s, �σ∨ +0 × �χ2)L(s, �χ1�χ−1 +2 )L(s, �χ2�χ−1 +1 ). + +20 +MAHDI ASGARI AND KWANGHO CHOIY +For non-supercuspidal �σ0 ∈ Irr(GL2), i.e., σ0 = StGL2 ⊗ �χ for some �χ ∈ (F ×)D, we have +ker + + + +ad + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + + + + + = +� + + +a +0 +0 +0 +0 +a +0 +0 +0 +0 +b +0 +0 +0 +0 +c + + , Lf1−f2, Lf1−f3, Lf1−f4, Lf3−f2, Lf3−f4, Lf4−f2, Lf4−f3 +� +. +(5.51) +It follows that +L(s, σ, Ad) += +L(s)2L(s + 1)L(s + 1, �χ�χ−1 +1 )L(s + 1, �χ�χ−1 +2 ) +·L(s, �χ−1�χ1)L(s, �χ−1�χ2)L(s, �χ1�χ−1 +2 )L(s, �χ2�χ−1 +1 ). +gnr-(III) Given M ∼= GL3 × GL1 and � +M = (GL3 × GL1) × GL1, for σ0 ∈ Irresq(GL3) and χ ∈ (F ×)D, we have +an irreducible induced representation +σ = iGSpin6 +M +(σ0 ⊠ χ) = ResGL4×GL1 +GSpin6 +� +iGL4×GL1 +GL3×GL1×GL1 (�σ0 ⊠ �χ ⊠ �η) +� +, +for �σ0 ∈ Irresq(GL3) and �χ, �η ∈ (F ×)D. If �σ0 ∈ Irresq(GL3) is supercuspidal, then we have +L(s, σ, Ad) = L(s)L(s, �σ0, Ad)L(s, �σ0 × �χ−1)L(s, �σ∨ +0 × �χ). +For non-supercuspidal �σ0 ∈ Irresq(GL3), i.e., σ0 = StGL3 ⊗ �χ0 for some �χ0 ∈ (F ×)D, we have +ker + + + +ad + + +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + + + + + = +� + + +a +c +0 +0 +0 +a +c +0 +0 +0 +a +0 +0 +0 +0 +b + + , Lf1−f3, Lf1−f4, Lf4−f3 +� +. +(5.52) +It follows that +L(s, σ, Ad) = L(s)L(s + 1)L(s + 2)L(s + 1, �χ�χ−1 +0 )L(s + 1, �χ−1�χ0). +gnr-(IV) Given M ∼= GL1 × GSpin4 and � +M = (GL2 × GL2) × GL1, we have the representation (5.24) +σ = iGSpin6 +M +(χ ⊠ σ0) +with σ0 ∈ Irresq(GSpin4), and χ ∈ (F ×)D. We have the irreducible iGL4 +GL2×GL2(�σ1 ⊠ �σ2) as in (5.26), +where χ ⊠ σ0 ⊂ (�σ1 ⊠ �σ2 ⊠ �η)|M with �σi ∈ Irresq(GL2), �η ∈ (F ×)D. Thus, if σ0 is supercuspidal (and +hence so are �σ1 and �σ2) we have +L(s, σ, Ad) = L(s)L(s, �σ1, Ad)L(s, �σ2, Ad)L(s, �σ1 × �σ∨ +2 )L(s, �σ∨ +1 × �σ1). +If σ0 is non-supercuspidal, with �σ1 supercuspidal and �σ2 non-supercuspidal, i.e., �σ2 = StGL2 ⊗ �χ +for some �χ ∈ (F ×)D, we have +ker + + + +ad + + +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + + + + + = +� + + +a +0 +0 +0 +0 +b +0 +0 +0 +0 +c +0 +0 +0 +0 +c + + , Lf1−f2, Lf1−f4, Lf2−f1, Lf2−f4, Lf3−f1, Lf3−f2, Lf3−f4 +� +, +(5.53) +and it then follows that +L(s, σ, Ad) = L(s)L(s + 1)L(s, �σ1, Ad)L(s + 1 +2, �σ∨ +1 × �χ)L(s + 1 +2, �σ1 × �χ−1). +If σ0 is non-supercuspidal, with �σ1 non-supercuspidal and �σ2 supercuspidal, i.e., �σ1 = StGL2 ⊗ �χ +for some �χ ∈ (F ×)D, then ker(ad(N)) is as in (5.51) and we have +L(s, σ, Ad) = L(s)L(s + 1)L(s, �σ2, Ad)L(s + 1 +2, �σ∨ +2 × �χ)L(s + 1 +2, �σ2 × �χ−1). + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +21 +If both �σ1 and �σ2 are non-supercuspidal, i.e., �σi = StGL2 ⊗ �χi with �χ1, �χ2 ∈ (F ×)D satisfying +�χ1 ̸= �χ2ν±1, we have +ker + + + +ad + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + + + + + = +� + + +a +0 +c +0 +0 +a +0 +c +d +0 +b +0 +0 +d +0 +b + + , Lf1−f2, Lf1−f4, Lf3−f2, Lf3−f4 +� +, +(5.54) +and it follows that +L(s, σ, Ad) = L(s)L(s + 1)2L(s + 1, �χ1�χ−1 +2 )L(s + 1, �χ−1 +1 �χ2)L(s, �χ−1 +1 �χ2)L(s, �χ1�χ−1 +2 ). +gnr-(V) Given M ∼= GL1 × GSpin4 and � +M = (GL2 × GL2) × GL1, we consider σ ∈ Irresq(GSpin6) and +�σ ∈ Irresq(GL4) and �η ∈ (F ×)D such that σ ⊂ (�σ ⊠ �η)|M. Then, �σ must be either (5.27) or (5.28). +For (5.27) (i.e., �σ = StGL4 ⊗ �χ), we have +ker + + + +ad + + +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + + + + + = +� + + +a +b +c +0 +0 +a +b +c +0 +0 +a +b +0 +0 +0 +a + + , Lf1−f4 +� +, +(5.55) +and it follows that +L(s, σ, Ad) = L(s + 3)L(s + 2)L(s + 1). +For (5.28) (i.e., �τ ∈ Irrsc(GL2)), we have +ker + + + +ad + + +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 + + + + + + = +� + + +a +c +0 +0 +d +b +0 +0 +0 +0 +a +c +0 +0 +d +b + + , Lf1−f3, Lf1−f4, Lf2−f3, Lf2−f4 +� +, +(5.56) +and it follows that +L(s, σ, Ad) = L(s, �τ, Ad)L(s, �τ × �τ ∨). +nongnr-(A) For Q([ν1/2�χ], [ν−1/2�χ], [�χ3], [�χ4]) (5.31), we have +L(s, σ, Ad) += +L(s)3L(s + 1)L(s − 1)L(s, �χ3�χ−1 +4 )L(s, �χ−1 +3 �χ4) +� +i=3,4 +� +L(s + 1 +2, �χ�χ−1 +i )L(s − 1 +2, �χ−1�χi)L(s − 1 +2, �χ�χ−1 +i )L(s + 1 +2, �χ−1�χi) +� +For Q +� +[ν �χ], [�χ], [ν−1�χ], [�χ4] +� +(5.32), we have +L(s, σ, Ad) = L(s)3L(s + 1)2L(s − 1)2L(s + 2)L(s − 2) +� +t=0,1,−1 +� +L(s + t, �χ�χ−1 +4 )L(s + t, �χ−1�χ4) +� +, +For Q([�χ, ν �χ], [ν−1�χ], [�χ4]) (5.33), we have ker(ad(N)) as in (5.51) and +L(s, σ, Ad) = L(s)2L(s − 1)2L(s − 2) +� +t=−1,0 +L(s + t, �χ�χ−1 +4 ) +� +t=±1 +L(s + t, �χ−1�χ4). +For Q +� +[ν �χ], [�χ, ν−1�χ], [�χ4] +� +(5.34), since +ker + + + +ad + + +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 + + + + + + = +� + + +a +0 +0 +0 +0 +b +0 +0 +0 +0 +b +0 +0 +0 +0 +c + + , Lf1−f3, Lf1−f4, Lf2−f1, Lf2−f3, Lf2−f4, Lf4−f1, Lf4−f3 +� +, +(5.57) +we have +L(s, σ, Ad) = L(s)2L(s + 2)L(s − 1)L(s + 1) +� +t=0,1 +L(s + t, �χ�χ−1 +4 ) +� +t=±1 +L(s + t, �χ−1�χ4). + +22 +MAHDI ASGARI AND KWANGHO CHOIY +For Q([ν3/2�χ], [ν1/2�χ], [ν−1/2�χ], [ν−3/2 �χ]) (5.35), we have +L(s, σ, Ad) = L(s)3L(s + 1)3L(s − 1)3L(s + 2)2L(s − 2)2L(s + 3)L(s − 3). +For Q +� +[ν1/2�χ, ν3/2�χ], [ν−1/2�χ], [ν−3/2�χ] +� +(5.36), we have ker(ad(N)) is as in (5.51) and +L(s, σ, Ad) = L(s)2L(s − 1)2L(s + 1)2L(s − 2)L(s + 2)L(s − 3). +For Q([ν3/2�χ], [ν−1/2�χ, ν1/2�χ], [ν−3/2�χ]) (5.37), we have ker(ad(N)) is as in (5.57) and +L(s, σ, Ad) = L(s)2L(s + 1)2L(s + 2)L(s − 1)2L(s − 3)L(s − 2). +For Q([ν3/2�χ], [ν1/2�χ], [ν−3/2�χ, ν−1/2�χ]) (5.38), we have ker(ad(N)) is as in (5.53) and +L(s, σ, Ad) = L(s)2L(s + 1)2L(s − 1)2L(s − 2)L(s + 2)L(s − 3). +For Q([ν1/2�χ, ν3/2�χ], [ν−3/2�χ, ν−1/2�χ]) (5.39), we have ker(ad(N)) is as in (5.54) and +L(s, σ, Ad) = L(s)L(s − 1)2L(s + 1)L(s + 2)L(s − 2)L(s − 3). +For Q([ν−1/2�χ, ν1/2�χ, ν3/2�χ], [ν−3/2�χ]) (5.40), we have ker(ad(N)) is as in (5.52) and +L(s, σ, Ad) = L(s)L(s − 1)L(s − 2)L(s + 1)L(s − 3). +Finally, for Q +� +[ν3/2�χ], [ν−3/2�χ, ν−1/2�χ, ν1/2�χ] +� +(5.41), since +ker + + + +ad + + +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 + + + + + + = +� + + +a +0 +0 +0 +0 +b +c +0 +0 +0 +b +c +0 +0 +0 +b + + , Lf1−f4, Lf2−f1, Lf2−f4 +� +, +(5.58) +we have +L(s, σ, Ad) = L(s)L(s + 1)L(s − 1)L(s − 2)L(s − 3). +nongnr-(B) For Q([∆�σ0], [ν1/2�χ], [ν−1/2�χ]) (5.45), with say [∆�σ0] = iGL2 +GL1×GL1(�η1 ⊠ �η2), �η1�η−1 +2 +̸= ν±1 we have +L(s, σ, Ad) += +L(s)3L(s + 1)L(s − 1)L(s, �η1�η−1 +2 )L(s, �η−1 +1 �η2) +� +i=1,2 +� +L(s − 1 +2, �ηi�χ−1)L(s + 1 +2, �ηi �χ−1)L(s + 1 +2, �η−1 +i +�χ)L(s − 1 +2, �η−1 +i +�χ) +� +. +nongnr-(C) As mentioned before, all the possibilities in this case were covered in (A) and (B) above. +nongnr-(D) For (5.49) with �σ supercuspidal, we have +L(s, σ, Ad) += +L(s)2L(s + 1)L(s − 1)L(s, σ, Ad) +L(s − 1 +2, σ × χ−1)L(s + 1 +2, σ × χ−1)L(s − 1 +2, σ∨ × χ)L(s + 1 +2, σ∨ × χ), +For (5.49) with non-supercuspidal �σ = StGL2 ⊗ η, η ∈ (F ×)D we have ker(ad(N)) as in (5.53) and +L(s, σ, Ad) += +L(s)2L(s + 1)2L(s − 1)L(s, χη−1)L(s + 1, χη−1)L(s + 1, χ−1η)L(s, χ−1η). +Recall that the remaining possibilities in this case were already covered in (A) above. +nongnr-(E) Finally, as mentioned before, all the possibilities in this case we also covered in (A). +6. Correction to [AC17] +We take this opportunity to correct the following errors in our earlier work [AC17]. They do not affect +the main results in that paper. + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +23 +6.1. Proposition 5.5 and 6.4. +• Change “1,2,4,8, if p = 2” to “1,2,4,8,..., 2[F :Q2]+2, if p = 2.” We have 2[F :Qp]+2 due to the fact that +��F ×/(F ×)2�� = 2[F :Q2]+2. +• For Proposition 5.5, using [GP92, Corollary 7.7], it follows that the case of p = 2 is bounded by +|(Z/2Z)4−1| = 8. Here 4 is coming from � +GSpin4 = GSO(4, C). +• For Proposition 6.4, using [GP92, Corollary 7.7], it follows that the case of p = 2 is bounded by +|(Z/2Z)6−1| = 32. Here 6 is coming from � +GSpin6 = GSO(6, C). +6.2. Remark 5.11. +• The formula (5.13) should read as follows: +��� Πϕ (GSpin4) +��� = +��� Πϕ(GSpin1,1 +4 ) +��� = 4, +��� Πϕ +� +GSpin2,1 +4 +���� = 1. +(5.13) +Also, in the following sentence change “in which case the multiplicity is 2” to “in which case the +multiplicity 2 could also occur”. We thank Hengfei Lu [Lu20] for bringing this error to our attention. +• In addition, it is more accurate that we use ‘not irreducible’ rather than ‘reducible’ in this Remark +since one may have indecomposable parameters. Alternatively, we may write �ϕi|WF is reducible. +Thus, at the beginning the Remark, change “When �ϕi is reducible,” to “When �ϕi is not irreducible,”. +References +[Art13] +J. Arthur. The endoscopic classification of representations, volume 61 of American Mathematical Society Collo- +quium Publications. American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups. +[AC17] +M. Asgari and K. Choiy. 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Takeda. The local Langlands conjecture for Sp(4). Int. Math. Res. Not. IMRN, (15):2987–3038, +2010. +[GT11] +W. T. Gan and S. Takeda. The local Langlands conjecture for GSp(4). Ann. of Math. (2), 173(3):1841–1882, 2011. +[GT14] +W. T. Gan and W. Tantono. The local Langlands conjecture for GSp(4), II: the case of inner forms. Amer. J. +Math., 136(3):761–805, 2014. +[GK82] +S. S. Gelbart and A. W. Knapp. L-indistinguishability and R groups for the special linear group. Adv. in Math., +43(2):101–121, 1982. +[GP92] +B. Gross and D. Prasad. On the decomposition of a representation of SOn when restricted to SOn−1. Canad. J. +Math., 44(5):974–1002, 1992. +[GR10] +B. Gross and M. Reeder. Arithmetic invariants of discrete Langlands parameters. Duke Math. J., 154(3):431–508, +2010. +[HT01] +M. Harris and R. Taylor. The geometry and cohomology of some simple Shimura varieties, volume 151 of Annals +of Mathematics Studies. Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. +Berkovich. +[Hen00] +G. Henniart. Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math., +139(2):439–455, 2000. +[HS12] +K. Hiraga and H. Saito. On L-packets for inner forms of SLn. Mem. Amer. Math. Soc., 215(1013):vi+97, 2012. +[KMSW14] T. Kaletha, A. Minguez, S. W. Shin, and P.-J. White. Endoscopic classification of representations: Inner forms of +unitary groups. preprint; arXiv:1409.3731v2 [math.NT], 2014. +[Kud94] +S. Kudla. The local Langlands correspondence: the non-Archimedean case. In Motives (Seattle, WA, 1991), vol- +ume 55 of Proc. Sympos. Pure Math., pages 365–391. Amer. Math. Soc., Providence, RI, 1994. +[Lab85] +J.-P. Labesse. Cohomologie, L-groupes et fonctorialit´e. Compositio Math., 55(2):163–184, 1985. +[Lu20] +H. Lu. Some applications of theta correspondence to branching laws. Math. Res. Lett., 27(1):243–263, 2020. + +24 +MAHDI ASGARI AND KWANGHO CHOIY +[Mok15] +C. P. Mok. Endoscopic classification of representations of quasi-split unitary groups. Mem. Amer. Math. Soc., +235(1108):vi+248, 2015. +[Rod82] +F. Rodier. Repr´esentations de GL(n, k) o`u k est un corps p-adique. In Bourbaki Seminar, Vol. 1981/1982, vol- +ume 92 of Ast´erisque, pages 201–218. Soc. Math. France, Paris, 1982. +[Rog90] +J. Rogawski. Automorphic representations of unitary groups in three variables, volume 123 of Annals of Mathe- +matics Studies. Princeton University Press, Princeton, NJ, 1990. +[Sch13] +P. Scholze. The local Langlands correspondence for GLn over p-adic fields. Invent. Math., 192(3):663–715, 2013. +[Tad92] +M. Tadi´c. Notes on representations of non-Archimedean SL(n). Pacific J. Math., 152(2):375–396, 1992. +[Tat79] +J. Tate. Number theoretic background. In Automorphic forms, representations and L-functions (Proc. Sympos. +Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. +Amer. Math. Soc., Providence, R.I., 1979. +[Wed08] +T. Wedhorn. The local Langlands correspondence for GL(n) over p-adic fields. In School on Automorphic Forms +on GL(n), volume 21 of ICTP Lect. Notes, pages 237–320. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008. +[Xu18] +B. Xu. L-packets of quasisplit GSp(2n) and GO(2n). Math. Ann., 370(1-2):71–189, 2018. +[Zel80] +A. V. Zelevinsky. Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). +Ann. Sci. ´Ecole Norm. Sup. (4), 13(2):165–210, 1980. +Mahdi Asgari, Department of Mathematics, Oklahoma State University, Stillwater, OK 74078-1058, U.S.A. +Email address: asgari@math.okstate.edu +Kwangho Choiy, School of Mathematical and Statistical Sciences, Southern Illinois University, Carbondale, +IL 62901-4408, U.S.A. +Email address: kchoiy@siu.edu + +REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 +25 +Table 1. Representations of GSpin4(F) +ResGL2×GL2 +GSpin4 +of +L-packet Structure +generic +(a) +(�σ1 ⊠ �σ2), +�σ2 ∼= �σ1�η, �σi ∈ Irrsc(GL2) +{1}, Z/2Z, (Z/2Z)2 +• +(b) +(�σ1 ⊠ �σ2), +�σ2 ̸∼= �σ1�η, �σi ∈ Irrsc(GL2) +{1}, Z/2Z +• +(i) +(StGL2 ⊠ StGL2) = StGSpin4 +(irreducible) +{1} +• +(ii) +(iGL2 +GL1×GL1(χGL2 +GL1×GL1(χ1 ⊗ χ2) ⊠ StGL2 ⊗ χ) +(irreducible) +{1} +• +(iii) +(iGL2 +GL1×GL1(χ1 ⊗ χ2) ⊠ iGL2 +GL1×GL1(χ3 ⊗ χ4)), χ1 ̸= ν±1χ2, χ3 ̸= ν±1χ4 +{1}, Z/2Z +• +(iv) +(�σ ⊠ StGL2 ⊗ χ), +�σ ∈ Irrsc(GL2) +(irreducible) +{1} +• +(v) +(�σ ⊠ iGL2 +GL1×GL1(χ1 ⊗ χ2)), +�σ ∈ Irrsc(GL2) +{1}, Z/2Z +• +nongnr +(χ ◦ det ⊠�σ), +�σ ∈ Irr(GL2) +(irreducible) +{1} +Table 2. The adjoint L-function L(s, σ, Ad) for GSpin4 +L(s, σ, Ad) +ords=1 +(a)&(b) +L(s, �σ1, Sym2 ⊗ω−1 +�σ1 )L(s, �σ2, Sym2 ⊗ω−1 +�σ2 ) +0 +(i) +L(s + 1)2 +0 +(ii) +L(s)L(s + 1)L(s, χ1χ−1 +2 )L(s, χ−1 +1 χ2) +0 +(iii) +L(s)2L(s, χ1χ−1 +2 )L(s, χ−1 +1 χ2)L(s, χ3χ−1 +4 )L(s, χ−1 +3 χ4) +0 +(iv) +L(s + 1)L(s, �σ2, Sym2 ⊗ω−1 +�σ2 ) +0 +(v) +L(s)L(s, χ1χ−1 +2 )L(s, χ−1 +1 χ2)L(s, �σ2, Sym2 ⊗ω−1 +�σ2 ) +0 +nongnr +L(s − 1)L(s)L(s + 1)L(s, �σ, Ad) +1 + ords=1 L(s, �σ, Ad) +Table 3. Representations of GSpin6(F) +ResGL4×GL1 +GSpin6 +of +generic +(a) +(�σ0 ⊠ �η), +�σ0 ∈ Irrsc(GL4) +• +(I) +iGL4×GL1 +(GL1×GL1×GL1×GL1)×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η), +�χi ̸= ν �χj +• +(II) +iGL4×GL1 +(GL2×GL1×GL1)×GL1(�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η), +�σ0 ∈ Irresq(GL2), �χ1 ̸= ν±1�χ2 +• +(III) +iGL4×GL1 +(GL3×GL1)×GL1(�σ0 ⊠ �χ ⊠ �η), +�σ0 ∈ Irresq(GL3) +• +(IV) +iGL4×GL1 +(GL2×GL2)×GL1(�σ1 ⊠ �σ2 ⊠ �η), +�σi ∈ Irresq(GL2), �σ1 ̸= ν±1�σ2 +• +(V) +(�σ ⊠ �η), +�σ ∈ Irresq(GL4) \ Irrsc(GL4) +• +(A) +iGL4×GL1 +(GL1×GL1×GL1×GL1)×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η), +�χi = ν �χj +(B) +iGL4×GL1 +(GL2×GL1×GL1)×GL1(�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η), +�σ0 ̸∈ Irresq(GL2), or �χ1 = ν±1�χ2 +(C) +iGL4×GL1 +(GL3×GL1)×GL1(�σ0 ⊠ �χ ⊠ �η), +non-generic �σ0 ∈ Irr(GL3) +(D) +iGL4×GL1 +(GL2×GL2)×GL1((χ ◦ det) ⊠ �σ ⊠ �η), +�σ ∈ Irr(GL2) +(E) +(�χ ◦ det ⊠�η), +�σ ∈ Irresq(GL4) \ Irrsc(GL4) + +26 +MAHDI ASGARI AND KWANGHO CHOIY +Table 4. The adjoint L-function L(s, σ, Ad) for GSpin6 +σ ∈ Irr(GSpin6(F)) determined by +L(s, σ, Ad) +ords=1 +(a) +(5.9) �σ0 ∈ Irrsc(GL4) +L(s, �σ0, Ad) +0 +(I) +(5.14) �χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η +L(s)3 � +i̸=j L(s, �χi�χ−1 +j ) +0 +(II) +(5.18) �σ0 ∈ Irrsc(GL2) +L(s)2L(s, �σ0, Ad)L(s, �σ0 × �χ−1 +1 )L(s, �σ∨ +0 × �χ1) +L(s, �σ0 × �χ−1 +2 )L(s, �σ∨ +0 × �χ2)L(s, �χ1�χ−1 +2 )L(s, �χ2�χ−1 +1 ) +0 +(II) +(5.18) �σ0 = StGL2 ⊗ �χ +L(s)2L(s + 1)L(s + 1, �χ�χ−1 +1 )L(s + 1, �χ�χ−1 +2 ) +L(s, �χ−1�χ1)L(s, �χ−1�χ2)L(s, �χ1�χ−1 +2 )L(s, �χ2�χ−1 +1 ) +0 +(III) +(5.22) �σ0 ∈ Irrsc(GL3) +L(s)L(s, �σ0, Ad)L(s, �σ0 × �χ−1)L(s, �σ∨ +0 × �χ) +0 +(III) +(5.22) �σ0 = StGL3 ⊗ �χ0 +L(s)L(s + 1)L(s + 2)L(s + 1, �χ�χ−1 +0 )L(s + 1, �χ−1�χ0) +0 +(IV) +(5.26) �σi ∈ Irrsc(GL2) +L(s)L(s, �σ1, Ad)L(s, �σ2, Ad) +L(s, �σ1 × �σ∨ +2 )L(s, �σ∨ +1 × �σ1) +0 +(IV) +(5.26) �σ1 ∈ Irrsc(GL2), �σ2 = StGL2 ⊗ �χ +L(s)L(s + 1)L(s, �σ1, Ad) +L(s + 1 +2, �σ∨ +1 × �χ)L(s + 1 +2, �σ1 × �χ−1) +0 +(IV) +(5.26) �σ2 ∈ Irrsc(GL2), �σ1 = StGL2 ⊗ �χ +L(s)L(s + 1)L(s, �σ2, Ad) +L(s + 1 +2, �σ∨ +2 × �χ)L(s + 1 +2, �σ2 × �χ−1) +0 +(IV) +(5.26) �σ1 = StGL2 ⊗ �χ1�σ2 = StGL2 ⊗ �χ2 +L(s)L(s + 1)2L(s, �χ−1 +1 �χ2)L(s, �χ1�χ−1 +2 ) +L(s + 1, �χ1�χ−1 +2 )L(s + 1, �χ−1 +1 �χ2) +0 +(V) +(5.27) �σ = StGL4 ⊗ �χ +L(s + 1)L(s + 2)L(s + 3) +0 +(V) +(5.28) �σ = ∆[ν1/2, ν−1/2], �τ ∈ Irrsc(GL2) +L(s, �τ, Ad)L(s, �τ × �τ ∨) +0 +(A) +(5.31) Q +� +[ν1/2�χ], [ν−1/2�χ], [�χ3], [�χ4] +� +L(s − 1)L(s)3L(s + 1)L(s, �χ3�χ−1 +4 )L(s, �χ−1 +3 �χ4) +� +i=3,4 +� L(s + 1 +2, �χ�χ−1 +i +)L(s − 1 +2, �χ−1�χi) +L(s − 1 +2, �χ�χ−1 +i +)L(s + 1 +2, �χ−1�χi) +� +≥ 1 +(A) +(5.32) Q +� +[ν �χ], [�χ], [ν−1�χ], [�χ4] +� +L(s − 2)L(s − 1)2L(s)3L(s + 1)2L(s + 2) +� +t=−1,0,1 +� +L(s + t, �χ�χ−1 +4 )L(s + t, �χ−1�χ4) +� +≥ 2 +(A) +(5.33) Q +� +[�χ, ν �χ], [ν−1�χ], [�χ4] +� +L(s − 2)L(s − 1)2L(s)2 +� +t=−1,0 +L(s + t, �χ�χ−1 +4 ) +� +t=−1,1 +L(s + t, �χ−1�χ4) +≥ 2 +(A) +(5.34) Q +� +[ν �χ], [�χ, ν−1�χ], [�χ4] +� +L(s − 1)L(s)2L(s + 1)L(s + 2) +� +t=0,1 +L(s + t, �χ�χ−1 +4 ) +� +t=−1,1 +L(s + t, �χ−1�χ4) +≥ 1 +(A) +(5.35) Q +� +[ν3/2�χ], [ν1/2�χ], [ν−1/2�χ], [ν−3/2�χ] +� +L(s − 3)L(s − 2)2L(s − 1)3L(s)3 +L(s + 1)3L(s + 2)2L(s + 3) +3 +(A) +(5.36) Q +� +[ν1/2�χ, ν3/2�χ], [ν−1/2�χ], [ν−3/2�χ] +� +L(s − 3)L(s − 2)L(s − 1)2L(s)2L(s + 1)2L(s + 2) +2 +(A) +(5.37) Q +� +[ν3/2�χ], [ν−1/2�χ, ν1/2�χ], [ν−3/2�χ] +� +L(s − 3)L(s − 2)L(s − 1)2L(s)2L(s + 1)2L(s + 2) +2 +(A) +(5.38) Q +� +[ν3/2�χ], [ν1/2�χ], [ν−3/2�χ, ν−1/2�χ] +� +L(s − 3)L(s − 2)L(s − 1)2L(s)2L(s + 1)2L(s + 2) +2 +(A) +(5.39) Q +� +[ν1/2�χ, ν3/2�χ], [ν−3/2�χ, ν−1/2�χ] +� +L(s − 3)L(s − 2)L(s − 1)2L(s)L(s + 1)L(s + 2) +2 +(A) +(5.40) Q +� +[ν−1/2�χ, ν1/2�χ, ν3/2�χ], [ν−3/2�χ] +� +L(s − 3)L(s − 2)L(s − 1)L(s)L(s + 1) +1 +(A) +(5.41) Q +� +[ν3/2�χ], [ν−3/2�χ, ν−1/2�χ, ν1/2�χ] +� +L(s − 3)L(s − 2)L(s − 1)L(s)L(s + 1) +1 +(B) +(5.45) Q +� +[iGL2 +B +(�η1 ⊠ �η2)], [�χν1/2], [�χν−1/2] +� +, +�η1�η−1 +2 +̸= ν±1 +L(s − 1)L(s)3L(s + 1)L(s, �η1�η−1 +2 )L(s, �η−1 +1 �η2) +� +t=± 1 +2 +� +i=1,2 +� +L(s + t, �ηi�χ−1)L(s + t, �η−1 +i +�χ) +� +≥ 1 +(B) +(5.45) (others covered in (A)) +(C) +(5.48) (covered in (A) and (B)) +(D) +(5.49) with �σ ∈ Irrsc(GL2) +L(s − 1)L(s)2L(s + 1)L(s, σ, Ad) +� +t=± 1 +2 +� +L(s + t, σ × χ−1)L(s + t, σ∨ × χ) +� +1 +(D) +(5.49) with �σ = StGL2 ⊗ η +L(s − 1)L(s)2L(s + 1)2 +L(s, χη−1)L(s + 1, χη−1)L(s + 1, χ−1η)L(s, χ−1η) +≥ 1 +(D) +(5.49) (others covered in (A)) +(E) +(5.50) (covered in (A)) + diff --git a/L9E4T4oBgHgl3EQf8g5c/content/tmp_files/load_file.txt b/L9E4T4oBgHgl3EQf8g5c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..326581560f5a2e4c4a4ae45ed982217ed644c9ad --- /dev/null +++ b/L9E4T4oBgHgl3EQf8g5c/content/tmp_files/load_file.txt @@ -0,0 +1,1241 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf,len=1240 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='05348v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='NT] 13 Jan 2023 REPRESENTATIONS OF THE p-ADIC GSpin4 AND GSpin6 AND THE ADJOINT L-FUNCTION MAHDI ASGARI AND KWANGHO CHOIY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We prove a conjecture of Gross-Prasad and Rallis about determination of generic L-packets in terms of the analytic properties of the adjoint L-function for p-adic general even spin groups of semi-simple ranks 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We also explicitly write the adjoint L-function for each L-packet in terms of the local Langlands L-functions for the general linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Introduction In this article, we provide further details on the local L-packets for the non-Archimedean split general spin groups GSpin4 and GSpin6, following our earlier work [AC17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We then use our explicit description of these L-packets to prove a conjecture of Gross-Prasad and Rallis, determining which of the L-packets are “generic” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', contain an irreducible representation with a Whittaker model) in terms of the analytic properties of the adjoint L-function of the packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We also write the adjoint L-function for each L-packet in terms of the local Langlands L-functions of the general linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In addition to details about the representations that our results provide, given that the adjoint L-functions have a significant role in the Gan-Gross-Prasad conjectures, we expect that our results in this paper would be helpful in that direction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let F be a p-adic field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Denote by WF the Weil group of F and let W ′ F = WF ×SL2(C) be the Weil-Deligne group of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let G be a connected, reductive, linear algebraic group over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The local Langlands Conjecture (LLC) predicts a surjective, finite-to-one map L from the set Irr(G) of equivalence classes of irreducible, smooth, complex representations of G(F) to the set Φ(G) of �G-conjugacy classes of L-parameters of G(F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', admissible homomorphisms φ : W ′ F −→ LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here, LG denotes the L-group of G with �G = LG0 its connected component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', the complex dual of G [Bor79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Among other properties, the map L is supposed to preserve the local L-, ǫ-, and γ-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Moreover, the (finite) fibers Πφ, for φ ∈ Φ(G), of the map L are called the L-packets of G and their structures are expected to be controlled by certain finite subgroups of �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Consider the split general spin groups G = GSpin4 and G = GSpin6, of type D2 = A1 × A2 and D3 = A3 respectively, whose algebraic structure we review in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We constructed most of the L-packets for these two groups in [AC17] and proved that they satisfy the expected properties of preservation of the local factors and their internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We review and complete the construction of these L-packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In particular, using the classification of representations of GLn, we give more explicit descriptions of the L- packets for GSpin4 and GSpin6 in terms of given representations of GL2 ×GL2 and GL4 ×GL1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' As a byproduct, we are able to give the criteria for determining the size of the L-packets for GSpin4 and GSpin6 (see Sections 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The known cases of the LLC for the p-adic groups include GLn[HT01, Hen00, Sch13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' SLn [GK82];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' non-quasi-split F-inner forms of GLn and SLn [HS12, ABPS16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' GSp4 and Sp4 [GT11, GT10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' non-quasi- split F-inner form GSp1,1 of GSp4 [GT14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Sp2n, SOn, and quasi-split SO∗ 2n [Art13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Un [Rog90, Mok15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' non quasi-split F-inner forms of Un [Rog90, KMSW14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' non-quasi-split F-inner form Sp1,1 of Sp4 [Cho17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' GSpin4, GSpin6 and their inner forms [AC17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' GSp2n and GO2n [Xu18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Going back to the case of general G, assume that ρ is a finite-dimensional complex representation of LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' When LLC is known, one can define the local Langlands L-functions L(s, π, ρ) = L(s, ρ ◦ φ) for each π ∈ Πφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here, the L-factors on the right hand side are the Artin local factors associated to the given representation of W ′ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 1 2 MAHDI ASGARI AND KWANGHO CHOIY B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Gross and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Prasad, following a remark of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Rallis, conjectured (in the generality of quasi-split groups) that the local L-packet Πφ(G) is generic if and only if the adjoint L-function L(s, Ad ◦ φ) is regular at s = 1 [GP92, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here, Ad denotes the adjoint representation of LG on the dual Lie algebra �g of �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (Note that in the body of this paper we use Ad exclusively for the restriction of the adjoint representation to the derived group of �g to distinguish it from the full adjoint L-function, which would have an extra factor of the L-function for the trivial character when �g has a one-dimensional center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=') We prove the above conjecture for the groups GSpin4 and GSpin6 as a consequence of our construction of the L-packets for these groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In fact, we prove the conjecture for a larger class of groups G = Gr,s m,n, which are given as subgroups of GLm × GLn satisfying a certain determinant equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We are able to work in the slightly larger generality because, as in the construction of the L-packets, we use the approach of restricting representations from GLm(F) × GLn(F) to the subgroup G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Moreover, we also give the adjoint L-function in all cases for G explicitly in terms of local Langlands L- functions of the general linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' While we are able to prove the Gross-Prasad-Rallis conjecture already without the explicit knowledge of the adjoint L-function, the explicit description of the adjoint L-function certainly also verifies the conjecture and we include it here since it may lead to other number theoretic or representation theoretic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Finally, we take this opportunity to correct a few inaccuracies in [AC17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' They do not affect the main results in that paper and fix some errors in our description of the L-packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The details are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We are grateful to Behrang Noohi and Ralf Schmidt for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Choiy was supported by a gift from the Simons Foundation (#840755).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Local Langlands Correspondence (LLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let p be a prime number and let F be a p-adic field of characteristic zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', a finite extension of Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We fix an algebraic closure ¯F of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Denote the ring of integers of F by OF and its unique maximal ideal by PF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Moreover, let q denote the cardinality of the residue field OF /PF and fix a uniformizer ̟ with |̟|F = q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Also, let WF denote the Weil group of F, W ′ F the Weil-Deligne group of F, and Γ the absolute Galois group Gal( ¯F/F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Throughout the paper, we will use the notation ν(·) = | · |F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let G be a connected, reductive, linear algebraic group over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Fixing Γ-invariant splitting data we define the L-group of G as a semi-direct product LG := �G ⋊ Γ, where �G = LG0 denotes the connected component of the L-group of G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', the complex dual of G (see [Bor79, §2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' LLC (still conjectural in this generality) asserts that there is a surjective, finite-to-one map from the set Irr(G) of isomorphism classes of irreducible smooth complex representations of G(F) to the set Φ(G) of �G-conjugacy classes of L-parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', admissible homomorphisms ϕ : W ′ F −→ LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given ϕ ∈ Φ(G), its fiber Πϕ(G), which is called an L-packet for G, is expected to be controlled by a certain finite group living in the complex dual group �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Furthermore, for π ∈ Πϕ(G) and ρ a finite dimensional algebraic representation of LG one defines the local factors L(s, π, ρ) = L(s, ρ ◦ φ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1) ǫ(s, π, ρ, ψ) = ǫ(s, ρ ◦ φ, ψ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2) γ(s, π, ρ, ψ) = γ(s, ρ ◦ φ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3) provided that LLC is known for the case in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here, the factors on the right are Artin factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The Adjoint L-Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' What we recall in this subsection holds for G quasi-split ([GP92, §2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' However, for simplicity we will take G to be split over F since the groups we are working with in this article are split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' When G is split over F, we may replace the L-group LG by its connected component �G = LG0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Take ρ to be the adjoint action of �G on its Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then we obtain the adjoint L-function L(s, π, Ad � G) = L(s, Ad � G ◦ φ) for all π ∈ Πϕ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The following is a conjecture of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Gross and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Prasad, suggested by a remark of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Rallis (see [GP92, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 3 Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Πϕ(G) contains a generic member if and only if L(s, Ad � G ◦ φ) is regular at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (Equivalently, π is generic if and only if L(s, π, Ad � G) is regular at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=') The conjecture is known in many cases in which the LLC is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' To mention a few, it was verified for GLn by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Gross and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Prasad [GP92], for GSp4 in [GT11] and, for non-supercuspidals, in [AS08], and for SO and Sp groups, it follows from the work of Arthur on endoscopic classification [Art13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We will verify this conjecture for the small rank split groups GSpin4 and GSpin6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The Groups GSpin4 and GSpin6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We gave detailed information about the structure of these two groups (as well as their inner forms) in [AC17, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For now we just recall the incidental isomorphisms GSpin4 ∼= {(g1, g2) ∈ GL2 × GL2 : det g1 = det g2} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4) GSpin6 ∼= � (g1, g2) ∈ GL1 × GL4 : g2 1 = det g2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='5) While our main interests in this article are the split general spin groups GSpin4 and GSpin6, for the purposes of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1 it is no more difficult, and perhaps also more natural, to consider a slightly more general setup as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Fix integers m, n ≥ 1 and r, s ≥ 1 and assume that gcd(r, s) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Define G = Gr,s m,n := {(g, h) ∈ GLm × GLn | (det g)r = (det h)s} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='6) Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The group Gr,s m,n is a split, connected, reductive, linear algebraic group over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let X = (Xij) and Y = (Ykl) be m×m and n×n matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' It is clear that Gr,s m,n, being an almost direct product of SLm×SLn and a torus, is reductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The only issue that requires justification is that the polynomial f(X, Y ) = (det X)r − (det Y )s is irreducible in F[Xij, Ykl] if and only if d = gcd(r, s) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' It is clear that if d > 1, then f is reducible since it would be divisible by (det X)(r/d) − (det Y )(s/d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' It remains to show that if d = 1, then f(X, Y ) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' This assertion should be easy to see via elementary arguments considering the polynomials in a possible factorization of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' However, we prove it below as a special case of a more general fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Assume that f(x, y) is an (arbitrary) irreducible polynomial in F[x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let p(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' , xa) ∈ F[x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' , xa] and p(y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' , yb) ∈ F[y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' , yb] be two polynomials such that p − α and q − α are irreducible for all constants α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then, f(p, q) is irreducible in F[x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' , xa, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' , yb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Our Proposition would clearly follow from the above assertion since (det −α) is always an irreducible polynomial and it is well-known that the two-variable polynomial xr − ys is irreducible in F[x, y] provided that d = gcd(r, s) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' To prove the assertion above, we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By base extension to an algebraic closure we may assume, without loss of generality, that F is algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let A be the subscheme of Spec F[x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' , xa, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' , yb] defined by f(p, q), and let B be the sub- scheme of Spec F[x, y] defined by xr − ys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The latter is irreducible since xr − ys is an irreducible polynomial by our assumption that d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' There is a natural map A → B which has irreducible (geometric) fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The result now follows from the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Claim: Let g : A → B be an open morphism of schemes of finite type over an algebraically closed field F such that the (geometric) fibers of g are irreducible and B is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then A is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' To see the claim let U be an open in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We want to show that for any other open V , we have that U ∩ V is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Since B is irreducible and g is open, we have that g(U) ∩ g(V ) is nonempty so there is a fiber F0 of g such that F0 ∩ U and F0 ∩ V are nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Hence, by irreducibility of F0, they have a nonempty intersection in F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In particular, U ∩ V is nonempty, which gives the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' It only remains to check that the map A → B above is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In fact, it is flat since it is a base extension of the cartesian product of two flat morphisms p : Spec F[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', xa] → Spec F[x] and q : Spec F[y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', yb] → Spec F[y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (Here, we are using the fact that Spec F[x] is a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=') This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' □ Of particular interest to us in this paper are the cases m = n = 2 and r = s = 1, when G = GSpin4, and 4 MAHDI ASGARI AND KWANGHO CHOIY m = 1, n = 4 and r = 2, s = 1, when G = GSpin6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The (connected) L-group of G is LGr,s 0 m,n = �G ∼= (GLm(C) × GLn(C))/{(z−rIm, zsIn) : z ∈ C×} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7) and we have the exact sequence 1 −→ {(z−rIm, zsIn) : z ∈ C×} ∼= C× −→ GLm(C) × GLn(C) prr,s m,n −−−−→ � Gr,s m,n −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='8) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Computation of the Adjoint L-Function for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let π be an irreducible admissible representation of G(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' There exist irreducible admissible representations πm and πn of GLm(F) and GLn(F), respectively, such that π ֒→ ResGLm(F )×GLn(F ) G(F ) (πm ⊗ πn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9) Let Ad � G denote the adjoint action of �G on its Lie algebra �g = {(X, Y ) ∈ glm(C) × gln(C) | r tr(X) = s tr(Y )} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) In what follows, let us write Ad � G = triv ⊕Ad (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='11) and for i ∈ {m, n} we similarly write Adi = Ad� GLi = triv ⊕Ad, where Ad here denotes the action of GLi(C) on the space of traceless i × i complex matrices sli(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let φπ : WF × SL2(C) → �G be the L-parameter of π and let φi : WF × SL2(C) → GLi(C), i = m, n, be the L-parameter of πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Recall by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='8) that we have a natural map pr = prr,s m,n : GLm(C) × GLn(C) −→ �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='12) Then we have φπ = pr ◦ (φm ⊗ φn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='13) Since the subgroup {(z−rIm, zsIn) : z ∈ C×} is central in GLm(C)×GLn(C) the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' GLm(C) × GLn(C) AutC (glm(C) × gln(C)) WF × SL2(C) �G AutC (�g) Adm⊗Adn pr φm⊗φn φπ Ad � G Note that the adjoint action Adm of GLm(C) on glm(C) preserves the trace, and similarly for n, so we obtain a right downward arrow by simply restricting any automorphism to the set of those pairs satisfying the trace equality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have L(s, 1F ×)L(s, π, Ad) · L(s, 1F ×) = L(s, π, Ad � G) · L(s, 1F ×) = L(s, Ad � G ◦ φπ) · L(s, 1F ×) = L (s, (Adm ⊗ Adn) ◦ (φm ⊗ φn)) = L(s, Adm ◦ φm)L(s, Adn ◦ φn) = L(s, πm, Adm)L(s, πn, Adn) = L(s, 1F ×)2L(s, πm, Ad)L(s, πn, Ad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='14) Therefore, we obtain the more convenient equality L(s, π, Ad) = L(s, πm, Ad)L(s, πn, Ad), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 5 which holds thanks to our choice of the notation Ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2 this relation helps verify Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1 for the groups of interest to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Genericity and The Conjecture of Gross-Prasad and Rallis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Restriction of Generic Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let us write □D for the group Hom(□, C×) of all contin- uous characters on a topological group □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Dente by □der the derived group of □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let G and �G be connected, reductive, linear, algebraic groups over F satisfying the property that Gder = �Gder ⊆ G ⊆ �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1) For any connected, reductive, linear, algebraic group □ over F, we write Irrsc(□) and Irresq(□) for the set of equivalence classes of supercuspidal and essentially square-integrable representations of □(F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Assume �G and G to be F-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let �B be a Borel subgroup of �G with Levi decomposition �B = �T �U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then B = �B ∩ G is a Borel subgroup of G with B = T U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Note that T = �T ∩ G and �U = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let ψ be a generic character of U(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' From [Tad92, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='8] we know that given a ψ-generic irreducible representation �σ of �G(F) we have a unique ψ-generic σ of G(F) such that σ ֒→ Res � G G(�σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The generic character associated with σ is not unique though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Each generic character associated with σ is determined up to the action of �T(F)/T (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We let �σ ∈ Irr( �G) be ψ-generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then there is a unique ψ-generic σψ ∈ Π�σ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' On the other hand, for each σ ∈ Π�σ(G) there exists t ∈ �T(F)/T (F) ∼= �G/G(F) such that σ = tσψ, where tσψ(g) = σ(t−1gt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' This implies that σ is tψ-generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here tψ is defined as tψ(u) = ψ(t−1ut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We say σ ∈ Irr(G), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ ∈ Irr( �G), is generic if it is ψ-generic with respect to some generic character ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' With this notation, σ ∈ Irr(G) is generic if and only if is �σ ∈ Irr( �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Criterion for Genericity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In this section we verify Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1 for the small rank general spin groups we are considering in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let G = Gr,s m,n be the group defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let π be an irreducible admissible representation of G(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then π is generic if and only if L(s, π, Ad) is regular at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given π there exist irreducible admissible representations πm of GLm(F) and πn of GLn(F) such that π is a subrepresentation of the restriction to G(F) of πm ⊗ πn as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Now, π is generic if and only if both πm and πn are generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By the truth of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1 for the general linear groups, the latter is equivalent to both L(s, πm, Ad) and L(s, πn, Ad) being regular at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Hence, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) and the fact that neither of the L-functions can have a zero at s = 1, we have that π is generic if and only if L(s, π, Ad) is regular at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' □ As we observed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3, the split groups GSpin4 and GSpin6 are special cases of Gr,s m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Therefore, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1 holds for the groups GSpin4 and GSpin6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Representations of GSpin4 In this section we list all the irreducible representations of GSpin4(F) and then calculate their associated adjoint L-function explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' To this end, we give the nilpotent matrix associated to their parameter in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The Reprsentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 6 MAHDI ASGARI AND KWANGHO CHOIY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Classification of representations of GSpin4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Following [AC17], we have 1 −→ GSpin4(F) −→ GL2(F) × GL2(F) −→ F × −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1) Recall that GSpin4(F) ∼= {(g1, g2) ∈ GL2(F) × GL2(F) : det g1 = det g2}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2) LGSpin4 = � GSpin4 = GSO4(C) ∼= (GL2(C) × GL2(C))/{(z−1, z) : z ∈ C×}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3) and 1 −→ C× −→ GL2(C) × GL2(C) pr4 −→ � GSpin4 −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4) When convenient, we view GSO4 as the group similitude orthogonal 4 × 4 matrices with respect to the anti-diagonal matrix J = J4 = \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='5) The Lie algebra of this group is also defined with respect to J and an element X in this Lie algebra satisfies tXJ + JX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Construction of the L-packets of GSpin4 (recalled from [AC17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given σ ∈ Irr(GSpin4) we have a lift �σ ∈ Irr(GL2 × GL2) such that σ ֒→ ResGL2×GL2 GSpin4 (�σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' It follows form the LLC for GLn [HT01, Hen00, Sch13] that there is a unique �ϕ�σ ∈ Φ(GL2 × GL2) corre- sponding to the representation �σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We now have a surjective, finite-to-one map L4 : Irr(GSpin4) −→ Φ(GSpin4) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='6) σ �−→ pr4 ◦ �ϕ�σ, which does not depend on the choice of the lifting �σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then, for each ϕ ∈ Φ(GSpin4), all inequivalent irreducible constituents of �σ constitutes the L-packet Πϕ(GSpin4) := Π�σ(GSpin4) = � σ ��� σ ֒→ ResGL2×GL2 GSpin4 (�σ) � � ∼= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7) Here, �σ is the member in the singleton Π�ϕ(GL2 × GL2) and �ϕ ∈ Φ(GL2 × GL2) is such that pr4 ◦ �ϕ = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We note that the construction does not depends on the choice of �ϕ, due to the LLC for GL2, [GK82, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4], [Tad92, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='5], and [HS12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Further details may be found in [AC17, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The L-parameters of GL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We recall the generic representations of GL2(F) in this paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We refer to [Wed08, Kud94, GR10] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let χ : F × → C× denote a continuous quasi-character of F ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By Zelevinski ([Zel80, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7] or [Kud94, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1]) we know that the generic representations of GL2 are: the supercuspidals, St ⊗ (χ ◦ det) where St denotes the Steinberg representation, and normally induced representations iGL2 GL1×GL1(χ1 ⊗ χ2) with χ1 ̸= χ2ν±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The only non-generic representation is χ◦ det .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Generic Representations of GSpin4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Following [AC17, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3], given ϕ ∈ Φ(GSpin4), fix the lift �ϕ = �ϕ1 ⊗ �ϕ2 ∈ Φ(GL2 × GL2) with �ϕi ∈ Φ(GL2) such that ϕ = pr4 ◦ �ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let �σ = �σ1 ⊠ �σ2 ∈ Π�ϕ(GL2 × GL2) be the unique member such that {�σi} = Π�ϕi(GL2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Recall the notation IGSpin4(�σ) := � χ ∈ (GL2(F) × GL2(F)/GSpin4(F))D ��� �σ ⊗ χ ∼= �σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then we have Πϕ(GSpin4) 1−1 ←→ IGSpin4(�σ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='8) REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 7 and we recall that, by [AC17, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7], we have IGSpin4(�σ) = � ISL2(�σ1), if �σ2 ∼= �σ1�η for some �η ∈ (F ×)D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' ISL2(�σ1) ∩ ISL2(�σ2), if �σ2 ̸∼= �σ1�η for any �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Irreducible Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let ϕ ∈ Φ(GSpin4) be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then �ϕ, �ϕ1, and �ϕ2 are all irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let ϕ ∈ Φ(GSpin6) be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then every member in Πϕ(GSpin4) is supercuspidal and generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' To study the internal structure of Πϕ(GSpin4), by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='8), we need to know the structure of IGSpin4(�σ), as we now recall from [AC17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(a) When �σ2 ∼= �σ1�η for some �η ∈ (F ×)D, we have IGSpin4(�σ) ∼= \uf8f1 \uf8f2 \uf8f3 {1}, if �ϕ1 (and hence also �ϕ2) is primitive or non-trivial on SL2(C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Z/2Z, if �ϕ1 (and hence also �ϕ2) is dihedral w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' one quadratic extension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (Z/2Z)2, if �ϕ1 (and hence also �ϕ2) is dihedral w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' three quadratic extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(b) When �σ2 ̸∼= �σ1�η for any �η ∈ (F ×)D, then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9) we have IGSpin4(�σ) ∼= {1} or Z/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Since �σ2 ̸∼= �σ1�η for any �η ∈ (F ×)D, the case of both �ϕ1 and �ϕ2 being diredral w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' three quadratic extensions is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Thus, we have the following list: If at least one of �ϕi is primitive, then IGSpin4(�σ) ∼= {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If both are dihedral, then IGSpin4(�σ) ∼= Z/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' From [AC17, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1], we recall the identification ∆∨ = {β∨ 1 = f ∗ 11 − f ∗ 12, β∨ 2 = f ∗ 21 − f ∗ 22} , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) using the notation fij and f ∗ ij, 1 ≤ i, j ≤ 2, for the usual Z-basis of characters and cocharacters of GL2 ×GL2 and β1, β2 denote the simple roots of GSpin4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We can use this identification to relate the nilpotent matrices associated to the parameters of GL2 × GL2 and GSpin4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For both (a) and (b) above, we have NGL2(C)×GL2(C) = ��0 0 0 0 � , �0 0 0 0 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO4(C) = 04×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We note that case (b) above was mentioned, less precisely, in [AC17, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Reducible Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If ϕ ∈ Φ(GSpin4) is reducible, then at least one �ϕi must be reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Since the number of irreducible constituents in ResGL2 SL2 (�σi) is at most 2, we have ISL2(�σi) ∼= {1}, or Z/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' This implies that IGSpin4(�σ) ∼= {1}, or Z/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If �ϕi is reducible and generic, then �σi is either the Steinberg representation twisted by a character or an irreducibly induced representation from the Borel subgroup of GL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We make case-by-case arguments as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(i) Note that the Steinberg representation of GL2 × GL2 is of the form StGL2 ⊠ StGL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have ResGL2×GL2 GSpin4 (StGL2 ⊠ StGL2) = StGSpin4 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='11) and ResGL2×GL2 GSpin4 (StGL2 ⊗ χ1 ⊠ StGL2 ⊗ χ2) = StGSpin4 ⊗ χ for some χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have IGSpin4(�σ) ∼= {1} as IG(StG) ∼= {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Thus, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9), the L-packet remains a singleton and the restriction is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' To determine χ, we use the required properties of χ1, χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Using T = ���a 0 0 b � , �c 0 0 d �� ���� ab = cd � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='12) we have χ1(ab) = χ2(cd) ⇔ χ1 = χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Denote χ1 = χ2 by χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 8 MAHDI ASGARI AND KWANGHO CHOIY For (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='11), we have NGL2(C)×GL2(C) = ��0 1 0 0 � , �0 1 0 0 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO4(C) = \uf8ee \uf8ef\uf8ef\uf8f0 0 1 1 0 0 0 0 −1 0 0 0 −1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb gnr-(ii) Next we consider ResGL2×GL2 GSpin4 � iGL2 GL1×GL1(χ1 ⊗ χ2) ⊠ StGL2 ⊗ χ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='13) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9), the fact that �σ2 ̸∼= �σ1�η for any �η ∈ (F ×)D, and since IG(StG) ∼= {1}, it follows that IGSpin4(�σ) ∼= {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Thus, the L-packet remains a singleton and the restriction (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='13) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' To describe the restriction (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='13), we proceed similarly as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have χ1(a)χ2(b) = χ(cd) = χ(ab) ⇔ χ1χ−1(a) = χ−1 2 χ(b) Specializing to a = b and c = d in the center, we have χ1χ2χ−2 = 1 For (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='13) , we have NGL2(C)×GL2(C) = �� 0 0 0 0 � , � 0 1 0 0 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO4(C) = \uf8ee \uf8ef\uf8ef\uf8f0 0 0 1 0 0 0 0 −1 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(iii) We consider ResGL2×GL2 GSpin4 � iGL2 GL1×GL1(χ1 ⊗ χ2) ⊠ iGL2 GL1×GL1(χ3 ⊗ χ4) � = iGSpin4 T � χ1 ⊗ χ2, χ3 ⊗ χ1χ2χ−1 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here, χ1 ̸= χ2ν±1 and χ3 ̸= χ4ν±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Note that by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9) this induced representation may be irre- ducible or consist of two irreducible inequivalent constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have NGL2(C)×GL2(C) = �� 0 0 0 0 � , � 0 0 0 0 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO4(C) = \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(iv) Given a supercuspidal �σ ∈ Irr(GL2), we consider ResGL2×GL2 GSpin4 (�σ ⊠ StGL2 ⊗ χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='14) Since IG(StG) ∼= {1}, due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9), the restriction (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='14) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We then have NGL2(C)×GL2(C) = ��0 0 0 0 � , �0 1 0 0 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO4(C) = \uf8ee \uf8ef\uf8ef\uf8f0 0 0 1 0 0 0 0 −1 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(v) Given supercuspidal �σ ∈ Irr(GL2), we next consider ResGL2×GL2 GSpin4 � �σ ⊠ iGL2 GL1×GL1(χ1 ⊗ χ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Note from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9) that this may be irreducible or consist of two irreducible inequivalent constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have NGL2(C)×GL2(C) = �� 0 0 0 0 � , � 0 0 0 0 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO4(C) = 04×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Non-Generic Representations of GSpin4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If σ ∈ Irr(GSpin4) is non-generic, then σ is of the form ResGL2×GL2 GSpin4 ((χ ◦ det) ⊠ �σ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) with �σ ∈ Irr(GL2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Note this restriction is irreducible due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9), and that as χ ◦ det is non-generic, so is the restriction σ for any �σ ∈ Irr(GL2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For �σ = St ∈ Irr(GL2), we have NGL2(C)×GL2(C) = ��0 0 0 0 � , �0 1 0 0 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO4(C) = \uf8ee \uf8ef\uf8ef\uf8f0 0 0 1 0 0 0 0 −1 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , and otherwise we have NGL2(C)×GL2(C) = ��0 0 0 0 � , �0 0 0 0 �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO4(C) = 04×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We summarize the above information about the representations of GSpin4 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Computation of the Adjoint L-function for GSpin4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We now give explicit expressions for the adjoint L-function for each of the representations of GSpin4(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We start by recalling that the adjoint L-functions of the representations �σ ∈ Irr(GL2) are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' L(s, �σ, Ad2) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 L(s)2L(s, χ1χ−1 2 )L(s, χ−1 1 χ2), if �σ = iGL2 GL1×GL1(χ1 ⊠ χ2) with χ1χ−1 2 ̸= ν±1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' L(s)L(s + 1), if �σ = StGL2 ⊗ χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' L(s)L(s, �σ, Sym2 ⊗ω−1 �σ ), if �σ is supercuspidal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' L(s)2L(s − 1)L(s + 1), if �σ = χ ◦ det .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here, L(s) = L(s, 1F ×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Recall our choice of notation L(s, �σ, Ad2) = L(s)L(s, �σ, Ad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Combining with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='14), Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(a)&(b) Given a supercuspidal σ ∈ Irr(GSpin4), we recall that σ ⊂ ResGL2×GL2 GSpin4 (�σ1 ⊠ �σ2) for some supercuspidal �σ1 ⊠ �σ2 ∈ Irr(GL2 × GL2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) we have L(s, σ, Ad) = L(s, �σ1, Sym2 ⊗ω−1 �σ1 )L(s, �σ2, Sym2 ⊗ω−1 �σ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(i) Given σ = StGSpin4 ⊗ χ ∈ Irr(GSpin4), by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) we have L(s, σ, Ad) = L(s + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(ii) Given σ ∈ Irr(GSpin4) such that σ = ResGL2×GL2 GSpin4 � iGL2 GL1×GL1(χ1 ⊗ χ2) ⊠ StGL2 ⊗ χ � , by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) we have L(s, σ, Ad) = L(s)L(s, χ1χ−1 2 )L(s, χ−1 1 χ2)L(s + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(iii) Given σ ∈ Irr(GSpin4) such that σ ⊂ ResGL2×GL2 GSpin4 � iGL2 GL1×GL1(χ1 ⊗ χ2) ⊠ iGL2 GL1×GL1(χ3 ⊗ χ4) � by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) we have L(s, σ, Ad) = L(s)2L(s, χ1χ−1 2 )L(s, χ−1 1 χ2)L(s, χ3χ−1 4 )L(s, χ−1 3 χ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 10 MAHDI ASGARI AND KWANGHO CHOIY gnr-(iv) Given σ ∈ Irr(GSpin4) such that σ = ResGL2×GL2 GSpin4 (�σ ⊠ StGL2 ⊗ χ) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) we have L(s, σ, Ad) = L(s, �σ2, Sym2 ⊗ω−1 �σ2 )L(s + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(v) Given σ ∈ Irr(GSpin4) such that σ ⊂ ResGL2×GL2 GSpin4 � �σ ⊠ iGL2 GL1×GL1(χ1 ⊗ χ2) � by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) we have L(s, σ, Ad) = L(s)L(s, �σ2, Sym2 ⊗ω−1 �σ2 )L(s, χ1χ−1 2 )L(s, χ−1 1 χ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr Given a non-generic σ ∈ Irr(GSpin4), from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15), we recall that σ = ResGL2×GL2 GSpin4 (χ ◦ det ⊠ �σ) and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) we have L(s, σ, Ad) = L(s)L(s − 1)L(s + 1)L(s, �σ, Ad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We summarize the explicit computations above in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Representations of GSpin6 We now list all the representations of GSpin6(F) and then calculate their associated adjoint L-function explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Again, we do this explicit calculation by finding the 6 × 6 nilpotent matrix in the complex dual group GSO6(C) in each case that is associated with the parameter of the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The Represenations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Classification of representations of GSpin6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Again, following [AC17], we have 1 −→ GSpin6(F) −→ GL1(F) × GL4(F) −→ F × −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1) Recall that GSpin6(F) ∼= � (g1, g2) ∈ GL1(F) × GL4(F) : g2 1 = det g2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2) LGSpin6 = � GSpin6 = GSO6(C) ∼= (GL1(C) × GL4(C))/{(z−2, z) : z ∈ C×}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3) and 1 −→ C× −→ GL1(C) × GL4(C) pr6 −→ � GSpin6 −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4) Just as the rank two case, here too we view GSO6 as the group similitude orthogonal 6 × 6 matrices with respect to the analogous 6 × 6, anti-diagonal, matrix J = J6 as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='5), and similarly define its Lie algebra with respect to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Construction of the L-packets of GSpin6 (recalled from [AC17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given σ ∈ Irr(GSpin6) we have a lift �σ ∈ Irr(GL1 × GL4) such that σ ֒→ ResGL1×GL4 GSpin6 (�σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' It follows from the LLC for GLn [HT01, Hen00, Sch13] that there is a unique �ϕ�σ ∈ Φ(GL1 × GL4) corre- sponding to the representation �σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We now have a surjective, finite-to-one map L6 : Irr(GSpin6) −→ Φ(GSpin6) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='5) σ �−→ pr6 ◦ �ϕ�σ, which does not depend on the choice of the lifting �σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then, for each ϕ ∈ Φ(GSpin6), all inequivalent irreducible constituents of �σ constitutes the L-packet Πϕ(GSpin6) := Π�σ(GSpin6) = � σ : σ ֒→ ResGL1×GL4 GSpin6 (�σ) � � ∼=, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='6) where �σ is the unique member of Π�ϕ(GL1 × GL4) and �ϕ ∈ Φ(GL1 × GL4) is such that pr6 ◦ �ϕ = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We note that the construction does not depends on the choice of �ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Further details may be found in [AC17, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 11 Following [AC17, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3], given ϕ ∈ Φ(GSpin6), fix the lift �ϕ = �η ⊗ �ϕ0 ∈ Φ(GL1 × GL4) with �ϕ0 ∈ Φ(GL4) such that ϕ = pr6 ◦ �ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let �σ = �η ⊠ �σ0 ∈ Π�ϕ(GL1 × GL4) be the unique member such that {�σ0} = Π�ϕ0(GL4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Recall that IGSpin6(�σ) := � �χ ∈ � GL1(F) × GL4(F)/GSpin6(F) �D : �σ ⊗ �χ ∼= �σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then we have Πϕ(GSpin6) 1−1 ←→ IGSpin6(�σ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7) and by [AC17, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='5 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='6] we have IGSpin6(�σ) ∼= {�χ ∈ ISL4(�σ0) : �χ2 = 1F ×} (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='8) and any �χ ∈ IGSpin6(�σ) is of the form �χ = (�χ′)−2 ⊠ �χ′, for some �χ′ ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Generic Representations of GSpin6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Thanks to the group structure (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2) and the relation of generic representations in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1, in order to classify the generic representations of GSpin6, it suffices to classify the generic representations of GL4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here are two key facts from the GL theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Recall from [Zel80, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7] and [Kud94, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1] that a generic representation of GL4 is of the form iGL4 M♭ (σ♭) where M♭ runs through any F-Levi subgroup of GL4 (including GL4 itself) and σ♭ is any essentially square-integrable representation of M♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For their L-parameters, we note from [Kud94, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2] that the generic representations of GL4 have Langlands parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', 4-dimensional Weil-Deligne representations (ρ, N)) of the form (ρ1 ⊗ sp(r1)) ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='. ⊗ (ρt ⊗ sp(rt)) with t ≤ 4, where ρi’s are irreducible and no two segments are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Irreducible Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let ϕ ∈ Φ(GSpin6) be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then �ϕ and �ϕ0 are also irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Let ϕ ∈ Φ(GSpin6) be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Every member in Πϕ(GSpin6) is supercuspidal and generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' To see the internal structure of Πϕ(GSpin6), we need, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7), to know the detailed structure of IGSpin6(�σ) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(a) Given σ ∈ Irrsc(GSpin6), we have �σ = �σ0 ⊠ �η ∈ Irrsc(GL4 × GL1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9) From [AC17, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1], we recall the identification: ∆∨ = {β∨ 1 = f ∗ 2 − f ∗ 3 , β∨ 2 = f ∗ 1 − f ∗ 2 , β∨ 3 = f ∗ 3 − f ∗ 4 } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) using the notation fij and f ∗ ij, 1 ≤ i, j ≤ 4, for the usual Z-basis of characters and cocharacters of GL4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Also, {β1, β2, β3} are the simple roots of GSpin6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have NGL4(C)×GL1(C) = (04×4, 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 12 MAHDI ASGARI AND KWANGHO CHOIY 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Reducible Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' When �ϕ0 is not irreducible, we have proper parabolic inductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' An exhaus- tive list of F-Levi subgroups M of GSpin6 (up to isomorphism) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' M ∼= GL1 × GL1 × GL1 × GL1 = � M ∩ GSpin6, where � M = (GL1 × GL1 × GL1 × GL1) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' M ∼= GL2 × GL1 × GL1 = � M ∩ GSpin6, where � M = (GL2 × GL1 × GL1) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' M ∼= GL3 × GL1 = � M ∩ GSpin6, where � M = (GL3 × GL1) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (Note: The factor GL1 of M is GSpin0 by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=') M ∼= GL1 × GSpin4 = � M ∩ GSpin6, where � M = (GL2 × GL2) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' M ∼= GSpin6 = � M ∩ GSpin6, where � M = GL4 × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (Note that M ∼= GL2 × GL2 does not occur on this list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=') We now consider each case and, by abuse of notation, conflate algebraic groups and their F-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(I) M ∼= GL1 × GL1 × GL1 × GL1 and � M = (GL1 × GL1 × GL1 × GL1) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given χi ∈ (F ×)D we consider iGSpin6 M (χ1 ⊠ χ2 ⊠ χ3 ⊠ χ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='11) Write χ1 ⊠ χ2 ⊠ χ3 ⊠ χ4 = (�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η)|M with �χi, �η ∈ (F ×)D so that �χ1�χ2�χ3�χ4 = �η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then we have the following relations χ1 = �χ1, χ2 = �χ2, χ3 = �χ3, χ4 = �η2(�χ2�χ3�χ4)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='12) By Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1, we know that the representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='11) is generic if and only if its lift iGL4×GL1 � M (�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='13) is generic if and only if iGL4 GL1×GL1×GL1×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='14) is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By the classification of the generic representations of GLn ([Zel80, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7] and [Kud94, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1]), this amounts to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='14) being irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By [Kud94, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1] and [BZ77, Zel80], the necessary and sufficient condition for this to occur is that there is no pair i, j with i ̸= j such that �χi = ν �χj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have NGL4(C)×GL1(C) = (04×4, 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6 gnr-(II) M ∼= GL2 × GL1 × GL1 and � M = (GL2 × GL1 × GL1) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given σ0 ∈ Irresq(GL2) and χ1, χ2 ∈ (F ×)D, we consider iGSpin6 M (σ0 ⊠ χ1 ⊠ χ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) Write σ0 ⊠ χ1 ⊠ χ2 = (�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η)|M with �σ0 ∈ Irresq(GL2), �χi, �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given (g, h1, h2, h3) ∈ � M with det(gh1h2) = h2 3, if we set (g, h1, h3) ∈ M, we have �σ0(g)�χ1(h1)�χ2(h2)�η(h3) = �σ0(g)�χ1(h1)�χ2(det g−1h−1 1 h2 3)�η(h3) = (�σ0 �χ−1 2 det)(g)(�χ1 �χ−1 2 )(h1)(�χ2 2�η)(h3) = σ(g)χ1(h1)χ2(h3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then we have �σ0 = σ0�χ2, �χ1 = χ1�χ2, �η = χ2�χ−2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 13 If we set (g, h2, h3) ∈ M, we have �σ0(g)�χ1(h1)�χ2(h2)�η(h3) = �σ0(g)�χ1(det g−1h−1 2 h2 3)�χ2(h2)�η(h3) = (�σ0 �χ−1 1 det)(g)(�χ2 �χ−1 1 )(h2)(�χ2 1�η)(h3) = σ(g)χ1(h2)χ2(h3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then we have �σ0 = σ0�χ1, �χ2 = χ2�χ1, �η = χ1�χ−2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='16) As before, the representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='15) is generic if and only if its lift iGL4×GL1 � M (�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='17) is generic if and only if iGL4 GL2×GL1×GL1(�σ0 ⊠ �χ1 ⊠ �χ2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='18) is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Again by the classification of the generic representations of GLn this amounts to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='18) being irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Hence, we must have �χ1 ̸= ν±1�χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In other words, given (g, h1, h2, h3) ∈ � M with det(gh1h2) = h2 3, if we set (g, h1, h3) ∈ M, then χ1 ̸= ν±1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' if we set (g, h2, h3) ∈ M, then χ2 ̸= ν±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If σ0 is supercuspidal, then NGL4(C)×GL1(C) = (04×4, 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If σ0 is non-supercuspidal, then NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(III) M ∼= GL3 × GL1 and � M = (GL3 × GL1) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given σ0 ∈ Irresq(GL3) and χ ∈ (F ×)D, we consider iGSpin6 M (σ0 ⊠ χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='19) Write σ0 ⊠ χ = (�σ0 ⊠ �χ ⊠ �η)|M with �σ0 ∈ Irresq(GL3), �χ, �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given (g, h1, h2) ∈ � M with det(gh1) = h2 2, if we set (g, h2) ∈ M, then we have �σ0(g)�χ(h1)�η(h2) = �σ0(g)�χ(det g−1h2 2)�η(h2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='20) = (�σ0�χ−1 ◦ det)(g)(�χ2�η)(h2) = σ(g)χ(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then, we have �σ0 = σ0�χ and �η = χ2�χ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' As before, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='19) is generic if and only if its lift iGL4×GL1 � M (�σ0 ⊠ �χ ⊠ �η) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='21) is generic if and only if iGL4 GL3×GL1(�σ0 ⊠ �χ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='22) 14 MAHDI ASGARI AND KWANGHO CHOIY is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' This amounts to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='22) being irreducible as before, which is always true since �σ0 is an essentially square integrable representation of GL3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Note that by the classification of essen- tially square-integrable representations of GL3 ([Kud94, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2]), �σ0 must be either supercuspidal or the unique subrepresentation of iGL3 GL1×GL1×GL1 � νχ ⊠ χ ⊠ ν−1χ � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='23) with any χ ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If σ0 is supercuspidal, then NGL4(C)×GL1(C) = (04×4, 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If σ0 is the non-supercuspidal, unique, subrepresentation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='23), then NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(IV) M ∼= GL1 × GSpin4 and � M = (GL2 × GL2) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given σ0 ∈ Irresq(GSpin4) and χ ∈ (F ×)D we consider iGSpin6 M (χ ⊠ σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='24) Write χ ⊠ σ0 ⊂ (�σ1 ⊠ �σ2 ⊠ �η)|M with �σi ∈ Irresq(GL2), �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' As before, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='24) is generic if and only if its lift iGL4×GL1 � M (�σ1 ⊠ �σ2 ⊠ �η) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='25) is generic if and only if iGL4 GL2×GL2(�σ1 ⊠ �σ2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='26) is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' This amounts to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='26) being irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Thus, we must have �σ1 ̸= ν±1�σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have several cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If σ0 is supercuspidal (so are �σ1 and �σ2), then NGL4(C)×GL1(C) = (04×40) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If σ0 is non-supercuspidal, then for supercuspidal �σ1 and non-supercuspidal �σ2 we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' for non-supercuspidal �σ1 and supercuspidal �σ2 we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 15 and for non-supercuspidal �σ1 and �σ2 we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(V) M ∼= GSpin6 and � M = GL4 × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given σ ∈ Irresq(GSpin6) \\ Irrsc(GSpin6), we consider σ ⊂ (�σ ⊠ �η)|M with �σ ∈ Irresq(GL4)\\Irrsc(GL4), �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here, we note that ϕ ∈ Φ(GSpin6) is not irreducible and neither �σ nor σ is supercuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' It is clear that σ is generic as �σ ⊠ �η is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By the classification of essentially square-integrable representations of GL4 ([Kud94, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2]), �σ must be the unique subrepresentation of either iGL4 GL1×GL1×GL1×GL1 � ν3/2 �χ ⊠ ν1/2�χ ⊠ ν−1/2�χ ⊠ ν−3/2�χ � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='27) with any �χ ∈ (F ×)D (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', �σ = StGL4 ⊗ �χ), or of iGL4 GL2×GL2 � ν1/2�τ ⊠ ν−1/2�τ � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='28) with any �τ ∈ Irrsc(GL2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Now, for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='27) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' and for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='28) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (We note, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' [Tat79, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='5)], that NGL4(C) is of the form O2×2 ⊗ I2×2 + � 0 1 0 0 � ⊗ I2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=') 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Non-Generic Representaions of GSpin6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Using the transitivity of the parabolic induction and the classification of generic representations of GLn, ([Zel80, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7] and [Kud94, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1]), the non-generic representations of GSpin6 are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(A) M ∼= GL1 × GL1 × GL1 × GL1 and � M = (GL1 × GL1 × GL1 × GL1) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given χi ∈ (F ×)D, by Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1 and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='12), the representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='11) contains a non-generic constituent if and only if the same is true for iGL4×GL1 � M (�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='29) if and only if iGL4 GL1×GL1×GL1×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='30) 16 MAHDI ASGARI AND KWANGHO CHOIY contains a non-generic constituent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' This amounts to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='30) being reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' As before, the necessary and sufficient condition for this to occur is that there is some pair i, j with i ̸= j such that �χi = ν �χj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By the Langlands classification and the description of constituents of the parabolic induction (see [Zel80, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1], [Rod82, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1], and [Kud94, Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1 §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1]), each constituent can be described as a Langlands quotient, denoted by Q(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='), as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The first case is when there is only one pair, say �χ1 = ν1/2�χ and �χ2 = ν−1/2�χ for some �χ ∈ (F ×)D while �χ3 ̸= ν±1�χj for j ̸= 3 and �χ4 ̸= ν±1�χj for j ̸= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then we have the non-generic constituent Q � [ν1/2�χ], [ν−1/2�χ], [�χ3], [�χ4] � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='31) which is the Langlands quotient of iGL4 GL2×GL1×GL1 � Q � [ν1/2�χ], [ν−1/2�χ] � ⊠ �χ3 ⊠ �χ4 � = iGL4 GL2×GL1×GL1 ((�χ ◦ det) ⊠ �χ3 ⊠ �χ4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have NGL4(C)×GL1(C) = (04×4, 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Note that the other constituent of this induced representation, which is generic, is Q � [ν−1/2�χ, ν1/2�χ], [�χ3], [�χ4] � = iGL4 GL2×GL1×GL1 � Q � [ν−1/2 �χ, ν1/2�χ] � ⊠ �χ3 ⊠ �χ4 � = iGL4 GL2×GL1×GL1 ((St ⊗ �χ) ⊠ �χ3 ⊠ �χ4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The next case is when there are two pairs, say �χ1 = ν �χ, �χ2 = �χ, and �χ3 = ν−1�χ for some �χ ∈ (F ×)D and �χ4 ̸= ν±1�χi for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then we have the following three non-generic constituents: Q � [ν �χ], [�χ], [ν−1�χ], [�χ4] � = iGL4 GL3×GL1((�χ ◦ det) ⊠ �χ3 ⊠ �χ4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='32) Q � [�χ, ν �χ], [ν−1�χ], [�χ4] � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='33) Q � [ν �χ], [�χ, ν−1�χ], [�χ4] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='34) For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='32) we have NGL4(C)×GL1(C) = (04×4, 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6, for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='33) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , and for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='34) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Finally, in the case where we have three pairs we are in the situation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then we have the following seven non-generic constituents: Q � [ν3/2�χ], [ν1/2�χ], [ν−1/2�χ], [ν−3/2�χ] � = �χ ◦ det;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='35) Q � [ν1/2�χ, ν3/2�χ], [ν−1/2�χ], [ν−3/2�χ] � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='36) REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 17 Q � [ν3/2�χ], [ν−1/2�χ, ν1/2�χ], [ν−3/2�χ] � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='37) Q � [ν3/2�χ], [ν1/2�χ], [ν−3/2 �χ, ν−1/2�χ] � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='38) Q � [ν1/2 �χ, ν3/2�χ], [ν−3/2 �χ, ν−1/2�χ] � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='39) Q � [ν−1/2�χ, ν1/2�χ, ν3/2�χ], [ν−3/2�χ] � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='40) Q � [ν3/2 �χ], [ν−3/2�χ, ν−1/2�χ, ν1/2�χ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='41) For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='35) we have NGL4(C)×GL1(C) = (04×4, 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6, for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='36) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='37) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='38) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='39) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='40) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , 18 MAHDI ASGARI AND KWANGHO CHOIY and for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='41) we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(B) M ∼= GL2 × GL1 × GL1 and � M = (GL2 × GL1 × GL1) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given σ0 ∈ Irr(GL2) and χ1, χ2 ∈ (F ×)D, we consider iGSpin6 M (σ0 ⊠ χ1 ⊠ χ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='42) Write σ0 ⊠ χ1 ⊠ χ2 = (�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η)|M with �σ0 ∈ Irr(GL2) and �χi, �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='16), it follows that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='42) contains a non-generic constituent if and only if its lift iGL4×GL1 � M (�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='43) contains a non-generic constituent if and only if iGL4 GL2×GL1×GL1(�σ0 ⊠ �χ1 ⊠ �χ2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='44) does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Recalling nongnr-(A), it is sufficient to consider the case of �σ0 ∈ Irr(GL2), �χ1 = ν1/2 �χ, and �χ2 = ν−1/2�χ for �χ ∈ (F ×)D, where the segment ∆�σ0 of �σ0 does not precede either �χ1 or �χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We then have the following sole non-generic constituent: Q([∆�σ0], [ν1/2�χ], [ν−1/2 �χ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='45) We have NGL4(C)×GL1(C) = (04×4, 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(C) M ∼= GL3 × GL1 and � M = (GL3 × GL1) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given a non-generic σ0 ∈ Irr(GL3) and any χ ∈ (F ×)D, we consider iGSpin6 M (σ0 ⊠ χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='46) Write σ0 ⊠ χ = (�σ0 ⊠ �χ ⊠ �η)|M with non-generic �σ0 ∈ Irr(GL3) and �χ, �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' As in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='20) we have �σ0 = σ0�χ, and �η = χ2�χ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' As before, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='46) contains a non-generic constituent if and only if its lift iGL4×GL1 � M (�σ0 ⊠ �χ ⊠ �η) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='47) also contains one if and only if iGL4 GL3×GL1(�σ0 ⊠ �χ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='48) does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' To have a non-generic �σ0 of GL3(F), the irreducible representation �σ0 must be some constituent in a reducible induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' This case has been covered in nongnr-(A) and (B) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(D) M ∼= GL1 × GSpin4 and � M = (GL2 × GL2) × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given a non-generic σ0 ∈ Irr(GSpin4), by Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3, we know that it must be of the form ResGL2×GL2 GSpin4 ((χ ◦ det) ⊠ �σ) for �σ ∈ Irr(GL2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For η ∈ (F ×)D, the induced representation iGSpin6 M ((χ ◦ det) ⊠ �σ ⊠ η) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='49) REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 19 contains a non-generic constituent if and only if so does iGL4 GL2×GL2((χ ◦ det) ⊠ �σ), which is always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Therefore, if �σ is supercuspidal, then NGL4(C)×GL1(C) = (04×4, 0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = 06×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If �σ is non-supercuspidal, then it suffices to consider the case �σ = StGL2 ⊗ η with η ∈ (F ×)D since the other case has been covered in nongnr-(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Thus, we have NGL4(C)×GL1(C) = \uf8eb \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='10) ⇐⇒ NGSO6(C) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(E) M ∼= GSpin6 and � M = GL4 × GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Given a non-generic σ ∈ Irr(GSpin6), it must be of the form ResGL4×GL1 GSpin6 (�χ ◦ det ⊠�η) = χ ◦ det, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='50) for some �χ, �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' This is the case Q([ν3/2�χ], [ν1/2�χ], [ν−1/2�χ], [ν−3/2�χ]) in nongnr-(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Computation of the Adjoint L-function for GSpin6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We now give explicit expressions for the adjoint L-function of each of the representations of GSpin6(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Recall that if we have a parameter (φ, N) with N a nilpotent matrix on the vector space V , then its adjoint L-function is L(s, φ, Ad) = det � 1 − q−sAd(φ)|V I N �−1 , where VN = ker(N), V I the vectors fixed by the inertia group, and V I N = V I ∩ VN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Below for the cases where N is non-zero, we write ker(Ad(N)) and we use Lα to denote the root group associated with the root α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We now consider each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='14) and Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='3, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(a) Given σ ∈ Irrsc(GSpin6), we have �σ = �σ0 ⊠ �η ∈ Irrsc(GL4 × GL1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then L(s, 1F ×)L(s, σ, Ad) = L(s, �σ0, Ad� GL4) or L(s, σ, Ad) = L(s, �σ0, Ad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(I) Given M ∼= GL1 × GL1 × GL1 × GL1 and � M = (GL1 × GL1 × GL1 × GL1) × GL1, we recall iGL4 GL1×GL1×GL1×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4) must be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Thus, given σ ∈ Irr(GSpin6) such that σ = iGSpin6 M (�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4), we have L(s, σ, Ad) = L(s)3 � i̸=j L(s, �χi�χ−1 j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(II) Given M ∼= GL2 × GL1 × GL1 and � M = (GL2 × GL1 × GL1) × GL1, for σ0 ∈ Irresq(GL2) and χ1, χ2 ∈ (F ×)D, we have an irreducible induced representation σ = iGSpin6 M (σ0 ⊠ χ1 ⊠ χ2) = ResGL4×GL1 GSpin6 � iGL4 GL2×GL1×GL1(�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η) � , for some �σ0 ∈ Irresq(GL2), and �χi, �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For supercuspidal �σ0 we have L(s, σ, Ad) = L(s)2L(s, �σ0, Ad)L(s, �σ0 × �χ−1 1 )L(s, �σ∨ 0 × �χ1) L(s, �σ0 × �χ−1 2 )L(s, �σ∨ 0 × �χ2)L(s, �χ1�χ−1 2 )L(s, �χ2�χ−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 20 MAHDI ASGARI AND KWANGHO CHOIY For non-supercuspidal �σ0 ∈ Irr(GL2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', σ0 = StGL2 ⊗ �χ for some �χ ∈ (F ×)D, we have ker \uf8eb \uf8ec \uf8ec \uf8edad \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � \uf8ee \uf8ef\uf8ef\uf8f0 a 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c \uf8f9 \uf8fa\uf8fa\uf8fb , Lf1−f2, Lf1−f3, Lf1−f4, Lf3−f2, Lf3−f4, Lf4−f2, Lf4−f3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='51) It follows that L(s, σ, Ad) = L(s)2L(s + 1)L(s + 1, �χ�χ−1 1 )L(s + 1, �χ�χ−1 2 ) L(s, �χ−1�χ1)L(s, �χ−1�χ2)L(s, �χ1�χ−1 2 )L(s, �χ2�χ−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(III) Given M ∼= GL3 × GL1 and � M = (GL3 × GL1) × GL1, for σ0 ∈ Irresq(GL3) and χ ∈ (F ×)D, we have an irreducible induced representation σ = iGSpin6 M (σ0 ⊠ χ) = ResGL4×GL1 GSpin6 � iGL4×GL1 GL3×GL1×GL1 (�σ0 ⊠ �χ ⊠ �η) � , for �σ0 ∈ Irresq(GL3) and �χ, �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If �σ0 ∈ Irresq(GL3) is supercuspidal, then we have L(s, σ, Ad) = L(s)L(s, �σ0, Ad)L(s, �σ0 × �χ−1)L(s, �σ∨ 0 × �χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For non-supercuspidal �σ0 ∈ Irresq(GL3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', σ0 = StGL3 ⊗ �χ0 for some �χ0 ∈ (F ×)D, we have ker \uf8eb \uf8ec \uf8ec \uf8edad \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � \uf8ee \uf8ef\uf8ef\uf8f0 a c 0 0 0 a c 0 0 0 a 0 0 0 0 b \uf8f9 \uf8fa\uf8fa\uf8fb , Lf1−f3, Lf1−f4, Lf4−f3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='52) It follows that L(s, σ, Ad) = L(s)L(s + 1)L(s + 2)L(s + 1, �χ�χ−1 0 )L(s + 1, �χ−1�χ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(IV) Given M ∼= GL1 × GSpin4 and � M = (GL2 × GL2) × GL1, we have the representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='24) σ = iGSpin6 M (χ ⊠ σ0) with σ0 ∈ Irresq(GSpin4), and χ ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We have the irreducible iGL4 GL2×GL2(�σ1 ⊠ �σ2) as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='26), where χ ⊠ σ0 ⊂ (�σ1 ⊠ �σ2 ⊠ �η)|M with �σi ∈ Irresq(GL2), �η ∈ (F ×)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Thus, if σ0 is supercuspidal (and hence so are �σ1 and �σ2) we have L(s, σ, Ad) = L(s)L(s, �σ1, Ad)L(s, �σ2, Ad)L(s, �σ1 × �σ∨ 2 )L(s, �σ∨ 1 × �σ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If σ0 is non-supercuspidal, with �σ1 supercuspidal and �σ2 non-supercuspidal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', �σ2 = StGL2 ⊗ �χ for some �χ ∈ (F ×)D, we have ker \uf8eb \uf8ec \uf8ec \uf8edad \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � \uf8ee \uf8ef\uf8ef\uf8f0 a 0 0 0 0 b 0 0 0 0 c 0 0 0 0 c \uf8f9 \uf8fa\uf8fa\uf8fb , Lf1−f2, Lf1−f4, Lf2−f1, Lf2−f4, Lf3−f1, Lf3−f2, Lf3−f4 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='53) and it then follows that L(s, σ, Ad) = L(s)L(s + 1)L(s, �σ1, Ad)L(s + 1 2, �σ∨ 1 × �χ)L(s + 1 2, �σ1 × �χ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' If σ0 is non-supercuspidal, with �σ1 non-supercuspidal and �σ2 supercuspidal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', �σ1 = StGL2 ⊗ �χ for some �χ ∈ (F ×)D, then ker(ad(N)) is as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='51) and we have L(s, σ, Ad) = L(s)L(s + 1)L(s, �σ2, Ad)L(s + 1 2, �σ∨ 2 × �χ)L(s + 1 2, �σ2 × �χ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 21 If both �σ1 and �σ2 are non-supercuspidal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', �σi = StGL2 ⊗ �χi with �χ1, �χ2 ∈ (F ×)D satisfying �χ1 ̸= �χ2ν±1, we have ker \uf8eb \uf8ec \uf8ec \uf8edad \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � \uf8ee \uf8ef\uf8ef\uf8f0 a 0 c 0 0 a 0 c d 0 b 0 0 d 0 b \uf8f9 \uf8fa\uf8fa\uf8fb , Lf1−f2, Lf1−f4, Lf3−f2, Lf3−f4 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='54) and it follows that L(s, σ, Ad) = L(s)L(s + 1)2L(s + 1, �χ1�χ−1 2 )L(s + 1, �χ−1 1 �χ2)L(s, �χ−1 1 �χ2)L(s, �χ1�χ−1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' gnr-(V) Given M ∼= GL1 × GSpin4 and � M = (GL2 × GL2) × GL1, we consider σ ∈ Irresq(GSpin6) and �σ ∈ Irresq(GL4) and �η ∈ (F ×)D such that σ ⊂ (�σ ⊠ �η)|M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Then, �σ must be either (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='27) or (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='27) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', �σ = StGL4 ⊗ �χ), we have ker \uf8eb \uf8ec \uf8ec \uf8edad \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � \uf8ee \uf8ef\uf8ef\uf8f0 a b c 0 0 a b c 0 0 a b 0 0 0 a \uf8f9 \uf8fa\uf8fa\uf8fb , Lf1−f4 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='55) and it follows that L(s, σ, Ad) = L(s + 3)L(s + 2)L(s + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='28) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', �τ ∈ Irrsc(GL2)), we have ker \uf8eb \uf8ec \uf8ec \uf8edad \uf8ee \uf8ef\uf8ef\uf8f0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � \uf8ee \uf8ef\uf8ef\uf8f0 a c 0 0 d b 0 0 0 0 a c 0 0 d b \uf8f9 \uf8fa\uf8fa\uf8fb , Lf1−f3, Lf1−f4, Lf2−f3, Lf2−f4 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='56) and it follows that L(s, σ, Ad) = L(s, �τ, Ad)L(s, �τ × �τ ∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(A) For Q([ν1/2�χ], [ν−1/2�χ], [�χ3], [�χ4]) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='31), we have L(s, σ, Ad) = L(s)3L(s + 1)L(s − 1)L(s, �χ3�χ−1 4 )L(s, �χ−1 3 �χ4) � i=3,4 � L(s + 1 2, �χ�χ−1 i )L(s − 1 2, �χ−1�χi)L(s − 1 2, �χ�χ−1 i )L(s + 1 2, �χ−1�χi) � For Q � [ν �χ], [�χ], [ν−1�χ], [�χ4] � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='32), we have L(s, σ, Ad) = L(s)3L(s + 1)2L(s − 1)2L(s + 2)L(s − 2) � t=0,1,−1 � L(s + t, �χ�χ−1 4 )L(s + t, �χ−1�χ4) � , For Q([�χ, ν �χ], [ν−1�χ], [�χ4]) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='33), we have ker(ad(N)) as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='51) and L(s, σ, Ad) = L(s)2L(s − 1)2L(s − 2) � t=−1,0 L(s + t, �χ�χ−1 4 ) � t=±1 L(s + t, �χ−1�χ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For Q � [ν �χ], [�χ, ν−1�χ], [�χ4] � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='34), since ker \uf8eb \uf8ec \uf8ec \uf8edad \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � \uf8ee \uf8ef\uf8ef\uf8f0 a 0 0 0 0 b 0 0 0 0 b 0 0 0 0 c \uf8f9 \uf8fa\uf8fa\uf8fb , Lf1−f3, Lf1−f4, Lf2−f1, Lf2−f3, Lf2−f4, Lf4−f1, Lf4−f3 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='57) we have L(s, σ, Ad) = L(s)2L(s + 2)L(s − 1)L(s + 1) � t=0,1 L(s + t, �χ�χ−1 4 ) � t=±1 L(s + t, �χ−1�χ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 22 MAHDI ASGARI AND KWANGHO CHOIY For Q([ν3/2�χ], [ν1/2�χ], [ν−1/2�χ], [ν−3/2 �χ]) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='35), we have L(s, σ, Ad) = L(s)3L(s + 1)3L(s − 1)3L(s + 2)2L(s − 2)2L(s + 3)L(s − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For Q � [ν1/2�χ, ν3/2�χ], [ν−1/2�χ], [ν−3/2�χ] � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='36), we have ker(ad(N)) is as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='51) and L(s, σ, Ad) = L(s)2L(s − 1)2L(s + 1)2L(s − 2)L(s + 2)L(s − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For Q([ν3/2�χ], [ν−1/2�χ, ν1/2�χ], [ν−3/2�χ]) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='37), we have ker(ad(N)) is as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='57) and L(s, σ, Ad) = L(s)2L(s + 1)2L(s + 2)L(s − 1)2L(s − 3)L(s − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For Q([ν3/2�χ], [ν1/2�χ], [ν−3/2�χ, ν−1/2�χ]) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='38), we have ker(ad(N)) is as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='53) and L(s, σ, Ad) = L(s)2L(s + 1)2L(s − 1)2L(s − 2)L(s + 2)L(s − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For Q([ν1/2�χ, ν3/2�χ], [ν−3/2�χ, ν−1/2�χ]) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='39), we have ker(ad(N)) is as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='54) and L(s, σ, Ad) = L(s)L(s − 1)2L(s + 1)L(s + 2)L(s − 2)L(s − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For Q([ν−1/2�χ, ν1/2�χ, ν3/2�χ], [ν−3/2�χ]) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='40), we have ker(ad(N)) is as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='52) and L(s, σ, Ad) = L(s)L(s − 1)L(s − 2)L(s + 1)L(s − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Finally, for Q � [ν3/2�χ], [ν−3/2�χ, ν−1/2�χ, ν1/2�χ] � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='41), since ker \uf8eb \uf8ec \uf8ec \uf8edad \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � \uf8ee \uf8ef\uf8ef\uf8f0 a 0 0 0 0 b c 0 0 0 b c 0 0 0 b \uf8f9 \uf8fa\uf8fa\uf8fb , Lf1−f4, Lf2−f1, Lf2−f4 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='58) we have L(s, σ, Ad) = L(s)L(s + 1)L(s − 1)L(s − 2)L(s − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(B) For Q([∆�σ0], [ν1/2�χ], [ν−1/2�χ]) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='45), with say [∆�σ0] = iGL2 GL1×GL1(�η1 ⊠ �η2), �η1�η−1 2 ̸= ν±1 we have L(s, σ, Ad) = L(s)3L(s + 1)L(s − 1)L(s, �η1�η−1 2 )L(s, �η−1 1 �η2) � i=1,2 � L(s − 1 2, �ηi�χ−1)L(s + 1 2, �ηi �χ−1)L(s + 1 2, �η−1 i �χ)L(s − 1 2, �η−1 i �χ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(C) As mentioned before, all the possibilities in this case were covered in (A) and (B) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(D) For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='49) with �σ supercuspidal, we have L(s, σ, Ad) = L(s)2L(s + 1)L(s − 1)L(s, σ, Ad) L(s − 1 2, σ × χ−1)L(s + 1 2, σ × χ−1)L(s − 1 2, σ∨ × χ)L(s + 1 2, σ∨ × χ), For (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='49) with non-supercuspidal �σ = StGL2 ⊗ η, η ∈ (F ×)D we have ker(ad(N)) as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='53) and L(s, σ, Ad) = L(s)2L(s + 1)2L(s − 1)L(s, χη−1)L(s + 1, χη−1)L(s + 1, χ−1η)L(s, χ−1η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Recall that the remaining possibilities in this case were already covered in (A) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' nongnr-(E) Finally, as mentioned before, all the possibilities in this case we also covered in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Correction to [AC17] We take this opportunity to correct the following errors in our earlier work [AC17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' They do not affect the main results in that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Change “1,2,4,8, if p = 2” to “1,2,4,8,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', 2[F :Q2]+2, if p = 2.” We have 2[F :Qp]+2 due to the fact that ��F ×/(F ×)2�� = 2[F :Q2]+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='5, using [GP92, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7], it follows that the case of p = 2 is bounded by |(Z/2Z)4−1| = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here 4 is coming from � GSpin4 = GSO(4, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' For Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='4, using [GP92, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='7], it follows that the case of p = 2 is bounded by |(Z/2Z)6−1| = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Here 6 is coming from � GSpin6 = GSO(6, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='13) should read as follows: ��� Πϕ (GSpin4) ��� = ��� Πϕ(GSpin1,1 4 ) ��� = 4, ��� Πϕ � GSpin2,1 4 ���� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='13) Also, in the following sentence change “in which case the multiplicity is 2” to “in which case the multiplicity 2 could also occur”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' We thank Hengfei Lu [Lu20] for bringing this error to our attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In addition, it is more accurate that we use ‘not irreducible’ rather than ‘reducible’ in this Remark since one may have indecomposable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Alternatively, we may write �ϕi|WF is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Thus, at the beginning the Remark, change “When �ϕi is reducible,” to “When �ϕi is not irreducible,”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' References [Art13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Arthur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The endoscopic classification of representations, volume 61 of American Mathematical Society Collo- quium Publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Orthogonal and symplectic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' [AC17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Asgari and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Choiy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} 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the p-adic GSp(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Number Theory, 128 (8):2340–2358, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' [ABPS16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Aubert, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Baum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Plymen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Solleveld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The local Langlands correspondence for inner forms of SLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', 3:Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' 32, 34, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' [BZ77] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Bernstein and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Zelevinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Induced representations of reductive p-adic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' ´Ecole Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (4), 10(4):441–472, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' [Bor79] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Automorphic L-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In Automorphic forms, representations and L-functions (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', Oregon State Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', Corvallis, Ore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', 1977), Part 2, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} 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+page_content=' The local Langlands correspondence for GL(n) over p-adic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' In School on Automorphic Forms on GL(n), volume 21 of ICTP Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Notes, pages 237–320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Abdus Salam Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=', Trieste, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' [Xu18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' L-packets of quasisplit GSp(2n) and GO(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Ann.' metadata={'source': 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+page_content=' Mahdi Asgari, Department of Mathematics, Oklahoma State University, Stillwater, OK 74078-1058, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Email address: asgari@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='okstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='edu Kwangho Choiy, School of Mathematical and Statistical Sciences, Southern Illinois University, Carbondale, IL 62901-4408, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Email address: kchoiy@siu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='edu REPRESENTATIONS AND ADJOINT L-FUNCTION FOR GSpin4 AND GSpin6 25 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Representations of GSpin4(F) ResGL2×GL2 GSpin4 of L-packet Structure generic (a) (�σ1 ⊠ �σ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ2 ∼= �σ1�η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σi ∈ Irrsc(GL2) {1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Z/2Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' (Z/2Z)2 (b) (�σ1 ⊠ �σ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ2 ̸∼= �σ1�η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σi ∈ Irrsc(GL2) {1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Z/2Z (i) (StGL2 ⊠ StGL2) = StGSpin4 (irreducible) {1} (ii) (iGL2 GL1×GL1(χGL2 GL1×GL1(χ1 ⊗ χ2) ⊠ StGL2 ⊗ χ) (irreducible) {1} (iii) (iGL2 GL1×GL1(χ1 ⊗ χ2) ⊠ iGL2 GL1×GL1(χ3 ⊗ χ4)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' χ1 ̸= ν±1χ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' χ3 ̸= ν±1χ4 {1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Z/2Z (iv) (�σ ⊠ StGL2 ⊗ χ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ ∈ Irrsc(GL2) (irreducible) {1} (v) (�σ ⊠ iGL2 GL1×GL1(χ1 ⊗ χ2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ ∈ Irrsc(GL2) {1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Z/2Z nongnr (χ ◦ det ⊠�σ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ ∈ Irr(GL2) (irreducible) {1} Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The adjoint L-function L(s, σ, Ad) for GSpin4 L(s, σ, Ad) ords=1 (a)&(b) L(s, �σ1, Sym2 ⊗ω−1 �σ1 )L(s, �σ2, Sym2 ⊗ω−1 �σ2 ) 0 (i) L(s + 1)2 0 (ii) L(s)L(s + 1)L(s, χ1χ−1 2 )L(s, χ−1 1 χ2) 0 (iii) L(s)2L(s, χ1χ−1 2 )L(s, χ−1 1 χ2)L(s, χ3χ−1 4 )L(s, χ−1 3 χ4) 0 (iv) L(s + 1)L(s, �σ2, Sym2 ⊗ω−1 �σ2 ) 0 (v) L(s)L(s, χ1χ−1 2 )L(s, χ−1 1 χ2)L(s, �σ2, Sym2 ⊗ω−1 �σ2 ) 0 nongnr L(s − 1)L(s)L(s + 1)L(s, �σ, Ad) 1 + ords=1 L(s, �σ, Ad) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' Representations of GSpin6(F) ResGL4×GL1 GSpin6 of generic (a) (�σ0 ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ0 ∈ Irrsc(GL4) (I) iGL4×GL1 (GL1×GL1×GL1×GL1)×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �χi ̸= ν �χj (II) iGL4×GL1 (GL2×GL1×GL1)×GL1(�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ0 ∈ Irresq(GL2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �χ1 ̸= ν±1�χ2 (III) iGL4×GL1 (GL3×GL1)×GL1(�σ0 ⊠ �χ ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ0 ∈ Irresq(GL3) (IV) iGL4×GL1 (GL2×GL2)×GL1(�σ1 ⊠ �σ2 ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σi ∈ Irresq(GL2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ1 ̸= ν±1�σ2 (V) (�σ ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ ∈ Irresq(GL4) \\ Irrsc(GL4) (A) iGL4×GL1 (GL1×GL1×GL1×GL1)×GL1(�χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �χi = ν �χj (B) iGL4×GL1 (GL2×GL1×GL1)×GL1(�σ0 ⊠ �χ1 ⊠ �χ2 ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ0 ̸∈ Irresq(GL2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' or �χ1 = ν±1�χ2 (C) iGL4×GL1 (GL3×GL1)×GL1(�σ0 ⊠ �χ ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' non-generic �σ0 ∈ Irr(GL3) (D) iGL4×GL1 (GL2×GL2)×GL1((χ ◦ det) ⊠ �σ ⊠ �η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ ∈ Irr(GL2) (E) (�χ ◦ det ⊠�η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' �σ ∈ Irresq(GL4) \\ Irrsc(GL4) 26 MAHDI ASGARI AND KWANGHO CHOIY Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content=' The adjoint L-function L(s, σ, Ad) for GSpin6 σ ∈ Irr(GSpin6(F)) determined by L(s, σ, Ad) ords=1 (a) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='9) �σ0 ∈ Irrsc(GL4) L(s, �σ0, Ad) 0 (I) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='14) �χ1 ⊠ �χ2 ⊠ �χ3 ⊠ �χ4 ⊠ �η L(s)3 � i̸=j L(s, �χi�χ−1 j ) 0 (II) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='18) �σ0 ∈ Irrsc(GL2) L(s)2L(s, �σ0, Ad)L(s, �σ0 × �χ−1 1 )L(s, �σ∨ 0 × �χ1) L(s, �σ0 × �χ−1 2 )L(s, �σ∨ 0 × �χ2)L(s, �χ1�χ−1 2 )L(s, �χ2�χ−1 1 ) 0 (II) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='18) �σ0 = StGL2 ⊗ �χ L(s)2L(s + 1)L(s + 1, �χ�χ−1 1 )L(s + 1, �χ�χ−1 2 ) L(s, �χ−1�χ1)L(s, �χ−1�χ2)L(s, �χ1�χ−1 2 )L(s, �χ2�χ−1 1 ) 0 (III) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='22) �σ0 ∈ Irrsc(GL3) L(s)L(s, �σ0, Ad)L(s, �σ0 × �χ−1)L(s, �σ∨ 0 × �χ) 0 (III) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='22) �σ0 = StGL3 ⊗ �χ0 L(s)L(s + 1)L(s + 2)L(s + 1, �χ�χ−1 0 )L(s + 1, �χ−1�χ0) 0 (IV) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='26) �σi ∈ Irrsc(GL2) L(s)L(s, �σ1, Ad)L(s, �σ2, Ad) L(s, �σ1 × �σ∨ 2 )L(s, �σ∨ 1 × �σ1) 0 (IV) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='26) �σ1 ∈ Irrsc(GL2), �σ2 = StGL2 ⊗ �χ L(s)L(s + 1)L(s, �σ1, Ad) L(s + 1 2, �σ∨ 1 × �χ)L(s + 1 2, �σ1 × �χ−1) 0 (IV) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='26) �σ2 ∈ Irrsc(GL2), �σ1 = StGL2 ⊗ �χ L(s)L(s + 1)L(s, �σ2, Ad) L(s + 1 2, �σ∨ 2 × �χ)L(s + 1 2, �σ2 × �χ−1) 0 (IV) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='26) �σ1 = StGL2 ⊗ �χ1�σ2 = StGL2 ⊗ �χ2 L(s)L(s + 1)2L(s, �χ−1 1 �χ2)L(s, �χ1�χ−1 2 ) L(s + 1, �χ1�χ−1 2 )L(s + 1, �χ−1 1 �χ2) 0 (V) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='27) �σ = StGL4 ⊗ �χ L(s + 1)L(s + 2)L(s + 3) 0 (V) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='28) �σ = ∆[ν1/2, ν−1/2], �τ ∈ Irrsc(GL2) L(s, �τ, Ad)L(s, �τ × �τ ∨) 0 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='31) Q � [ν1/2�χ], [ν−1/2�χ], [�χ3], [�χ4] � L(s − 1)L(s)3L(s + 1)L(s, �χ3�χ−1 4 )L(s, �χ−1 3 �χ4) � i=3,4 � L(s + 1 2, �χ�χ−1 i )L(s − 1 2, �χ−1�χi) L(s − 1 2, �χ�χ−1 i )L(s + 1 2, �χ−1�χi) � ≥ 1 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='32) Q � [ν �χ], [�χ], [ν−1�χ], [�χ4] � L(s − 2)L(s − 1)2L(s)3L(s + 1)2L(s + 2) � t=−1,0,1 � L(s + t, �χ�χ−1 4 )L(s + t, �χ−1�χ4) � ≥ 2 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='33) Q � [�χ, ν �χ], [ν−1�χ], [�χ4] � L(s − 2)L(s − 1)2L(s)2 � t=−1,0 L(s + t, �χ�χ−1 4 ) � t=−1,1 L(s + t, �χ−1�χ4) ≥ 2 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='34) Q � [ν �χ], [�χ, ν−1�χ], [�χ4] � L(s − 1)L(s)2L(s + 1)L(s + 2) � t=0,1 L(s + t, �χ�χ−1 4 ) � t=−1,1 L(s + t, �χ−1�χ4) ≥ 1 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='35) Q � [ν3/2�χ], [ν1/2�χ], [ν−1/2�χ], [ν−3/2�χ] � L(s − 3)L(s − 2)2L(s − 1)3L(s)3 L(s + 1)3L(s + 2)2L(s + 3) 3 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='36) Q � [ν1/2�χ, ν3/2�χ], [ν−1/2�χ], [ν−3/2�χ] � L(s − 3)L(s − 2)L(s − 1)2L(s)2L(s + 1)2L(s + 2) 2 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='37) Q � [ν3/2�χ], [ν−1/2�χ, ν1/2�χ], [ν−3/2�χ] � L(s − 3)L(s − 2)L(s − 1)2L(s)2L(s + 1)2L(s + 2) 2 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='38) Q � [ν3/2�χ], [ν1/2�χ], [ν−3/2�χ, ν−1/2�χ] � L(s − 3)L(s − 2)L(s − 1)2L(s)2L(s + 1)2L(s + 2) 2 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='39) Q � [ν1/2�χ, ν3/2�χ], [ν−3/2�χ, ν−1/2�χ] � L(s − 3)L(s − 2)L(s − 1)2L(s)L(s + 1)L(s + 2) 2 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='40) Q � [ν−1/2�χ, ν1/2�χ, ν3/2�χ], [ν−3/2�χ] � L(s − 3)L(s − 2)L(s − 1)L(s)L(s + 1) 1 (A) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='41) Q � [ν3/2�χ], [ν−3/2�χ, ν−1/2�χ, ν1/2�χ] � L(s − 3)L(s − 2)L(s − 1)L(s)L(s + 1) 1 (B) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='45) Q � [iGL2 B (�η1 ⊠ �η2)], [�χν1/2], [�χν−1/2] � , �η1�η−1 2 ̸= ν±1 L(s − 1)L(s)3L(s + 1)L(s, �η1�η−1 2 )L(s, �η−1 1 �η2) � t=± 1 2 � i=1,2 � L(s + t, �ηi�χ−1)L(s + t, �η−1 i �χ) � ≥ 1 (B) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='45) (others covered in (A)) (C) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='48) (covered in (A) and (B)) (D) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='49) with �σ ∈ Irrsc(GL2) L(s − 1)L(s)2L(s + 1)L(s, σ, Ad) � t=± 1 2 � L(s + t, σ × χ−1)L(s + t, σ∨ × χ) � 1 (D) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='49) with �σ = StGL2 ⊗ η L(s − 1)L(s)2L(s + 1)2 L(s, χη−1)L(s + 1, χη−1)L(s + 1, χ−1η)L(s, χ−1η) ≥ 1 (D) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='49) (others covered in (A)) (E) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} +page_content='50) (covered in (A))' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E4T4oBgHgl3EQf8g5c/content/2301.05348v1.pdf'} diff --git a/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf b/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c92be7b3d613cb9c9204c9381ec03c6b50b49b7e --- /dev/null +++ b/L9E4T4oBgHgl3EQfiw1J/content/2301.05136v1.pdf @@ -0,0 +1,3 @@ +version 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mode 100644 index 0000000000000000000000000000000000000000..163e12ca59a73795f7cd968c69b113e222ca695e --- /dev/null +++ b/LtAzT4oBgHgl3EQfkf1m/content/tmp_files/2301.01532v1.pdf.txt @@ -0,0 +1,1206 @@ +arXiv:2301.01532v1 [math.PR] 4 Jan 2023 +ver_mv06122022preprint_degenerate_d.tex +On weak existence of solutions of +degenerate McKean–Vlasov equations +A.Yu. Veretennikov∗ +January 5, 2023 +Abstract +A new weak existence result for degenerate multi-dimensional stochastic +McKean–Vlasov equation is established under relaxed regularity conditions. +Keywords: McKean-Vlasov equations; degenerate diffusion; weak solutions. +MSC: 60J60 +1 +Introduction +The subject of this paper is solutions of the stochastic Itˆo-McKean-Vlasov (McKean- +Vlasov) equation in R2d +dXt = Ytdt, +dYt = B[t, Zt, µt]dt + Σ[t, Zt, µt]dWt, +X0 = x0, Y0 = y0, +(1) +where Zt = (Xt, Yt) ∈ R2d, in a particular situation called the true McKean-Vlasov +case under the convention +B[t, z, µ] = +� +b(t, z, ζ)µ(dζ), Σ[t, z, µ] = +� +σ(t, z, ζ)µ(dζ), +(2) +∗Institute for Information Transmission Problems, +Moscow, +Russian Federation; +email: +ayv@iitp.ru. This research was funded by the RFBR grant 20-01-00575a. +1 + +where z = (x, y) ∈ R2d and ζ = (ξ, η) ∈ R2d, and under certain non-degeneracy +assumptions on σ. Here W is a standard d-dimensional Wiener process, b and σ are +vector and matrix Borel functions of corresponding dimensions d and d × d, µt is +the distribution of the process X at time t. The initial data z0 = (x0, y0) may be +random and in this case it is independent of W. Vlasov’s proposal was a substitution +of a real multiparticle interaction by a certain “mean field” [13]. The introduction +to the whole topic in its stochastic version may be found in [11]; one more important +reference, although devoted soleyly to the deterministic setting is [1]. In this paper +we investigate only the problem of weak existence for genuinly degenerate stochastic +McKean – Vlasov equations. The equations like (1) naturally arise in mechanical +systems with stochastic forces or noise. The aim of this paper is to show weak exis- +tence for a such a degenerate SDE system simultaneously minimizing the regularity +assumptions on both coefficients with respect to all variables. We assume the non- +degeneracy of σ and highlight that this non-degeneracy only holds for the second +component Y of the system (1). The interest to the minimal regularity is mainly +due to the control problems where the optimal strategies are usually discontinuous. +Among the most important works on the subject there is the paper [3]. The study +in the present paper is based on Krylov’s bounds [6] and on the approach proposed +in [9] for the ordinary Itˆo SDEs which was further generalised to some extent in [12] +also for the ordinary Itˆo SDEs. Other useful references may be found in the cited +papers. +The structure of the paper is as follows. In the section 2 weak existence is stated +under appropriate conditions. The section 3 containts its proof based on a combi- +nation of Krylov’s bounds and on Krylov’s existence results for the nondegenerate +Ito’s equations [5], [6], and on Nisio’s weak existence proof also for Ito’s SDEs [9]. +The degeneracy of the diffusion is overcome still by using Krylov’s bounds for non- +degenerate Itˆo processes. No regularity of the coefficients b and σ is assumed with +respect to the variables t, y, and η. Uniform continuity is assumed for both b and σ +with respect to the variables x and ξ. The abbreviation by CBS signifies the Cauchy- +Buniakovsky-Schwarz inequality and BCM stands the Bienaym´e-Chebyshev-Markov +inequality. +2 +Weak existence +2.1 +Main results +Let us recall a fact from functional analysis useful for the case (1)– (2), see, for +example, (see [7, Theorem 1.5.5]). The proposition 1 and its corollary are stated in +2 + +a slightly more general form than what is needed for bounded coefficients. +Proposition 1. For any Borel function f(z, ζ) and any probability measure µ(dζ) +such that f(z, ·) is integrable with respect to this measure, the function f[z, µ] := +� +f(z, ζ) µ(dζ) is Borel measurable in z. +Corollary 1. Suppose for each (t, z) the Borel coefficients b(t, z, ζ) and σ(t, z, ζ) +are bounded in ζ and integrable in z with respect to all (µt), t ≥ 0, where µt are +marginal distributions of any weak solution of the equation (1). Then the functions +˜b(t, z) := B[t, z, µt] and ˜σ(t, z) := Σ[t, z, µt] are Borel measurable in (t, z). +Theorem 1. Let the initial value z0 have a finite fourth moment and assume that the +following three conditions are satisfied. Firstly, the functions b and σ are uniformly +bounded, i.e., there exists C > 0 such that for any s, x, y, +|b(s, z, ζ)| + ∥σ(s, z, ζ)∥ ≤ C, +(3) +where |·| stands for the Euclidean norm in Rd for b and ∥·∥ for the ∥σ∥ = +�� +i,j σ2 +ij . +Secondly, the diffusion matrix σ(s, z, ζ) is symmetric and uniformly nondegenerate +in the following sense: there is a value ν > 0 such that +inf +s,z,ζ inf +|λ|=1 λ∗σ(s, z, ζ)λ ≥ ν. +(4) +Thirdly, b(t, x, y, ξ, η) and σ(t, x, y, ξ, η) are continuous with respect to (x, ξ) for each +(t, y, η) with a uniform modulus of continuity ρ(·). Then the equation (1) has a weak +solution on some probability space with a standard d-dimensional Wiener process with +respect to some filtration (Ft, t ≥ 0). +Denote +A[t, z, µ] := ΣΣ∗[t, z, µ]. +2.2 +Proof +1. Let us mollify both coefficients b and σ with respect to all variables by convolutions +in such a way that they become globally Lipschitz in z, ζ, and t. Namely, let +bn(t, z, ζ) = b(t, z, ζ) ∗ ψn(t) ∗ ϕn(x) ∗ ϕn(y) ∗ ϕn(ξ) ∗ ϕn(η), +and +σn(t, z, ζ) = σ(t, z, ζ) ∗ ψn(t) ∗ ϕn(x) ∗ ϕn(y) ∗ ϕn(ξ) ∗ ϕn(η), +3 + +where the sequences ϕn(·) and ψn are defined in a standard way, i.e., as non-negative +C∞ functions with a compact support, integrated to one, and so that this compact +support squeezes to the origin of the corresponding variable as n → ∞; or, in other +words, that they are delta-sequences in the corresponding variables. +Note that, +of course, for every n the smoothed coefficients remain uniformly bounded and all +have the same uniform modulus of continuity with respect to the variables y, η; +also, the smoothed diffusion σ remains uniformly non-degenerate with ellipticity +constants independent of n (however, recall that this nondegeneracy is only valid +along the variable y). While performing the convolution with ψn, it is assumed that +σ(t, z, ζ) ≡ Id×d for t < 0 (this is needed to leave the mollified diffusion acting on +the variable y uniformly nondegenerate for t ≥ 0 near zero), and that b(t, z, ζ) ≡ 0 +for t < 0. +The equation with smoothed coefficients has a strong solution. Even under weaker +linear growth conditions it is explained, for example, in [8, proof of proposition 1], as +well as in many other sources; this is not linked to the non-degeneracy in any way. +2. In a standard way (see, e.g., the proof of [4, theorem 1.6.4]), the estimates uniform +in n follow, +E sup +0≤t≤T +|Zn +t |4 ≤ CT(1 + E|z0|4), +(5) +and +sup +0≤s≤t≤T; t−s≤h +E|Zn +t − Zn +s |4 ≤ CTh2, +(6) +with some constants CT which may be different for different inequalities but do not +depend on n. (In fact, in [4] the assumptions allow a linear growth in x; overall, it +is a very standard material.) +3. Let us introduce new processes (ξn, ηn) =: ζn which are the copies of (Xn, Y n) =: +Zn, that satisfy similar SDEs on some independent probability spaces. In the se- +quel by E3σn(s, Xn +s , ξn +s ) we denote expectation with respect to the third variable ζn +s +conditional on Xn +s , that is, +E3σn(s, Zn +s , ζn +s ) = +� +σn(s, Zn +s , ζ)µζn +s (dζ), +where µζn +s = L(ζn +s ) (here naturally ζ is the variable of integration); likewise, +E3(σn(s, Zn +s , ζn +s ) − σn(s, Z0 +s, ζ0 +s)) +4 + +is another notation for +� +σn(s, Zn +s , ζ)µζn +s (dζ) − +� +σn(s, Zs, ζ)µζ0 +s (dζ), +where µζ +s = L(ζs) for any random variable ζ ∈ R2d; the integral +E3∥σn(s, Zn +s , ηn +s ) − σ(s, Z0 +s, η0 +s)∥2 +is understood as +� +∥σn(s, Zn +s , ζ) − σn(s, Zs, ζ′)∥2P(ηn +s ∈ dζ, η0 +s ∈ dζ′), +if ζn and ζ0 are defined on the same probability space. +Due to the estimates (5)–(6) and by virtue of Skorokhod’s Lemma about a single +probability space and convergence in probability (see [10, §6, ch. 1], or [6, Lemma +2.6.2], or [8, Lemma 4 in the Appendix]) without loss of generality we may and will +assume that not only µn =⇒ µ, but also on some probability space +( ˜Zn +t , ˜ζn +t , ˜W n +t ) +P→ ( ˜Z0 +t , ˜ζ0 +t , ˜W 0 +t ), +n → ∞, +for any t and for some equivalent random processes ( ˜Zn, ˜ζn, ˜W n), generally speaking, +over a sub-sequence. Slightly abusing notations, we denote initial values still by z0 +without tilde. Also, without loss of generality we assume that each process (˜ζn +t , t ≥ 0) +for any n ≥ 1 is independent of ( ˜Zn, ˜W n), as well as their limit ˜ζ0 +t may be chosen to +be independent of the limits ( ˜Z0, ˜W 0) (this follows from the fact that on the original +probability space ηn is independent of (Zn, W n) and on the new probability space +their joint distribution remains the same; hence, independence of ˜ζn is also valid and +in the limit this is still true). See the details in the proof of the Theorem 2.6.1 in [6]. +On independent probability spaces we have, +dξn +t = ηn +t dt, +dηn +t = Bn[t, ζn +t , µt]dt+Σn[t, ζn +t , µt]dW ′,n +t , t ≥ 0, +L(ζn +0 ) = L(z0), (7) +and +d˜ξn +t = ˜ηn +t dt, +d˜ηn +t = Bn[t, ˜ζn +t , µt]dt + Σn[t, ˜ζn +t , µt]d ˜W ′,n +t , t ≥ 0, +L(˜ζn +0 ) = L(z0). +Due to the inequality (6), the same inequality holds for ˜Zn and ˜W n, in particular, +sup +0≤s≤t≤T; t−s≤h +E| ˜Zn +t − ˜Zn +s |4 ≤ CTh2. +(8) +5 + +Due to Kolmogorov’s continuity theorem, it means that all processes ˜Zn may be +regarded as continuous, and ˜W n can be assumed also continuous by the same reason. +Note for the sequel that the bound (5) is also applicable to the process ˜Z: +E sup +0≤t≤T +| ˜Zn +t |4 ≤ CT(1 + E|z0|4) +(9) +because of the equivalence of Z and ˜Z. +Further, due to the independence of the increments of W n after time t of +the sigma-algebra σ(Zn +s , W n +s , s ≤ t), the same property holds true for ˜W n and +σ( ˜Zn +s , ˜W n +s , s ≤ t), as well as for ˜W n and for the completions of the sigma-algebras +σ( ˜Zn +s , ˜W n +s , s ≤ t) which we denote by F (n) +t +. Also, the processes ˜Zn are adapted to +the filtration (F (n) +t +). So, all stochastic integrals which involve ˜Zn and ˜W n are well +defined. The same relates to the processes ˜ζn. +Hence, again by using Skorokhod’s lemma we may choose a subsequence n′ → ∞ +so that we may hope to pass to the limit in the equation +˜Xn′ +t = x0 + +� t +0 +˜Y n′ +s ds, +˜Y n′ +t += y0 + +� t +0 +E3bn′(s, ˜Zn′ +s , ˜ζn′ +s ) ds + +� t +0 +E3σn′(s, ˜Zn′ +s , ˜ζn′ +s )d ˜W n′ +s , +in order to get +˜X0 +t = x0 + +� t +0 +˜Y 0 +s ds, +˜Y 0 +t = y0 + +� t +0 +E3b(s, ˜Z0 +s, ˜ζ0 +s) ds + +� t +0 +E3σ(s, ˜Z0 +s, ˜ζ0 +s)d ˜W 0 +s , +as n′ → ∞, or, equivalently, +˜X0 +t = x0 + +� t +0 +˜Y 0 +s ds, +(10) +˜Y 0 +t = y0 + +� t +0 +B(s, ˜Z0 +s, µs) ds + +� t +0 +Σ(s, ˜Z0 +s, µs)d ˜W 0 +s , +µs = L( ˜Z0 +s). +6 + +First of all, recall that a priori bounds (5) – (6) and (8) hold true with constants +not depending on n. Now, by Skorokhod’s lemma ([10, §6, ch. 1], or [6, Lemma +2.6.2], or [8, Lemma 4 in the Appendix]), we have a sequence of equivalent processes +( ˜Zn′ +t , ˜ζn′ +t , ˜W n′ +t ) on some probability space and a limiting triple ( ˜Z0 +t , ˜ζ0 +t , ˜W 0 +t ) such that +for any t, +( ˜Zn′ +t , ˜ζn′ +t , ˜W n′ +t ) +P→ ( ˜Z0 +t , ˜ζ0 +t , ˜W 0 +t ). +By virtue of the a priori estimates for ˜W n, the process ˜W is continuous and it +is, naturally, a d-dimensional Wiener process. Also, the limits are adapted to the +corresponding filtration ˜Ft := � +n≥1 F (n) +t +and ˜W is a Wiener process with respect to +this filtration. Moreover, by virtue of the uniform estimates (6), the limit ( ˜Z0 +t , ˜ζ0 +t ) +may be also regarded as continuous due to Kolmogorov’s continuity theorem, because +a priori bounds (5) – (6) remain valid for the limiting processes ˜Z, ˜ζ. +4. We have to show that +� t +0 +E3bn′(s, ˜Zn′ +s , ˜ζn′ +s )ds +P→ +� t +0 +E3b(s, ˜Z0 +s, ˜ζ0 +s)ds, +(11) +and +� t +0 +E3σn′(s, ˜Zn′ +s , ˜ζn′ +s )d ˜W n′ +s +P→ +� t +0 +E3σ(s, ˜Z0 +s, ˜ζ0 +s)d ˜W 0 +s , +n′ → ∞. +(12) +We start with the drift term. Let us fix some n0 and let n > n0. We have for any +t ≤ T, +P +����� +� t +0 +� +(E3bn(s, ˜Xn +s , ˜Y n +s , ˜ξn +s , ˜ηn +s )) − (E3b(s, ˜X0 +s, ˜Y 0 +s , ˜ξ0 +s, ˜η0 +s)) +� +ds +���� > c +� +≤P +����� +� t +0 +� +E3bn(s, ˜Xn +s , ˜Y n +s , ˜ξn +s , ˜ηn +s )−(E3bn0(s, ˜Xn +s , ˜Y n +s , ˜ξn +s , ˜ηn +s ) +� +ds +����> c +3 +� ++ P +����� +� t +0 +� +(E3bn0(s, ˜Zn +s , ˜ζn +s )) − (E3bn0(s, ˜Z0 +s, ˜ζ0 +s)) +� +ds +���� > c +3 +� ++P +����� +� t +0 +� +E3bn0(s, ˜X0 +s, ˜Y 0 +s , ˜ξ0 +s, ˜ξ0 +s)))−E3b(s, ˜X0 +s , ˜Y 0 +s , ˜ξ0 +s, ˜ξ0 +s)) +� +ds +����> c +3 +� +=: I1 + I2 + I3. +7 + +Now the idea is that on a finite interval of time on each ω the components ˜Xn +s +and ˜ξn +s are close to certain trajectories of some countable epsilon-net of continuous +(even differentiable) functions in C([0, T]; Rd); denote this net by Nǫ and the union +of its first N elements by NN,ǫ. More than that, since ˜Zn +s and ˜ζn +s are bounded in +probability (uniformly in n) on any bounded interval [0, T], we may take into account +only finitely many elements of this epsilon-net, up to a small enough probability, that +is, for any ǫ > 0 there exists M > 0 such that +sup +n≥1 +P( sup +0≤t≤T +| ˜Zn +t | ∨ |˜ζn +t | > M +� +�� +� +=:An +M,ǫ +) < ǫ, +and there exists N > 0 such that +sup +n≥1 +P( +N +� +k,j=1 +sup +0≤t≤T +| ˜Xn +t − φk +t | ∨ |˜ξn +t − φj +t| > ǫ +� +�� +� +=:Bn +N,ǫ +) < ǫ, +where all φk ∈ Nǫ. The value N may be chosen uniformly with respect to n due to the +a priori bounds (8)–(9) and, moreover, because the trajectories ( ˜Xn +t ) and (˜ξn +t ) admit +H¨older type bounds by the Kolmogorov continuity theorem, see, e.g., [7, theorems +4.8 and 4.6]. Outside these two events An +M,ǫ and Bn +N,ǫ of the total probability not +exceeding 2ǫ we may assume that +sup +0≤t≤T +| ˜Zn +t | ∨ |˜ζn +t | ≤ M, +and +inf +k,j≤N sup +0≤t≤T +| ˜Xn +t − φk +t | ∨ |˜ξn +t − φj +t| ≤ ǫ. +(13) +On the event An +k,j,ǫ := (sup0≤t≤T | ˜Y n +t − φk +t | ∨ |˜ηn +t − φj +t| ≤ ǫ) we have, +���� +� t +0 +E31(An +k,j,ǫ) +� +bn(s, ˜Xn +s , ˜Y n +s , ˜ξn +s , ˜ηn +s )−bn(s, φk +s, ˜Y n +s , φj +s, ˜ηn +s ) +� +ds +����≤tρ(ǫ). +where ρ is the joint modulus of continuity of both coefficients b(s, x, y, ξ, η) and +σ(s, x, y, ξ, η) in x and in ξ. +Similar bounds hold true for the pair ( ˜X0 +t , ˜ξ0 +t ) due to the convergence and because +of the a priori bounds (5). Therefore, there exists M such that +P( sup +0≤t≤T +| ˜Z0 +t | ∨ |˜ζ0 +t | > M +� +�� +� +A0 +M,ǫ +) < ǫ, +8 + +and there exists N > 0 such that +P( +N +� +k,j=1 +sup +0≤t≤T +| ˜X0 +t − φk +t | ∨ |˜ξ0 +t − φj +t| > ǫ +� +�� +� +B0 +N,ǫ +) < ǫ, +where all φk, φj ∈ NN,ǫ. +Replacing ˜Xn +s and ˜ξn +s by nonrandom φ, ψ ∈ NN,ǫ in the integrals like +P +����� +� t +0 +� +E3bn(s, φs, ˜Y n +s , ψs, ˜ηn +s ) − (E3bn0(s, φs, ˜Y n +s , ψs, ˜ηn +s ) +� +ds +���� > c +3 +� +, +(14) +we will be able to apply Krylov’s bounds to show convergence due to the nondegen- +eracy of σ; a similar approach is applicable to the probability +P +����� +� t +0 +� +E3bn0(s, φs, ˜Y 0 +s , ψs, ˜ξ0 +s) − (E3b(s, φs, ˜Y 0 +s , ψs, ˜ξ0 +s) +� +ds +���� > c +3 +� +, +(15) +with the help of Fatou’s lemma, while the difference due to this replacement can be +evaluated by using the modulus of continuity of b in the variables x, ξ. Similarly the +stochastic integrals can be tackled, which is explained in what follows (in the next +steps of the proof). Denote +Dn +M,N,ǫ := Ω \ (An +M,ǫ ∪ Bn +N,ǫ), +n ≥ 0. +Notice that infn≥0 P(Dn +M,N,ǫ) > 1 − 2ǫ and that +1(Dn +M,N,ǫ) sup +0≤t≤T +| ˜Zn +t | ∨ |˜ζn +t | ≤ M, +n ≥ 0, +and +1(Dn +M,N,ǫ) 1 +� +N +� +k,j=1 +sup +0≤t≤T +| ˜Xn +t − φk +t | ∨ |˜ξn +t − φj +t| ≤ ǫ +� += 1(Dn +M,N,ǫ). +Let +Dn,k,j +M,N,ǫ := +� +ω : sup +0≤t≤T +| ˜Xn +t − φk +t | ∨ |˜ξn +t − φj +t| ≤ ǫ +� +. +Denote for a chosen couple (φk, φj) +gn,n0,k,j(s, y, η) := bn(s, φk +s, y, φj +s, η) − bn0(s, φk +s, y, φj +s, η), +9 + +gn,k,j(s, y, η) := bn(s, φk +s, y, φj +s, η) − b(s, φk +s, y, φj +s, η). +Then the first summand I1 may be estimated by the BCM inequality as follows: +I1 ≤ 3 +c E +� T +0 +C E3|bn(s, ˜Zn +s , ˜ζn +s ) − bn0(s, ˜Zn +s , ˜ζn +s )| ds += C E(1(Dn +M,N,ǫ) + 1(An +M,ǫ ∪ Bn +N,ǫ)) +� T +0 +|bn(s, ˜Zn +s , ˜ζn +s ) − bn0(s, ˜Zn +s , ˜ζn +s )| ds. +Due to () and () we have +E1(An +M,ǫ ∪ Bn +N,ǫ) +� T +0 +|bn(s, ˜Zn +s , ˜ζn +s ) − bn0(s, ˜Zn +s , ˜ζn +s )| ds ≤ Cǫ. +So, it remains to evaluate the term +E1(Dn +M,N,ǫ) +� T +0 +|bn(s, ˜Zn +s , ˜ζn +s ) − bn0(s, ˜Zn +s , ˜ζn +s )| ds +≤ +N +� +k,j=1 +E1(Dn,k,j +M,N,ǫ) +� T +0 +|bn(s, ˜Zn +s , ˜ζn +s ) − bn0(s, ˜Zn +s , ˜ζn +s )| ds +We have for any k, j ≤ N +E1(Dn,k,j +M,N,ǫ) +� T +0 +|bn(s, ˜Zn +s , ˜ζn +s ) − bn0(s, ˜Zn +s , ˜ζn +s )| ds +≤ E1(Dn,k,j +M,N,ǫ) +� t +0 +���bn(s, ˜Xn +s , ˜Y n +s , ˜ξn +s , ˜ηn +s ) − bn(s, φk +s, ˜Y n +s , φj +s, ˜ηn +s ) +��� ds ++ E1(Dn,k,j +M,N,ǫ) +� t +0 +���bn0(s, ˜Xn +s , ˜Y n +s , ˜ξn +s , ˜ηn +s ) − bn0(s, φk +s, ˜Y n +s , φj +s, ˜ηn +s ) +��� ds ++ E1(Dn,k,j +M,N,ǫ) +� t +0 +���bn(s, φk +s, ˜Y n +s , φj +s, ˜ηn +s ) − bn0(s, φk +s, ˜Y n +s , φj +s, ˜ηn +s ) +��� ds +10 + +≤ Cǫ + E1(Dn,k,j +M,N,ǫ) +� t +0 +���bn(s, φk +s, ˜Y n +s , φj +s, ˜ξn +s ) − bn0(s, φk +s, ˜Y n +s , φj +s, ˜ηn +s ) +��� ds. +By virtue of Krylov’s estimate (see the Theorems 2.4.1 or 2.3.4 in [6]) +E1(Dn,k,j +M,N,ǫ) +� t +0 +���bn(s, φk +s, ˜Y n +s , φj +s, ˜ηn +s ) − bn0(s, φk +s, ˜Y n +s , φj +s, ˜ηn +s ) +��� ds += E1(Dn,k,j +M,N,ǫ) +� t +0 +|g|n,n0,k,j(s, ˜Y n +s , ˜ηn +s )ds +(16) +≤ NR +� +∥gn,k,j∥L2d+1([0,T]×BR×BR) + ∥gn0,k,j∥L2d+1([0,T]×BR×BR) +� +→ 0, +as n, n0 → ∞ for each R, because of the well-known property of mollified functions. +Hence, overall, we obtain that +I1 → 0, +n, n0 → ∞. +Further, the second term admits the bound +I2 = P +����� +� t +0 +� +(E3bn0(s, ˜Zn +s , ˜ζn +s )) − (E3bn0(s, ˜Z0 +s, ˜ζ0 +s)) +� +ds +���� > c +3 +� +≤ CE +� t +0 +E3 ���bn0(s, ˜Zn +s , ˜ζn +s )) − (bn0(s, ˜Z0 +s, ˜ζ0 +s)) +��� ds +≤ CE +� t +0 +���bn0(s, ˜Zn +s , ˜ζn +s )) − (bn0(s, ˜Z0 +s, ˜ζ0 +s)) +��� ds → 0, +n → ∞, +due to the Lebesgue bounded convergence theorem. Hence, for each n0 +lim +n→∞ I2 = 0, +and therefore +lim +n0→∞ lim +n→∞ I2 = 0. +11 + +The term I3 can be considered similarly to I1 with just one nuance that it is +not known in advance whether or not the limiting processes ˜X0, ˜ζ0 are diffusions. +However, it is explained in [6, section II.6]; see also some details in [8, proof of +inequality (2.17)]. The main point is the extension to the limiting process ( ˜Z0, ˜ζ0) +of Krylov’s bound for diffusions ( ˜Zn, ˜ζn), n ≥ 1: (1◦) as a first step these bounds +are proved for the limiting process in the argument of continuous functions from +L2d+1, and (2◦) as a second step this extension is generalised to any nonnegative +Borel measurable functions using the property of regularity of probability measures +in finite-dimensional Euclidean spaces. The details may be read in the cited sources. +Hence, by the properties of the mollified functions it follows that +lim +n0→∞ I3 = 0. +The convergence (11) is, thus, proved. +5. Let us show for stochastic integrals in (12) that for any c, ǫ > 0 there exists C > 0 +such that +P +����� +� t +0 +(E3σn(s, ˜Zn +s , ˜ζn +s ))d ˜W n +s − +� t +0 +(E3σ(s, ˜Z0 +s, ˜ζ0 +s))d ˜W 0 +s +���� > c +� +< Cǫ, +(17) +if n is large enough. The task is similar to the convergence of Lebesgue integrals +related to the coefficient b studied in the previous steps of the proof. The additional +obstacle is to show convergence of the difference of stochastic integrals driven by dif- +ferent Wiener processes ˜W n and ˜W 0 in (17) with continuous and bounded integrands +f n +s := E3σn0(s, ˜Xn +s , ˜ξn +s ) and f 0 +s := E3σn0(s, ˜Xs, ˜ξs) in Skorokhod’s lemma, which does +require such a continuity and boundedness. Some details are as follows: +P +����� +� t +0 +(E3σn(s, ˜Xn +s , ˜ξn +s ))d ˜W n +s − +� t +0 +(E3σ(s, ˜Xs, ˜ξs))d ˜Ws +���� > c +� +≤ P +����� +� t +0 +(E3σn(s, ˜Xn +s , ˜ξn +s ))d ˜W n +s − +� t +0 +(E3σn0(s, ˜Xn +s , ˜ξn +s ))d ˜W n +s +���� > c/3 +� ++ P +����� +� t +0 +(E3σn0(s, ˜Xn +s , ˜ξn +s ))d ˜W n +s − +� t +0 +(E3σn0(s, ˜Xs, ˜ξs))d ˜Ws +���� > c/3 +� +12 + ++ P +����� +� t +0 +(E3σn0(s, ˜Xs, ˜ξs))d ˜Ws − +� t +0 +(E3σ(s, ˜Xs, ˜ξs))d ˜Ws +���� > c/3 +� +≤ CE +���� +� t +0 +E3(σn(s, ˜Xn +s , ˜ξn +s )) − σn0(s, ˜Xn +s , ˜ξn +s ))d ˜W n +s +���� +2 ++ CE +���� +� t +0 +(E3σn0(s, ˜Xn +s , ˜ξn +s ))d ˜W n +s − +� t +0 +(E3σn0(s, ˜Xs, ˜ξs))d ˜Ws +���� ++ CE +���� +� t +0 +E3(σn0(s, ˜Xs, ˜ξs)) − σ(s, ˜Xs, ˜ξs))d ˜Ws +���� +2 +≤ CE +� t +0 +E3 ���(σn(s, ˜Xn +s , ˜ξn +s )) − σn0(s, ˜Xn +s , ˜ξn +s )) +��� +2 +ds ++ CE +������� +� t +0 +E3σn0(s, ˜Zn +s , ˜ζn +s ) +� +�� +� +:=fn(s,ω) +d ˜W n +s − +� t +0 +E3σn0(s, ˜Z0 +s, ˜ζ0 +s) +� +�� +� +:=f0(s,ω) +d ˜W 0 +s +������� ++ CE +� t +0 +E3 ���(σn0(s, ˜Xs, ˜ξs)) − σ(s, ˜Xs, ˜ξs)) +��� +2 +ds +=: J1 + J2 + J3. +The terms J1 and J3 are tackled similarly to I1 and I3 from the previous steps. +The additional difficulty discussed above relates to the term J2. Skorokhod’s lemma +(see [8, Lemma 4 in the Appendix]) is applicable if for bounded and (stochastically) +continuous in s integrands f n := E3σn0(s, ˜Zn +s , ˜ζn +s ) uniformly with respect to n ≥ 0 it +holds that +f n(s, ω) +P→ f 0(s, ω), +a.e. s ≤ T +n → ∞. +(18) +The fact that f n is uniformly bounded follows straightforwardly from the bounded- +ness of the function σ. Further, f n(s, ω) is continuous in s a.s. uniformly in n ≥ 0 +because of the continuity of σn0 in all variables and due to the uniform stochastic +continuity of all processes ˜Xn +s for n ≥ 0 (see (8)). Finally, the convergence (18) in +13 + +probability for all s ≤ T follows from the following little calculus: +E∥f n(s, ω) − f 0(s, ω)∥ = E∥E3σn0(s, ˜Zn +s , ˜ζn +s ) − E3σn0(s, ˜Z0 +s, ˜ζ0 +s)∥ += E∥E3 � +σn0(s, ˜Zn +s , ˜ζn +s ) − σn0(s, ˜Z0 +s, ˜ζ0 +s) +� +∥ ≤ EE3∥σn0(s, ˜Zn +s , ˜ζn +s ) − E3σn0(s, ˜Z0 +s, ˜ζ0 +s)∥ += E∥σn0(s, ˜Zn +s , ˜ζn +s ) − E3σn0(s, ˜Z0 +s, ˜ζ0 +s)∥ → 0, +n → ∞, +the latter convergence by virtue of Lebesgue’s bounded convergence theorem. Hence, +in this way we obtain that +J1 + J2 + J3 → 0, +n, n0 → ∞. +So, the convergence (12) holds true, which along with (11) leads to the equation (10) +and, hence, completes the proof of the theorem. +QED +References +[1] R. Dobrushin, Vlasov equations. Funct. Anal. Appl., 1979, 13, 115–123. +https://doi.org/10.1007/BF01077243 +[2] H. Dong, T. Yastrzhembsky, Global Lp estimates for kinetic Kolmogorov-Fokker- +Planck equations in nondivergence form, https://arxiv.org/abs/2107.08568 +[3] T. Funaki, A certain class of diffusion processes associated with nonlin- +ear parabolic equations. Z. Wahrsch. Verw. Gebiete, 1984, 67(3), 331–348. +https://doi.org/10.1007/BF00535008 +[4] I.I. Gihman, A.V. Skorohod, Stochastic differential equations, Springer, Berlin, +1972. https://books.google.ru/books?id=UZ7fMQAACAAJ +[5] N.V. Krylov. On Ito’s stochastic integral equations. Theory Probab. Appl., 14 +(1969), 330-336; Addendum: On Ito’s stochastic integral equations, ibid., 17(2) +(1973), 373-374. https://doi.org/10.1137/1114042 +[6] N.V. Krylov, Controlled diffusion processes, Springer, New York, 2009 (the +reprint of the 1st edition 1980). https://doi.org/10.1007/978-3-540-70914-5 1 +[7] N.V. Krylov, Introduction to the Theory of Random Processes, AMS, Provi- +dence, R.I., 2002. DOI:10.1090/gsm/043 +14 + +[8] Yu.S. Mishura, A.Yu. Veretennikov, Existence and uniqueness theorems for so- +lutions of McKean–Vlasov stochastic equations, Theor. Probability and Math. +Statist. 2020, 103, 59–101. DOI: https://doi.org/10.1090/tpms/1135 +[9] M. +Nisio, +On +the +existence +of +solutions +of +stochastic +dif- +ferential +equations, +Osaka +J. +Math. +1973, +10(1), +185–208. +https://projecteuclid.org/download/imagefirstpage 1/euclid.ojm/1200694133 +[10] A.V. +Skorokhod, +Studies +in +the +theory +of +random +processes. +Addison-Wesley +Publishing +Co., +Inc., +Reading, +Mass., +1965. +https://books.google.ru/books/about/Studies in the Theory of Random +Processe.html?id=4X0zDwAAQBAJ&redir esc=y +[11] A.-S. Sznitman, Topics in propagation of chaos. In ´Ecole d’´Et´e de Probabilit´es +de Saint-Flour XIX—1989, volume 1464 of Lecture Notes in Math., pages 165– +251. Springer, Berlin, 1991. https://doi.org/10.1007/BFb0085169 +[12] A.Yu. +Veretennikov, +On +Weak +Solutions +of +Highly +Degenerate +SDEs, +Automation +and +Remote +Control, +2020, +81(3), +398-410. +DOI +10.1134/S0005117920030029 +[13] A.A. Vlasov, The vibrational properties of an electron gas. Physics-Uspekhi, +1968, 10(6):721–733. DOI 10.1070/PU1968v010n06ABEH003709 +15 + diff --git a/LtAzT4oBgHgl3EQfkf1m/content/tmp_files/load_file.txt b/LtAzT4oBgHgl3EQfkf1m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1edd74dd1c90c88e033809951a6d894759e63dd --- /dev/null +++ b/LtAzT4oBgHgl3EQfkf1m/content/tmp_files/load_file.txt @@ -0,0 +1,444 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf,len=443 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='01532v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='PR] 4 Jan 2023 ver_mv06122022preprint_degenerate_d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='tex On weak existence of solutions of degenerate McKean–Vlasov equations A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Veretennikov∗ January 5, 2023 Abstract A new weak existence result for degenerate multi-dimensional stochastic McKean–Vlasov equation is established under relaxed regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Keywords: McKean-Vlasov equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' degenerate diffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' MSC: 60J60 1 Introduction The subject of this paper is solutions of the stochastic Itˆo-McKean-Vlasov (McKean- Vlasov) equation in R2d dXt = Ytdt, dYt = B[t, Zt, µt]dt + Σ[t, Zt, µt]dWt, X0 = x0, Y0 = y0, (1) where Zt = (Xt, Yt) ∈ R2d, in a particular situation called the true McKean-Vlasov case under the convention B[t, z, µ] = � b(t, z, ζ)µ(dζ), Σ[t, z, µ] = � σ(t, z, ζ)µ(dζ), (2) ∗Institute for Information Transmission Problems, Moscow, Russian Federation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' email: ayv@iitp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' This research was funded by the RFBR grant 20-01-00575a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 1 where z = (x, y) ∈ R2d and ζ = (ξ, η) ∈ R2d, and under certain non-degeneracy assumptions on σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Here W is a standard d-dimensional Wiener process, b and σ are vector and matrix Borel functions of corresponding dimensions d and d × d, µt is the distribution of the process X at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The initial data z0 = (x0, y0) may be random and in this case it is independent of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Vlasov’s proposal was a substitution of a real multiparticle interaction by a certain “mean field” [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The introduction to the whole topic in its stochastic version may be found in [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' one more important reference, although devoted soleyly to the deterministic setting is [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' In this paper we investigate only the problem of weak existence for genuinly degenerate stochastic McKean – Vlasov equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The equations like (1) naturally arise in mechanical systems with stochastic forces or noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The aim of this paper is to show weak exis- tence for a such a degenerate SDE system simultaneously minimizing the regularity assumptions on both coefficients with respect to all variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' We assume the non- degeneracy of σ and highlight that this non-degeneracy only holds for the second component Y of the system (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The interest to the minimal regularity is mainly due to the control problems where the optimal strategies are usually discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Among the most important works on the subject there is the paper [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The study in the present paper is based on Krylov’s bounds [6] and on the approach proposed in [9] for the ordinary Itˆo SDEs which was further generalised to some extent in [12] also for the ordinary Itˆo SDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Other useful references may be found in the cited papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' In the section 2 weak existence is stated under appropriate conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The section 3 containts its proof based on a combi- nation of Krylov’s bounds and on Krylov’s existence results for the nondegenerate Ito’s equations [5], [6], and on Nisio’s weak existence proof also for Ito’s SDEs [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The degeneracy of the diffusion is overcome still by using Krylov’s bounds for non- degenerate Itˆo processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' No regularity of the coefficients b and σ is assumed with respect to the variables t, y, and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Uniform continuity is assumed for both b and σ with respect to the variables x and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The abbreviation by CBS signifies the Cauchy- Buniakovsky-Schwarz inequality and BCM stands the Bienaym´e-Chebyshev-Markov inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 2 Weak existence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='1 Main results Let us recall a fact from functional analysis useful for the case (1)– (2), see, for example, (see [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The proposition 1 and its corollary are stated in 2 a slightly more general form than what is needed for bounded coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' For any Borel function f(z, ζ) and any probability measure µ(dζ) such that f(z, ·) is integrable with respect to this measure, the function f[z, µ] := � f(z, ζ) µ(dζ) is Borel measurable in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Suppose for each (t, z) the Borel coefficients b(t, z, ζ) and σ(t, z, ζ) are bounded in ζ and integrable in z with respect to all (µt), t ≥ 0, where µt are marginal distributions of any weak solution of the equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Then the functions ˜b(t, z) := B[t, z, µt] and ˜σ(t, z) := Σ[t, z, µt] are Borel measurable in (t, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Let the initial value z0 have a finite fourth moment and assume that the following three conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Firstly, the functions b and σ are uniformly bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=', there exists C > 0 such that for any s, x, y, |b(s, z, ζ)| + ∥σ(s, z, ζ)∥ ≤ C, (3) where |·| stands for the Euclidean norm in Rd for b and ∥·∥ for the ∥σ∥ = �� i,j σ2 ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Secondly, the diffusion matrix σ(s, z, ζ) is symmetric and uniformly nondegenerate in the following sense: there is a value ν > 0 such that inf s,z,ζ inf |λ|=1 λ∗σ(s, z, ζ)λ ≥ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' (4) Thirdly, b(t, x, y, ξ, η) and σ(t, x, y, ξ, η) are continuous with respect to (x, ξ) for each (t, y, η) with a uniform modulus of continuity ρ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Then the equation (1) has a weak solution on some probability space with a standard d-dimensional Wiener process with respect to some filtration (Ft, t ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Denote A[t, z, µ] := ΣΣ∗[t, z, µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='2 Proof 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Let us mollify both coefficients b and σ with respect to all variables by convolutions in such a way that they become globally Lipschitz in z, ζ, and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Namely, let bn(t, z, ζ) = b(t, z, ζ) ∗ ψn(t) ∗ ϕn(x) ∗ ϕn(y) ∗ ϕn(ξ) ∗ ϕn(η), and σn(t, z, ζ) = σ(t, z, ζ) ∗ ψn(t) ∗ ϕn(x) ∗ ϕn(y) ∗ ϕn(ξ) ∗ ϕn(η), 3 where the sequences ϕn(·) and ψn are defined in a standard way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=', as non-negative C∞ functions with a compact support, integrated to one, and so that this compact support squeezes to the origin of the corresponding variable as n → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' or, in other words, that they are delta-sequences in the corresponding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Note that, of course, for every n the smoothed coefficients remain uniformly bounded and all have the same uniform modulus of continuity with respect to the variables y, η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' also, the smoothed diffusion σ remains uniformly non-degenerate with ellipticity constants independent of n (however, recall that this nondegeneracy is only valid along the variable y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' While performing the convolution with ψn, it is assumed that σ(t, z, ζ) ≡ Id×d for t < 0 (this is needed to leave the mollified diffusion acting on the variable y uniformly nondegenerate for t ≥ 0 near zero), and that b(t, z, ζ) ≡ 0 for t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The equation with smoothed coefficients has a strong solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Even under weaker linear growth conditions it is explained, for example, in [8, proof of proposition 1], as well as in many other sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' this is not linked to the non-degeneracy in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' In a standard way (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=', the proof of [4, theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='4]), the estimates uniform in n follow, E sup 0≤t≤T |Zn t |4 ≤ CT(1 + E|z0|4), (5) and sup 0≤s≤t≤T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' t−s≤h E|Zn t − Zn s |4 ≤ CTh2, (6) with some constants CT which may be different for different inequalities but do not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' (In fact, in [4] the assumptions allow a linear growth in x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' overall, it is a very standard material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Let us introduce new processes (ξn, ηn) =: ζn which are the copies of (Xn, Y n) =: Zn, that satisfy similar SDEs on some independent probability spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' In the se- quel by E3σn(s, Xn s , ξn s ) we denote expectation with respect to the third variable ζn s conditional on Xn s , that is, E3σn(s, Zn s , ζn s ) = � σn(s, Zn s , ζ)µζn s (dζ), where µζn s = L(ζn s ) (here naturally ζ is the variable of integration);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' likewise, E3(σn(s, Zn s , ζn s ) − σn(s, Z0 s, ζ0 s)) 4 is another notation for � σn(s, Zn s , ζ)µζn s (dζ) − � σn(s, Zs, ζ)µζ0 s (dζ), where µζ s = L(ζs) for any random variable ζ ∈ R2d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' the integral E3∥σn(s, Zn s , ηn s ) − σ(s, Z0 s, η0 s)∥2 is understood as � ∥σn(s, Zn s , ζ) − σn(s, Zs, ζ′)∥2P(ηn s ∈ dζ, η0 s ∈ dζ′), if ζn and ζ0 are defined on the same probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Due to the estimates (5)–(6) and by virtue of Skorokhod’s Lemma about a single probability space and convergence in probability (see [10, §6, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 1], or [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='2], or [8, Lemma 4 in the Appendix]) without loss of generality we may and will assume that not only µn =⇒ µ, but also on some probability space ( ˜Zn t , ˜ζn t , ˜W n t ) P→ ( ˜Z0 t , ˜ζ0 t , ˜W 0 t ), n → ∞, for any t and for some equivalent random processes ( ˜Zn, ˜ζn, ˜W n), generally speaking, over a sub-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Slightly abusing notations, we denote initial values still by z0 without tilde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Also, without loss of generality we assume that each process (˜ζn t , t ≥ 0) for any n ≥ 1 is independent of ( ˜Zn, ˜W n), as well as their limit ˜ζ0 t may be chosen to be independent of the limits ( ˜Z0, ˜W 0) (this follows from the fact that on the original probability space ηn is independent of (Zn, W n) and on the new probability space their joint distribution remains the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' hence, independence of ˜ζn is also valid and in the limit this is still true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' See the details in the proof of the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='1 in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' On independent probability spaces we have, dξn t = ηn t dt, dηn t = Bn[t, ζn t , µt]dt+Σn[t, ζn t , µt]dW ′,n t , t ≥ 0, L(ζn 0 ) = L(z0), (7) and d˜ξn t = ˜ηn t dt, d˜ηn t = Bn[t, ˜ζn t , µt]dt + Σn[t, ˜ζn t , µt]d ˜W ′,n t , t ≥ 0, L(˜ζn 0 ) = L(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Due to the inequality (6), the same inequality holds for ˜Zn and ˜W n, in particular, sup 0≤s≤t≤T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' t−s≤h E| ˜Zn t − ˜Zn s |4 ≤ CTh2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' (8) 5 Due to Kolmogorov’s continuity theorem, it means that all processes ˜Zn may be regarded as continuous, and ˜W n can be assumed also continuous by the same reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Note for the sequel that the bound (5) is also applicable to the process ˜Z: E sup 0≤t≤T | ˜Zn t |4 ≤ CT(1 + E|z0|4) (9) because of the equivalence of Z and ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Further, due to the independence of the increments of W n after time t of the sigma-algebra σ(Zn s , W n s , s ≤ t), the same property holds true for ˜W n and σ( ˜Zn s , ˜W n s , s ≤ t), as well as for ˜W n and for the completions of the sigma-algebras σ( ˜Zn s , ˜W n s , s ≤ t) which we denote by F (n) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Also, the processes ˜Zn are adapted to the filtration (F (n) t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' So, all stochastic integrals which involve ˜Zn and ˜W n are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The same relates to the processes ˜ζn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' again by using Skorokhod’s lemma we may choose a subsequence n′ → ∞ so that we may hope to pass to the limit in the equation ˜Xn′ t = x0 + � t 0 ˜Y n′ s ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n′ t = y0 + � t 0 E3bn′(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn′ s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn′ s ) ds + � t 0 E3σn′(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn′ s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn′ s )d ˜W n′ s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' in order to get ˜X0 t = x0 + � t 0 ˜Y 0 s ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y 0 t = y0 + � t 0 E3b(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Z0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζ0 s) ds + � t 0 E3σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Z0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζ0 s)d ˜W 0 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' as n′ → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' equivalently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜X0 t = x0 + � t 0 ˜Y 0 s ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' (10) ˜Y 0 t = y0 + � t 0 B(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Z0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' µs) ds + � t 0 Σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Z0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' µs)d ˜W 0 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' µs = L( ˜Z0 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 6 First of all, recall that a priori bounds (5) – (6) and (8) hold true with constants not depending on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Now, by Skorokhod’s lemma ([10, §6, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 1], or [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='2], or [8, Lemma 4 in the Appendix]), we have a sequence of equivalent processes ( ˜Zn′ t , ˜ζn′ t , ˜W n′ t ) on some probability space and a limiting triple ( ˜Z0 t , ˜ζ0 t , ˜W 0 t ) such that for any t, ( ˜Zn′ t , ˜ζn′ t , ˜W n′ t ) P→ ( ˜Z0 t , ˜ζ0 t , ˜W 0 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' By virtue of the a priori estimates for ˜W n, the process ˜W is continuous and it is, naturally, a d-dimensional Wiener process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Also, the limits are adapted to the corresponding filtration ˜Ft := � n≥1 F (n) t and ˜W is a Wiener process with respect to this filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Moreover, by virtue of the uniform estimates (6), the limit ( ˜Z0 t , ˜ζ0 t ) may be also regarded as continuous due to Kolmogorov’s continuity theorem, because a priori bounds (5) – (6) remain valid for the limiting processes ˜Z, ˜ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' We have to show that � t 0 E3bn′(s, ˜Zn′ s , ˜ζn′ s )ds P→ � t 0 E3b(s, ˜Z0 s, ˜ζ0 s)ds, (11) and � t 0 E3σn′(s, ˜Zn′ s , ˜ζn′ s )d ˜W n′ s P→ � t 0 E3σ(s, ˜Z0 s, ˜ζ0 s)d ˜W 0 s , n′ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' (12) We start with the drift term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Let us fix some n0 and let n > n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' We have for any t ≤ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' P ����� � t 0 � (E3bn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s )) − (E3b(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜X0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y 0 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξ0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜η0 s)) � ds ���� > c � ≤P ����� � t 0 � E3bn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s )−(E3bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s ) � ds ����> c 3 � + P ����� � t 0 � (E3bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn s )) − (E3bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Z0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζ0 s)) � ds ���� > c 3 � +P ����� � t 0 � E3bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜X0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y 0 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξ0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξ0 s)))−E3b(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜X0 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y 0 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξ0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξ0 s)) � ds ����> c 3 � =: I1 + I2 + I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 7 Now the idea is that on a finite interval of time on each ω the components ˜Xn s and ˜ξn s are close to certain trajectories of some countable epsilon-net of continuous (even differentiable) functions in C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Rd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' denote this net by Nǫ and the union of its first N elements by NN,ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' More than that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' since ˜Zn s and ˜ζn s are bounded in probability (uniformly in n) on any bounded interval [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' we may take into account only finitely many elements of this epsilon-net,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' up to a small enough probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' for any ǫ > 0 there exists M > 0 such that sup n≥1 P( sup 0≤t≤T | ˜Zn t | ∨ |˜ζn t | > M � �� � =:An M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ǫ ) < ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' and there exists N > 0 such that sup n≥1 P( N � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='j=1 sup 0≤t≤T | ˜Xn t − φk t | ∨ |˜ξn t − φj t| > ǫ � �� � =:Bn N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ǫ ) < ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' where all φk ∈ Nǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The value N may be chosen uniformly with respect to n due to the a priori bounds (8)–(9) and, moreover, because the trajectories ( ˜Xn t ) and (˜ξn t ) admit H¨older type bounds by the Kolmogorov continuity theorem, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=', [7, theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='8 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Outside these two events An M,ǫ and Bn N,ǫ of the total probability not exceeding 2ǫ we may assume that sup 0≤t≤T | ˜Zn t | ∨ |˜ζn t | ≤ M, and inf k,j≤N sup 0≤t≤T | ˜Xn t − φk t | ∨ |˜ξn t − φj t| ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' (13) On the event An k,j,ǫ := (sup0≤t≤T | ˜Y n t − φk t | ∨ |˜ηn t − φj t| ≤ ǫ) we have, ���� � t 0 E31(An k,j,ǫ) � bn(s, ˜Xn s , ˜Y n s , ˜ξn s , ˜ηn s )−bn(s, φk s, ˜Y n s , φj s, ˜ηn s ) � ds ����≤tρ(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' where ρ is the joint modulus of continuity of both coefficients b(s, x, y, ξ, η) and σ(s, x, y, ξ, η) in x and in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Similar bounds hold true for the pair ( ˜X0 t , ˜ξ0 t ) due to the convergence and because of the a priori bounds (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Therefore, there exists M such that P( sup 0≤t≤T | ˜Z0 t | ∨ |˜ζ0 t | > M � �� � A0 M,ǫ ) < ǫ, 8 and there exists N > 0 such that P( N � k,j=1 sup 0≤t≤T | ˜X0 t − φk t | ∨ |˜ξ0 t − φj t| > ǫ � �� � B0 N,ǫ ) < ǫ, where all φk, φj ∈ NN,ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Replacing ˜Xn s and ˜ξn s by nonrandom φ, ψ ∈ NN,ǫ in the integrals like P ����� � t 0 � E3bn(s, φs, ˜Y n s , ψs, ˜ηn s ) − (E3bn0(s, φs, ˜Y n s , ψs, ˜ηn s ) � ds ���� > c 3 � , (14) we will be able to apply Krylov’s bounds to show convergence due to the nondegen- eracy of σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' a similar approach is applicable to the probability P ����� � t 0 � E3bn0(s, φs, ˜Y 0 s , ψs, ˜ξ0 s) − (E3b(s, φs, ˜Y 0 s , ψs, ˜ξ0 s) � ds ���� > c 3 � , (15) with the help of Fatou’s lemma, while the difference due to this replacement can be evaluated by using the modulus of continuity of b in the variables x, ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Similarly the stochastic integrals can be tackled, which is explained in what follows (in the next steps of the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Denote Dn M,N,ǫ := Ω \\ (An M,ǫ ∪ Bn N,ǫ), n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Notice that infn≥0 P(Dn M,N,ǫ) > 1 − 2ǫ and that 1(Dn M,N,ǫ) sup 0≤t≤T | ˜Zn t | ∨ |˜ζn t | ≤ M, n ≥ 0, and 1(Dn M,N,ǫ) 1 � N � k,j=1 sup 0≤t≤T | ˜Xn t − φk t | ∨ |˜ξn t − φj t| ≤ ǫ � = 1(Dn M,N,ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Let Dn,k,j M,N,ǫ := � ω : sup 0≤t≤T | ˜Xn t − φk t | ∨ |˜ξn t − φj t| ≤ ǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Denote for a chosen couple (φk, φj) gn,n0,k,j(s, y, η) := bn(s, φk s, y, φj s, η) − bn0(s, φk s, y, φj s, η), 9 gn,k,j(s, y, η) := bn(s, φk s, y, φj s, η) − b(s, φk s, y, φj s, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Then the first summand I1 may be estimated by the BCM inequality as follows: I1 ≤ 3 c E � T 0 C E3|bn(s, ˜Zn s , ˜ζn s ) − bn0(s, ˜Zn s , ˜ζn s )| ds = C E(1(Dn M,N,ǫ) + 1(An M,ǫ ∪ Bn N,ǫ)) � T 0 |bn(s, ˜Zn s , ˜ζn s ) − bn0(s, ˜Zn s , ˜ζn s )| ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Due to () and () we have E1(An M,ǫ ∪ Bn N,ǫ) � T 0 |bn(s, ˜Zn s , ˜ζn s ) − bn0(s, ˜Zn s , ˜ζn s )| ds ≤ Cǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' it remains to evaluate the term E1(Dn M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ǫ) � T 0 |bn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn s ) − bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn s )| ds ≤ N � k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='j=1 E1(Dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='j M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ǫ) � T 0 |bn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn s ) − bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn s )| ds We have for any k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' j ≤ N E1(Dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='j M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ǫ) � T 0 |bn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn s ) − bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn s )| ds ≤ E1(Dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='j M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ǫ) � t 0 ���bn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s ) − bn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φk s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φj s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s ) ��� ds + E1(Dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='j M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ǫ) � t 0 ���bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s ) − bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φk s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φj s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s ) ��� ds + E1(Dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='j M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ǫ) � t 0 ���bn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φk s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φj s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s ) − bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φk s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φj s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s ) ��� ds 10 ≤ Cǫ + E1(Dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='j M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ǫ) � t 0 ���bn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φk s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φj s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ) − bn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φk s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Y n s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' φj s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ηn s ) ��� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' By virtue of Krylov’s estimate (see the Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='4 in [6]) E1(Dn,k,j M,N,ǫ) � t 0 ���bn(s, φk s, ˜Y n s , φj s, ˜ηn s ) − bn0(s, φk s, ˜Y n s , φj s, ˜ηn s ) ��� ds = E1(Dn,k,j M,N,ǫ) � t 0 |g|n,n0,k,j(s, ˜Y n s , ˜ηn s )ds (16) ≤ NR � ∥gn,k,j∥L2d+1([0,T]×BR×BR) + ∥gn0,k,j∥L2d+1([0,T]×BR×BR) � → 0, as n, n0 → ∞ for each R, because of the well-known property of mollified functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Hence, overall, we obtain that I1 → 0, n, n0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Further, the second term admits the bound I2 = P ����� � t 0 � (E3bn0(s, ˜Zn s , ˜ζn s )) − (E3bn0(s, ˜Z0 s, ˜ζ0 s)) � ds ���� > c 3 � ≤ CE � t 0 E3 ���bn0(s, ˜Zn s , ˜ζn s )) − (bn0(s, ˜Z0 s, ˜ζ0 s)) ��� ds ≤ CE � t 0 ���bn0(s, ˜Zn s , ˜ζn s )) − (bn0(s, ˜Z0 s, ˜ζ0 s)) ��� ds → 0, n → ∞, due to the Lebesgue bounded convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Hence, for each n0 lim n→∞ I2 = 0, and therefore lim n0→∞ lim n→∞ I2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 11 The term I3 can be considered similarly to I1 with just one nuance that it is not known in advance whether or not the limiting processes ˜X0, ˜ζ0 are diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' However, it is explained in [6, section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' see also some details in [8, proof of inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='17)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The main point is the extension to the limiting process ( ˜Z0, ˜ζ0) of Krylov’s bound for diffusions ( ˜Zn, ˜ζn), n ≥ 1: (1◦) as a first step these bounds are proved for the limiting process in the argument of continuous functions from L2d+1, and (2◦) as a second step this extension is generalised to any nonnegative Borel measurable functions using the property of regularity of probability measures in finite-dimensional Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The details may be read in the cited sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Hence, by the properties of the mollified functions it follows that lim n0→∞ I3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The convergence (11) is, thus, proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Let us show for stochastic integrals in (12) that for any c, ǫ > 0 there exists C > 0 such that P ����� � t 0 (E3σn(s, ˜Zn s , ˜ζn s ))d ˜W n s − � t 0 (E3σ(s, ˜Z0 s, ˜ζ0 s))d ˜W 0 s ���� > c � < Cǫ, (17) if n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The task is similar to the convergence of Lebesgue integrals related to the coefficient b studied in the previous steps of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The additional obstacle is to show convergence of the difference of stochastic integrals driven by dif- ferent Wiener processes ˜W n and ˜W 0 in (17) with continuous and bounded integrands f n s := E3σn0(s, ˜Xn s , ˜ξn s ) and f 0 s := E3σn0(s, ˜Xs, ˜ξs) in Skorokhod’s lemma, which does require such a continuity and boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Some details are as follows: P ����� � t 0 (E3σn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ))d ˜W n s − � t 0 (E3σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξs))d ˜Ws ���� > c � ≤ P ����� � t 0 (E3σn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ))d ˜W n s − � t 0 (E3σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ))d ˜W n s ���� > c/3 � + P ����� � t 0 (E3σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ))d ˜W n s − � t 0 (E3σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξs))d ˜Ws ���� > c/3 � 12 + P ����� � t 0 (E3σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξs))d ˜Ws − � t 0 (E3σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξs))d ˜Ws ���� > c/3 � ≤ CE ���� � t 0 E3(σn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s )) − σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ))d ˜W n s ���� 2 + CE ���� � t 0 (E3σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s ))d ˜W n s − � t 0 (E3σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξs))d ˜Ws ���� + CE ���� � t 0 E3(σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξs)) − σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξs))d ˜Ws ���� 2 ≤ CE � t 0 E3 ���(σn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s )) − σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξn s )) ��� 2 ds + CE ������� � t 0 E3σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Zn s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζn s ) � �� � :=fn(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ω) d ˜W n s − � t 0 E3σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Z0 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ζ0 s) � �� � :=f0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='ω) d ˜W 0 s ������� + CE � t 0 E3 ���(σn0(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξs)) − σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' ˜ξs)) ��� 2 ds =: J1 + J2 + J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The terms J1 and J3 are tackled similarly to I1 and I3 from the previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' The additional difficulty discussed above relates to the term J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Skorokhod’s lemma (see [8, Lemma 4 in the Appendix]) is applicable if for bounded and (stochastically) continuous in s integrands f n := E3σn0(s, ˜Zn s , ˜ζn s ) uniformly with respect to n ≥ 0 it holds that f n(s, ω) P→ f 0(s, ω), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' s ≤ T n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' (18) The fact that f n is uniformly bounded follows straightforwardly from the bounded- ness of the function σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Further, f n(s, ω) is continuous in s a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' uniformly in n ≥ 0 because of the continuity of σn0 in all variables and due to the uniform stochastic continuity of all processes ˜Xn s for n ≥ 0 (see (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Finally, the convergence (18) in 13 probability for all s ≤ T follows from the following little calculus: E∥f n(s, ω) − f 0(s, ω)∥ = E∥E3σn0(s, ˜Zn s , ˜ζn s ) − E3σn0(s, ˜Z0 s, ˜ζ0 s)∥ = E∥E3 � σn0(s, ˜Zn s , ˜ζn s ) − σn0(s, ˜Z0 s, ˜ζ0 s) � ∥ ≤ EE3∥σn0(s, ˜Zn s , ˜ζn s ) − E3σn0(s, ˜Z0 s, ˜ζ0 s)∥ = E∥σn0(s, ˜Zn s , ˜ζn s ) − E3σn0(s, ˜Z0 s, ˜ζ0 s)∥ → 0, n → ∞, the latter convergence by virtue of Lebesgue’s bounded convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Hence, in this way we obtain that J1 + J2 + J3 → 0, n, n0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' So, the convergence (12) holds true, which along with (11) leads to the equation (10) and, hence, completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' QED References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Dobrushin, Vlasov equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAzT4oBgHgl3EQfkf1m/content/2301.01532v1.pdf'} +page_content=' Anal.' metadata={'source': 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0000000000000000000000000000000000000000..cbdf560dd7c342762023e5c9148e63082f87606e --- /dev/null +++ b/MtE3T4oBgHgl3EQfYwqL/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9cc9971e8fa940da27170542b1ade7979b5dcae5c0f509f6ac68b0073a24ab50 +size 7209005 diff --git a/NtAzT4oBgHgl3EQfWPy8/content/tmp_files/2301.01299v1.pdf.txt b/NtAzT4oBgHgl3EQfWPy8/content/tmp_files/2301.01299v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..55c020d8d97e05fd38e9fff98746c70bc432f629 --- /dev/null +++ b/NtAzT4oBgHgl3EQfWPy8/content/tmp_files/2301.01299v1.pdf.txt @@ -0,0 +1,2140 @@ +Recent Advances on Federated Learning: A Systematic Survey +Bingyan Liu +bingyanliu@bupt.edu.cn +Beijing University of Posts and Telecommunications +Beijing, China +Nuoyan Lv +lvnuoyan@bupt.edu.cn +Beijing University of Posts and Telecommunications +Beijing, China +Yuanchun Guo +gyc2001@bupt.edu.cn +Beijing University of Posts and Telecommunications +Beijing, China +Yawen Li +warmly0716@126.com +Beijing University of Posts and Telecommunications +Beijing, China +ABSTRACT +Federated learning has emerged as an effective paradigm to achieve +privacy-preserving collaborative learning among different parties. +Compared to traditional centralized learning that requires collecting +data from each party, in federated learning, only the locally trained +models or computed gradients are exchanged, without exposing any +data information. As a result, it is able to protect privacy to some +extent. In recent years, federated learning has become more and +more prevalent and there have been many surveys for summarizing +related methods in this hot research topic. However, most of them +focus on a specific perspective or lack the latest research progress. +In this paper, we provide a systematic survey on federated learn- +ing, aiming to review the recent advanced federated methods and +applications from different aspects. Specifically, this paper includes +four major contributions. First, we present a new taxonomy of fed- +erated learning in terms of the pipeline and challenges in federated +scenarios. Second, we summarize federated learning methods into +several categories and briefly introduce the state-of-the-art meth- +ods under these categories. Third, we overview some prevalent +federated learning frameworks and introduce their features. Finally, +some potential deficiencies of current methods and several future +directions are discussed. +KEYWORDS +Decentralized AI, Federated Learning, Neural Networks, Survey +ACM Reference Format: +Bingyan Liu, Nuoyan Lv, Yuanchun Guo, and Yawen Li. 2023. Recent +Advances on Federated Learning: A Systematic Survey. In Proceedings +of ACM Conference (Conference’17). ACM, New York, NY, USA, 18 pages. +https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +Over the past few years, deep neural networks (DNNs) have re- +ceived a lot of attention due to their remarkable performance on +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +Conference’17, July 2017, Washington, DC, USA +© 2023 Association for Computing Machinery. +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +various tasks such as Computer Vision (CV) [68, 89, 91, 125], Nat- +ural Language Processing (NLP) [27, 139, 151], Recommendation +Systems (RS) [16–18] and Data Mining (DM) [85, 93, 121]. However, +the superiority of DNNs depends on the support of big data, which +is hard to access in a certain party considering the limitation of the +storage space and the difficulty of data collection. Gathering data +from different parties to a central server for training is a direct solu- +tion to the issue. Nevertheless, data in each party may be sensitive +or include some user privacy information. For example, medical +images in a hospital are prohibited from outsourcing due to their +privacy property. Besides, policies such as General Data Protection +Regulation (GDPR) [4] also highlight the importance of protecting +privacy when sharing information among different organizations. +Thus, how to aggregate the data knowledge from different parties +while ensuring privacy is an important and practical problem in +real-world scenarios. +Federated learning (FL) [104], which enables multiple parties to +collaboratively train a DNN with the help of a central server, can +be regarded as an effective solution to the aforementioned problem. +Different from the traditional centralized learning that needs to +collect data from each party, in FL, data do not need to upload for a +joint training. Instead, the local trained models are exchanged with +a central server, which are used to aggregate the knowledge from +all of the uploaded models and then distribute the global model +to each party. As a result, each party is able to benefit from other +parties, improving the model accuracy. In recent years, there have +been many applications based on FL in practice, such as loan status +prediction, health situation assessment, and next-word prediction +[48, 153, 154]. +We take Fig. 1 as an example to illustrate a typical FL pipeline. +First, each hospital (party) trains the local model distributed from a +central cloud. The training process is usually implemented based on +SGD with local data and then generates corresponding local updates. +Second, the local updates rather than local data are transferred to +the cloud, where the updates are sampled in terms of some heuristic +rules to ensure the overhead and some aggregation algorithms +(e.g., FedAvg [104]) are conducted to achieve effective knowledge +integration. In this way, the cloud can get an improved new global +model and distributes it to each hospital for further tuning. These +steps may repeat several times until the healthcare service can be +satisfied (e.g., the accuracy of the learned model is acceptable for +practical deployment). +arXiv:2301.01299v1 [cs.LG] 3 Jan 2023 + +Conference’17, July 2017, Washington, DC, USA +Liu et al. +There have been other surveys on FL over the past few years. +For instance, Li et al. [78] summarized related FL methods from the +system perspective, where the authors provided the definition of +federated learning systems and analyzed the system components. +Lim et al. [87] focused on the FL application in mobile edge net- +works. Lyu et al. [99] paid more attention to the security and privacy +issues existed in current FL schemes. However, these surveys only +review a specific aspect of federated learning, failing to give read- +ers a comprehensive understanding on FL. Towards the general FL +overviews, most of them are out of date and cannot catch the latest +trend in FL research. For example, Yang et al. [153] divided FL meth- +ods into three categories (i.e., horizontal federated learning, vertical +federated learning and federated transfer learning) and described +their features respectively. Kairouz et al. [65] gave a comprehensive +introduction of federated learning theory and application. Notice +that both of the surveys mainly cited papers published before 2020, +which is impossible to track the latest research progress on FL +considering the rapid development in this field. As shown in Fig. +2, we can clearly see that the number of accepted FL papers in +top-tier conferences increases dramatically after 2020, which calls +for a timely survey to summarize the advances in the FL commu- +nity. Besides, the rapid update of FL frameworks also requires us to +highlight their latest features. +In this paper, we attempt to provide a systematic survey on feder- +ated learning, targeting at reviewing the recent advanced federated +methods and applications from different aspects. Specifically, the +key contributions of this survey are as follows: (1) we present a +new taxonomy based on the federated learning pipeline and chal- +lenges, which includes four typical aspects: aggregation optimiza- +tion, heterogeneity, privacy protection, fairness. We will give detailed +explanation in the following sections. (2) we summarize different +federated learning methods into the proposed categories and briefly +describe the state-of-the-art methods under these categories. (3) we +overview the latest federated learning frameworks and introduce +their features. (4) we discuss some potential deficiencies of current +methods and several future directions. +The remainder of this survey is structured as follows. In Section +2, we first introduce preliminaries of federated learning. In Section +3, we propose the taxonomy of federated learning according to +different aspects, in which various federated learning approaches +are discussed and categorized. Then, in Section 4, we introduce +some prevalent frameworks to show the practical deployment of +federated learning. Finally, Section 5 and Section 6 discuss the future +work and concludes this paper. +2 +PRELIMINARIES +2.1 +Problem formulation +In this section, we first introduce some notations and symbols used +in this survey to formally define federated learning. In general, there +are two ends participated in the round of federated learning: client +end and server end. The client end holds a series of local private +data D = {D1, D2, ..., D𝑁 }, which are then used to train the model +in each client and generate local models M = {𝑀1, 𝑀2, ..., 𝑀𝑁 }. +Here 𝑁 denotes the number of clients. After the local training +process, the local models M, rather than the data D, are uploaded +to the server end, where aggregation algorithms are implemented +to obtain a global model 𝑀𝑔𝑙𝑜𝑏𝑎𝑙. The process can be defined as +𝑀𝑔𝑙𝑜𝑏𝑎𝑙 = 𝐴𝐺𝐺(𝑀1, 𝑀2, ..., 𝑀𝑁 ), +(1) +where 𝐴𝐺𝐺 represents the aggregation algorithms. In this way, we +finish one round of federated learning and distribute the global +model to each client side for further local training. The concrete +number of round is usually determined by the model performance +(i.e., we stop the process until the model can achieve desirable ac- +curacy). In addition, to provide a more rigorous privacy protection, +each client may enforce some encryption techniques to the models +before uploading them. Differential privacy (DP) [30] and homo- +morphic encryption (HE) [39] are widely used to conduct such +protection. +Based on the aforementioned statement, we can see that the per- +formance of federated learning largely depends on the aggregation +algorithm in the server end. Formally, the goal of federated learning +is to optimize the following objective function +min +𝑤 +𝐿(𝑤), 𝑤ℎ𝑒𝑟𝑒 𝐿(𝑤) = +𝑁 +∑︁ +𝑖=1 +𝑓𝑖𝐿𝑖 (𝑤), +(2) +where 𝑤 is the weights of DNNs, 𝐿(𝑤) is the global loss function +and 𝐿𝑖 (𝑤) is the local loss function in the 𝑖𝑡ℎ client. 𝑓𝑖 represents the +importance of the 𝑖𝑡ℎ client and �𝑁 +𝑘=1 = 1. In federated learning, the +aggregation algorithm determines the value allocation for 𝑓𝑖. Many +research papers that try to improve the accuracy performance of +federated learning are focused on this aspect. +2.2 +Key challenges +Different from traditional centralized learning or distributed learn- +ing, federated learning faces the following key challenges: +• Heterogeneity problem. In federated learning, the hetero- +geneity comes from three aspects:(1) Data heterogeneity. +Considering that each participator collects data from its lo- +cal end, the overall data distribution inevitably conforms to +the non-independent identically distribution (non-iid) situ- +ation. For example, the same object image collected from +different environments, or the same activity coming from dif- +ferent people, can lead to different data distributions, which +will further affect the performance of federated aggrega- +tion [174]. (2) Model heterogeneity. In real-world scenarios, +it is hard to limit the federated clients to use an identical +model architecture. Instead, each client may prefer a dis- +tinctive model architecture for improved task performance. +Therefore, how to aggregate these heterogeneous models +is challenging in practical federated learning conditions. (3) +System heterogeneity. Because of the variability in hardware, +different parties may have different storage space, computa- +tion power, and communication capabilities. As a result, the +server end needs to decide whether to wait for all parties +to upload their models for better accuracy or remove strag- +glers (i.e., the parties with weak hardware performance) for +accelerating the federation process. +• Privacy leakage. The key idea of federated learning is to +achieve collaborative learning in a privacy-preserving man- +ner, which differs from the traditional paradigm that ex- +changes data or other sensitive information. Keeping data + +Recent Advances on Federated Learning: A Systematic Survey +Conference’17, July 2017, Washington, DC, USA +Figure 1: An example of the FL pipeline [80]. +AAAI +AISTATS +ICLR +ICML +KDD +NeurIPS +0 +10 +20 +30 +40 +50 +0 +0 +0 +3 +0 +0 +6 +3 +4 +6 +2 +17 +14 +8 +10 +18 +6 +26 +19 +19 +21 +37 +8 +57 +year +2019 +2020 +2021 +2022 +Figure 2: The number of pulished FL papers in top-tie con- +ference from 2019-2022. +in the local end and transferring corresponding models is +the original privacy protection design in federated learn- +ing. However, the parameters of the uploaded models may +also be exploited by attackers to infer the user privacy in- +formation [180]. So we require more rigorous encryption or +obfuscation methods to ensure privacy. +• Unfairness. In traditional centralized learning or distributed +learning, the unfairness problem does not exist since the +participants belong to a same organization. However, the +participants in federated learning come from various parties +with different data resources. According to a previous work +[31], if individuals with similar preferences and character- +istics receive substantially different outcomes, then we say +that the model violates individual fairness. Thus, it is neces- +sary to generate federated models that go beyond average +accuracy to further consider the fairness performance. +3 +APPROACHES OF FEDERATED LEARNING +In this section, we first present a taxonomy of federated learning +and allocate different federated approaches into different categories +according to the taxonomy. Then for each category, we describe in +detail how various methods achieve their goal. +3.1 +Taxonomy +In this survey, we propose a new taxonomy to classify the existing +federated learning methods (Fig. 3). Our taxonomy is motivated by +the pipeline and challenges in federated learning. As stated in the +previous section, the key step in the federated learning pipeline +is the aggregation algorithm and the key challenges come from +three different aspects. Therefore, in our taxonomy, federated learn- +ing approaches can be summarized into four cases: aggregation +optimization, heterogeneous federated learning, secure federated +learning and fair federated learning. +• Aggregation optimization. Considering that the number +of participants in a federated learning system is usually large, +it is essential to implement an effective aggregation optimiza- +tion for outputting a better global model compared to the +ones with local training. This survey investigates various +aggregation methods such as FedAvg [104, 109, 174], FedMA +[142] and FedProx [81], with a focus on how to combine +local models into an improved global model. +• Heterogeneous federated learning. In real-world scenar- +ios, federated clients may come from different environments + +local data +local data ++ +local +local ++! +updates. +updates +/new global +learntmodel: +model +personalhealthcare ++ +local data +local dataConference’17, July 2017, Washington, DC, USA +Liu et al. +Federated learning +Aggregation optimization +Weight-level aggregation +Feature-level aggregation +Other aggregation +Heterogeneous federated learning +Data heterogeneity +Model heterogeneity +System heterogeneity +Secure federated learning +Attack methods +Defense methods +Fair federated learning +Fair client selection +Fair model optimization +Fair contribution evaluation +Figure 3: Our taxonomy of different federated learning methods. +or equip with various hardware, leading to the heterogeneity +problem. In the following sections, we respectively explore +how related research efforts address the issue of data hetero- +geneity, model heterogeneity and system heterogeneity. In +particular, techniques such as meta-learning [1, 19, 33, 63, +66, 176], multi-task learning [21, 24, 51, 60, 79, 103, 127, 138, +163, 177], transfer learning [112, 116, 143, 159] and cluster- +ing [42, 43, 97, 119, 120, 167] are incorporated to achieve our +goal. +• Secure federated learning. Although traditional federated +learning has attempted to protect data privacy by only ex- +changing parameters of the local trained models, malicious +attackers can still design some scheme to infer the properties +of raw data. In our survey, we first summarize a series of +attacks targeting federated learning, where we describe how +backdoor attack [7, 111, 130, 141, 149, 150, 172], gradients +attack [38, 55, 72, 86, 156, 173, 179, 180] and poison attack +[10, 114, 129, 147] are applied to compromise federated learn- +ing. Then we introduce how to combine federated learning, +differential privacy (DP) [3, 40, 44, 64, 105, 142, 146, 168, 176], +homomorphic encryption (HE) [50, 165], trusted execution +environment (TEE) [106, 107] and other algorithms [8, 61, +133, 149] to defend aforementioned attacks. +• Fair federated learning. During federated learning, it is +possible that the performance of the global model varies sig- +nificantly across the devices, resulting in the fairness prob- +lem. This survey reviews literature about how to ensure fair +federated learning, such as designing minimax optimization +strategies [123, 134] and sample reweighting approaches +[32, 175]. +3.2 +Aggregation optimization +The goal of aggregation optimization is to improve the performance +of the final global model, which is the core output in federated +learning. There have been a large number of aggregation algorithms +proposed to combine these local models to a better global one. In +the following parts, we will describe in detail how different types +of aggregation methods work. +3.2.1 +Weight-level aggregation. A typical and prevalent weight- +level aggregation method called FedAvg [104] is mostly adopted +by developers. The key idea of FedAvg is to aggregate these local +models in a coordinate-based weight averaging manner, which can +be denoted as +𝑊 𝑟 +𝑔 = 1 +𝑁 +𝑁 +∑︁ +𝑘=1 +𝑤𝑟 +𝑘, +(3) +where N is the number of federated clients. 𝑤𝑘 denotes the weight +parameters of the 𝑘𝑡ℎ client and𝑊 𝑟𝑔 is the final aggregated model at +the𝑟𝑡ℎ round. Researchers have shown the remarkable performance +of FedAvg on a variety of public datasets (e.g., MNIST [73] and +CIFAR-10 [67]) and provided some theoretical analyses to prove +why FedAvg works well [83]. +Despite being widely applied, FedAvg still suffers from the weight +divergence problem [174]: the weight in the same coordinates (e.g., +same layer or same filter) may have a large mismatching due to + +Recent Advances on Federated Learning: A Systematic Survey +Conference’17, July 2017, Washington, DC, USA +Figure 4: The illustration of PFNM [162]. +the highly skewed data distribution in each distinctive client/party. +Therefore, directly averaging them will degrade the accuracy of the +generated global model. To solve the issue, researchers leverage a +particular DNN principle, weight permutation invariance, which has +been mentioned and discussed by recent works [142, 161, 162]. The +key idea of this principle is that the weights in a DNN can be spe- +cially shuffled without incurring much accuracy drop. Concretely, +suppose 𝑙𝑗 and 𝑙𝑗+1 are the weight of two continuous layers in a +DNN model, where the output function can be denoted as +𝑂𝑗+1 = 𝑙𝑗+1𝑙𝑗𝐼, +(4) +where 𝐼 is the input and 𝑂𝑗+1 is the output of the 𝑗 + 1𝑡ℎ layer. +Note that for each weight matrix 𝑙, it can be further decomposed as +follows +𝑙 = 𝑙1 = 𝑙ΠΠ𝑇, +(5) +where Π represents the permutation matrix. In terms of this equa- +tion, we can transform Eq. 4 to the following form +𝑂𝑗+1 = (𝑙𝑗+1Π𝑗+1Π𝑇 +𝑗+1)𝑙𝑗𝐼 = (𝑙𝑗+1Π𝑗+1)(Π𝑇 +𝑗+1𝑙𝑗)𝐼, +(6) +Based on Eq. 6, we can clearly see that the original layer weight can +be losslessly transformed with a pair of well-designed permutation +matrices, which we call it weight permutation invariance. +In federated learning, traditional aggregation methods fuse lo- +cal models according to their weight location, which may be sub- +optimal since the weight permutation invariance principle indicates +that we can change the weight value in a specific location while +ensuring the same performance. Thus, the location-based aggre- +gation cannot achieve accurate knowledge fusion, leading to the +weight mismatching problem. +To address this problem, a large number of federated optima- +tion works attempt to achieve weight-level alignment. For example, +Yurochkin et al. [162] developed Probabilistic Federated Neural +Matching (PFNM). As shown in Fig. 4, the key idea is to identify +subsets of neurons in each local model that matches neurons in +other local models and then combine the matched neurons to an +improved global model by leveraging Bayesian nonparametric ma- +chinery. For single-layer neural matching, they presented a Beta +Bernoulli Process [169] based model of MLP weight parameters, +Figure 5: Comparison between FedAvg and 𝐹𝑒𝑑2 [158]. +where the corresponding neurons in the output layer are used to +convert the neurons in each batch and form a cost matrix. Then +the matched neurons can be aggregated to generate the final global +model. For multilayer neural matching, they extended the single +strategy by defining a generative model of deep neural network +weights from outputs back to inputs. In this way, they could adopt +a greedy inference procedure that first infers the matching of the +top layer and then proceeds down the layers of the model. +Unfortunately, PFNM only performs well on simple architec- +tures (e.g. fully connected feedforward networks). For more com- +plex CNNs and LSTMs, it just receives minor improvements over +location-based methods (e.g., FedAvg). To further achieve the weight +alignment goal, Wang et al. [142] proposed Federated Matched Av- +eraging (FedMA) to effectively align advanced CNNs and LSTMs in +a layer-wise manner. The key idea is to search for the best permu- +tation matrices by addressing the following optimization problem +min +� +𝜋 𝑗 +𝑙𝑖 +� +𝐿 +∑︁ +𝑖=1 +∑︁ +𝑗,𝑙 +min +𝜃𝑖 +𝜋 𝑗 +𝑙𝑖𝑐 +� +𝑤𝑗𝑙,𝜃𝑖 +� +(7) +s.t. +∑︁ +𝑖 +𝜋 𝑗 +𝑙𝑖 = 1∀𝑗,𝑙; +∑︁ +𝑙 +𝜋 𝑗 +𝑙𝑖 = 1∀𝑖, 𝑗, +(8) +where 𝜃𝑖 is the 𝑖𝑡ℎ neuron in the current global model, 𝑤𝑗𝑙 is the +output weights processed by permutation matrix 𝜋 𝑗 +𝑙𝑖. 𝑐() is the dis- +tance metric served as determining the similarity between neurons. +To solve this optimization problem, unlike PFNM that used heuristic +choices, FedMA addressed it by the Hungarian matching algorithm +[69]. +3.2.2 +Feature-level aggregation. Despite effectiveness, the perfor- +mance of weight-level aggregation/alignment largely depends on +the selection of distance metric, which may not fully reflect the +inherent feature information embedded in the neurons. In addition, +the computation cost of the matching process is significantly heavy. +To address these limitations, Yu et al. [158] designed a feature-level +alignment method, named 𝐹𝑒𝑑2 which is composed of a feature- +oriented structure adaptation and a model fusion algorithm. As +shown in Fig. 5, compared with traditional weight alignment, 𝐹𝑒𝑑2 +paid more attention to the neuron features and then aggregated +the corresponding neurons. As a result, similar knowledge can be +fused to achieve better performance. +Concretely, the authors developed two schemes to accomplish +feature-based federated learning. Fig. 6 shows the pipeline of the + +Server 1 +Server 2 +Server 3 +Outputs +Hidden layers +Input +Match and merge neurons to form aggregate layer +Outputs +Global hidden layer +InputNeuron Coordinates +Neuron Group Coordinates +Collaborative Nodes +Model Average +Model Average +(a) FedAvg (IID) +(b) Our Framework (IID)Conference’17, July 2017, Washington, DC, USA +Liu et al. +Figure 6: The illustration of 𝐹𝑒𝑑2 [158]. +proposed 𝐹𝑒𝑑2. The first scheme is model structure adaptation, +where 𝐹𝑒𝑑2 takes advantage of the group-convolution technique to +allocate and learn the distinctive neuron features. Next, a feature +paired averaging policy is presented to aggregate different neu- +rons according to the partitioned group features. In this way, 𝐹𝑒𝑑2 +enables more accurate feature alignment as well as avoiding the +expensive distance-based optimization. +3.2.3 +Other aggregation. The aforementioned works mainly focus +on alignment, in fact, there are also many other literatures target- +ing federated aggregation. For example, Yin et al. [155] proposed a +robust aggregation method for distributed learning. In the begin- +ning, this work mainly analyzed two robust distributed gradient +descent (GD) algorithms, including the coordinate-wise median and +the coordinate-wise trimmed mean. They proved statistical error +rates for three kinds of population loss functions: strongly convex, +non-strongly convex, and smooth non-convex. Furthermore, to re- +duce the communication cost, the authors designed a median-based +distributed algorithm and demonstrate its effectiveness by exten- +sive experiments. Chen et al. [22] further considered the federated +learning scenario, and found that heterogeneous data in different +nodes will harm the training convergence to some degree. Based +on this observation, they developed a novel gradient correction +mechanism that can perturb the local gradients with noise. The +main advantage of the proposed scheme is that it offers a provable +convergence guarantee even when data are non-iid. +Besides, Yurochkin et al. [160] leveraged Bayesian nonpara- +metrics to design a meta-model that can potentially capture the +global structure through statistical parameter matching. The au- +thors pointed out that their approach is model-independent and is +applicable to a wide range of model types. Chen et al. [20] proposed +FEDBE, a novel method to apply bayesian model ensemble into +conventional federated learning, aiming at making the aggregation +more robust. Motivated by prior work [100], the authors utilized +bayesian inference to construct an improved global model. In addi- +tion, stochastic weight average (SWA) [62] is also used to further +boost the performance. +3.3 +Heterogeneous federated learning +Heterogeneous federated learning aims to effectively aggregate +models generated from heterogeneous environments. Here the +heterogeneous property could be reflected from data, models or +device systems. We will dive into each aspect in the next parts. +3.3.1 +Data heterogeneity. Data heterogeneity indicates that collab- +orative clients might be in different situations, resulting in various +data distributions. For example, the dog images collected from in- +doors and outdoors display highly heterogeneous data distribution. +To address the issue, the research community borrows the idea +from other AI techniques to alleviate the heterogeneity influence, +which we list as follows. +Multi-task learning based methods. Multi-task learning enables +learning models for multiple related tasks at the same time [11, 102, +118]. The core design principle is to capture the relationship among +tasks and leverage the relationship to facilitate the learning process. +In federated learning, clients with different data distributions could +also be considered as a type of multi-task learning, where each task +has a distinctive statistical representation [46, 47, 60, 132, 138, 163]. +For instance, Smith et al. [127] first proposed to combine federated +learning and multi-task learning. By a series of concept formula- +tions and theoretical analyses, they suggested multi-task learning +is a natural choice to handle the statistical problem in the federated +setting. Based on the combination, they further developed a novel +approach MOCHA, in order to accomplish their goal. Specifically, +the authors formulated the problem as a dual optimization problem +as follows +min +𝜶 +� +D(𝜶) := +𝑚 +∑︁ +𝑡=1 +𝑛𝑡 +∑︁ +𝑖=1 +ℓ∗ +𝑡 +� +−𝜶𝑖 +𝑡 +� ++ R∗(X𝜶) +� +, +(9) +where 𝑙∗ +𝑡 and R∗ are the conjugate dual functions of 𝑙𝑡 and R, +respectively. To solve 9, they carefully designed the quadratic ap- +proximation of the dual problem to separate computation across +the nodes. +Despite federated multi-task learning being demonstrated effec- +tive, it has been applied only on convex models. To address the +limitation, Corinzia et al. [24] proposed a more general approach, +named VIRTUAL, to achieve federation on non-convex models. The +key idea is to construct a hierarchical Bayesian network in terms of +the central server and the clients, such that the inference could be +performed with variational methods. In this way, each client can +obtain a task-specific model that benefits from the server model in +a transfer learning manner. + +Feature-to-Structure +- +国C +Allocation +Feature +- + Paired Averaging +- +AoA +Different Data +Distribution +Collaboration +Shared Decoupled +- +(a) Model Structure Adaptation +(b) Feature Paired AveragingRecent Advances on Federated Learning: A Systematic Survey +Conference’17, July 2017, Washington, DC, USA +Marfoq et al. [103] further proposed to study federated multi-task +learning under the flexible assumption that each local data distri- +bution is a mixture of unknown underlying distributions, which is +a more challenging and practical scenario. In the beginning, the au- +thors showed the fact that t federated learning is impossible without +assumptions on local data distributions. Then they made the flexible +assumption and developed Federated Expectation-Maximization +to accomplish their objective. Besides, the proposed approach is +proven generalizable to unseen clients. +Meta-learning based methods. Meta-learning is commonly consid- +ered as learning to learn [135]. Compared with conventional deep +learning algorithms that learn specific feature knowledge, meta- +learning focus more on learning the learning ability. In the field of +federated learning, meta-learning techniques can also be applied +to generate a more personalized federation model. Jiang et al. [63] +first proposed to combine them, where they believed meta-learning +had a number of similarities with the objective of addressing the +statistical challenge in FL. Concretely, they developed a novel algo- +rithm to further combine FedAvg [104] and Reptile [110], with two +modifications: the first one is to decrease the local learning rate to +make training more stable; another is to design a fine-tuning stage +based on Reptile with smaller K and Adam as the server optimizer, +which could improve the initial model as well as preserving and +stabilizing the personalized model. +Khodak et al. [66] built a theoretical framework to further char- +acterize meta-learning methods and apply them into federated +learning. They introduced Average Regret-Upper-Bound Analysis +(ARUBA), which enables meta-learning to leverage more sophis- +ticated structures. With ARUBA, researchers could improve the +results of many ML tasks, including adapting to the task-similarity, +adapting to dynamic environments, adapting to the inter-task geom- +etry and statistical learning-to-Learn. Towards FL, they improved +meta-test-time performance on few-shot learning and effectively +added user-personalization to FedAvg. +Fallah et al. [33] aims to find an initial shared model that can be +easily fitted to their local data with one or a few steps of gradient +descent. They achieved their objective by incorporating Model- +Agnostic Meta-Learning (MAML) [36, 37] into current FL pipelines. +Specifically, the authors proposed a personalized variant of the +FedAvg algorithm, named Per-FedAvg, which can be formulated as +optimizing the following equation +min +𝑤∈R𝑑 𝐹 (𝑤) := 1 +𝑛 +𝑛 +∑︁ +𝑖=1 +𝑓𝑖 (𝑤 − 𝛼∇𝑓𝑖 (𝑤)) , +(10) +where 𝑛 is the number of clients and 𝛼 is the learning rate. The +detailed solution for the optimization problem can be seen in the +paper if readers have an interest. +Acar et al. [1] further modified meta-learning to benefit federated +learning. As shown in Fig. 7, they proposed PFL, a gradient correc- +tion method based on prior works, which explicitly de-biased the +meta-model in the distributed heterogeneous data setting to learn a +more personalized device model. During the process, convergence +guarantees of PFL for strongly convex, convex and nonconvex meta +objectives are provided. +Transfer learning based methods. Transfer learning aims to trans- +fer the information learned from a source task to a target task [113]. +Figure 7: The illustration of PFL [1]. +A large number of research works have been proposed to advance +this promising field [82, 84, 157, 170]. In federated learning, trans- +ferring the knowledge of the federated model to each client model +will significantly facilitate the personalization performance under +the data heterogeneity environment. Wang et al. [143] proposed +to use fine-tuning, a typical transfer learning algorithm to achieve +personalization. They first conducted traditional FL to obtain a +global model. Then the federated model is regarded as the source +model and further retrained using individual client’s training cache +data. In this way, each client model can acquire and benefit the +transferred knowledge, outputting an improved customized model. +Based on the aforementioned work, Yu et al. [159] extended the +simple fine-tuning strategy. They investigated how three adapta- +tion mechanisms: fine-tuning, multi-task learning, and knowledge +distillation affect the personalization performance. The authors +characterized these mechanisms as local adaptation. In addition, +different model protection techniques such as differential privacy +and robust aggregation were applied to further validate the effec- +tiveness of local adaptation. Finally, they used both CV and NLP +datasets to demonstrate the superiority and necessity to conduct +local adaptation. +Peng et al. [115] considered a new FL+TL scenario beyond fine- +tuning. Instead, they paid more attention to domain shift, which +means that the labeled data collected by source nodes statistically +differ from the target node’s unlabeled data. Based on this setting, +they proposed the problem of federated domain adaptation and +address it by Federated Adversarial Domain Adaptation (FADA). +The key idea is to apply adversarial adaptation and representation +disentanglement to FL settings. +Ozkara et al. [112] introduced a quantized and personalized FL +algorithm to deal with the data issue. The quantized training process +is conducted via knowledge distillation (KD) among clients who + +2 +0 +-VfioTi(w) +unbiased +-6 +Device 1 +-8 +Device 2 +debiasing +Device 3 +-10 +Global +Server w +8 +6 +-4 +2 +0 +2 +4 +6 +8Conference’17, July 2017, Washington, DC, USA +Liu et al. +Figure 8: The illustration of IFCA [42]. +have access to heterogeneous data and resources. Besides, they +developed an alternating proximal gradient update to address this +compressed personalization challenge and analyzed its convergence +properties. +Clustering-based methods. Clustering-based FL attempts to tackle +the data heterogeneity issue via partitioning clients into different +clusters, each of which conforms to a similar distribution. In terms +of this key idea, much research effort is made to explore cluster- +based FL. Sattler et al. [120] proposed Clustered Federated Learning +(CFL), to utilize geometric properties of the FL loss surface, in order +to group the client population into clusters with jointly trainable +data distributions. It is worth noting that CFL is orthogonal to the +current FL communication protocol and can be applied to general +non-convex objectives beyond DNNs. +Ghosh et al. [42] proposed the Iterative Federated Clustering Al- +gorithm (IFCA), which alternately estimated the cluster identities of +the users and optimized model parameters for the user clusters via +gradient descent. As shown in Fig. 8, the server broadcasted models +and the workers dynamically identified their cluster memberships +and run local updates. This process will continue to operate until +the clusters become stable. +To train high-quality cluster models, Ruan et al. [119] suggested +FedSoft, which uses proximal updates to restrict client burden by +asking a subset of clients to complete just one optimization task +per communication round. +Liu et al. [92] proposed a framework to accomplish privacy- +preserving federated adaptation. The key idea is to group the clients +with similar distribution to collaboratively adapt the federated +model, rather than just adapting it with the data in a single device. +PFA leveraged the sparsity property of neural networks to generate +privacy-preserving representations and used them to efficiently +identify clients with similar data distributions. In this way, PFA can +conduct an FL process in a group-wise way on the federated model +to achieve adaptation. +Besides, in order to achieve clustering without uploading any +extra information, Liu et al. [90] further proposed DistFL, targeting +at finishing accurate, automated and efficient cluster-based FL in +terms of distribution feature. Specifically, they extracted the distri- +bution knowledge from the uploaded model via existing synthesis +techniques [101] and then compared them to obtain the clustering +results. Finally, they aggregated models in each cluster, getting rid +of the influence of heterogeneous data. +Figure 9: The illustration of HeteroFL [28]. +Figure 10: The illustration of Oort [71]. +3.3.2 +Model heterogeneity. Model heterogeneity means that the +federated model might not be identical due to the different hard- +ware and data distributions of clients. For example, in order to fit +various computation capabilities of clients, we require deploying +different model architectures to match each client. On the other +hand, NAS techniques [181] have been widely used to search a +crafted architecture based on the data in each device, thus leading +to the model heterogeneity situation. +To tackle the problem, Li et al. [75] used transfer learning and +knowledge distillation to develop a universal framework, which +enabled federated learning with uniquely designed models. Lin et +al. [88] further proposed a distillation framework for robust feder- +ated model fusion and leveraged entropy-reduction to accelerate +convergence. Diao et al. [28] designed HeteroFL to address hetero- +geneous clients equipped with highly different computation and +communication capabilities. As shown in Fig. 9, the federation is +achieved by aggregating parameters on the same location while +unlearning the other non-overlapping area. +3.3.3 +System heterogeneity. System heterogeneity is a practical +property in FL scenarios because different clients/parties naturally +own heterogeneous hardware and memory limitation. Therefore, +how to accomplish FL under the condition of system heterogeneity +is worth exploring. +A key design for system acceleration is to develop different client +selection strategies for avoiding the influence of latency stragglers. + +Coordinator +① Job + Info update +Oort +Execution +submission +Metastore +Selector +Driver +Selection +3 Execution + Aggregation +Participants +Participants +Client Pool(01,.02) +(a) +(c) +02 +(d) +2Global model parameters W +Local model parameters W3 +Local model parameters W? +Local model parameters W,Recent Advances on Federated Learning: A Systematic Survey +Conference’17, July 2017, Washington, DC, USA +Figure 11: The illustration of model replacement [7]. +Here stragglers refer to the clients with weak computing power +and thus could slow down the overall FL process. Lai et al. [71] +proposed Oort, a system to improve the performance of federated +training and testing with guided participant selection. As shown in +Fig. 10, Oort cherry-picked participants according to the tradeoff +between statistical and system efficiency. Specifically, they defined +"Client Statistical Utility" to measure the importance of each client. +Shin et al. [124] developed FedBalancer, a framework to actively +select clients’ training samples in terms of the more “informative" +data. Besides, they introduced an adaptive deadline control scheme +to predict the optimal deadline for each round, in order to further +speed up global training. Li et al. [74] observed that current client +selection was coarse-grained due to their under-exploitation on the +clients’ data and system heterogeneity. Based on this finding, they +proposed PyramidFL, a fine-grained client selection framework to +speed up the FL training. The key idea is to not only focus on the +divergence of those selected participants but also fully exploited the +data and system heterogeneity within selected clients to profile their +utility more efficiently. As a result, PyramidFL is able to achieve +better performance compared to other baselines. +3.4 +Secure federated learning +The original design of federated learning considers the security +problem via exchanging parameters while keeping raw data in their +own devices. However, recent studies have proved that attackers +might steal the privacy information from the uploaded models. +Therefore, more rigorous secure FL should be investigated. In the +following parts, we will introduce the attack methods and defense +methods in FL scenarios. +3.4.1 +Attack methods. Backdoor attack. The goal of backdoor at- +tacks is to manipulate a subset of training data by injecting adversar- +ial triggers such that DNN models will output incorrect prediction +on the test set when the same trigger occurs. In federated learning, +directly applying current backdoor attacks is unsuitable since the +aggregation process might destroy the triggers. Bagdasaryan et +al. [7] is the first to backdoor federated learning. They achieved +their objective by proposing model replacement, which means the +backdoor is injected to the joint model rather than raw data. As +shown in Fig. 11, the attacker trained a model on the backdoor +data using the constrain-and-scale technique. In this way, the av- +eraging function is largely affected by this attack model. Wang et +al. [141] proposed edge-case backdoors, which forced a model to +misclassify on seemingly easy inputs that are unlikely to be part +of the training or testing data. For example, they may exist on the +tail of the input distribution. As a result, it is extremely hard to +detect them. Xie et al. [150] further developed distributed backdoor +attack (DBA) to compromise FL. They mainly took advantage of +the distributed nature of FL, decomposing a global trigger pattern +into separate local patterns and introducing them into the training +set of different adversarial parties respectively. Therefore, DBA is +more persistent and stealthy compared to centralized ones. In FL +models, backdoors can be inserted, but these backdoors are often +not durable, i.e., they do not remain in the model after poisoned +updates stop being uploaded. Since training occurs gradually in FL +systems, an inserted backdoor may not survive until deployment. +Zhang et al.[172] proposed Neurotoxin, which is a simple modifi- +cation to existing backdoor attacks that target parameters that are +not changed in magnitude as much during training. +Gradients attack. Gradients attack targets at reverse some pri- +vacy information from gradients. In federated learning, exchanging +gradients is a typical step for knowledge update and aggregation. +Therefore, gradient attack poses a high risk to the federal partic- +ipants. Zhu et al. [180] found since training occurs gradually in +FL systems, an inserted backdoor may not survive until deploy- +ment. that it is possible to obtain the private training data from the +publicly shared gradients. They first randomly generated a pair of +“dummy” inputs and labels and used them to compute correspond- +ing gradients. Then the gradients were compared to the shared ones +and continually optimize the dummy inputs and labels to minimize +the distance between them. As a result, the dummy data are close +to the original ones and can peek into user privacy. Lam et al. [72] +further realized gradients attack from the aggregated model up- +dates/gradients. The authors leveraged the summary information +from device analytics and reconstructed the user participant matrix, +which invalided the current secure aggregation protocols [12]. Zhu +et al. [179] proposed Recursive Gradient Attack on Privacy (R-GAP), +an approach to analyze how and when the target gradients can lead +to the unique recovery of original data. Concretely, the authors +designed a recursive, depth-wise algorithm for recovering training +data from the gradient information, which is the first closed-form +algorithm that works on both CNN layers and FC layers. Li et al. +[86] found that under certain defense settings, generative gradient +leakage can still leak private training information. +Model poison attack. The goal of poison attacks is to induce the +FL model to output the target label specified by the adversary. For +example, Tolpegin et al. [136] implemented data poison attack by +flipping the labels of training data from one class to another class +in the local training epoch to mislead the global model output. +Although the aggregation process in FL can mitigate the attack +to some extent, when the number of malicious clients becomes +large, FL is inevitably poisoned. Fang et al. [34] conducted the first +systematic study on local model poisoning attacks to federated +learning. Based on this study, they proposed local model poisoning +attacks to Byzantine robust federated learning via manipulating the +local model parameters on compromised worker devices during the + +benign participants +user C +user B +userA +Diocal +train +Federated +Gt +Averaging +User M +constrain +It+1 +and +Dbackdoor +scaleConference’17, July 2017, Washington, DC, USA +Liu et al. +Figure 12: The illustration of BatchCrypt [165]. +learning process. Besides, the authors further stated two defense +strategies and test their performance on the proposed attack. +3.4.2 +Defense methods. DP-based defense. Differential privacy (DP) +[30] has been widely used to prevent information leakage. The key +idea is to add some noises to obfuscate the original information. As +a result, attackers are hard to infer the privacy properties. Federated +learning also requires this type of protection since the uploaded +model parameters can be easily exploited to extract sensitive infor- +mation. Wei et al. [146] proposed NbAFL, a framework that applied +DP into FL. Specifically, they added noises to parameters of the local +model at the client side before aggregation. Besides, the authors +theoretically analyzed the convergence property of differentially +private FL algorithms and proved the effectiveness of the proposed +framework. +Kairouz et al. [64] presented a comprehensive end-to-end sys- +tem, where they discretized the data and added discrete Gaussian +noise before conducting secure aggregation. In addition, the authors +provided a novel privacy analysis for sums of discrete Gaussians +and carefully analyzed the effects of data quantization and mod- +ular summation arithmetic. Experiments demonstrated that their +method can achieve comparable performance with 16 bits of pre- +cision per value. Agarwal et al. [3] proposed a multi-dimensional +Skellam mechanism, where two independent poisson random vari- +ables are used to measure the difference. The authors applied their +mechanism to FL and provided a novel algorithm that appropriately +discretized the data and used the Skellam mechanism along with +modular arithmetic to bound the range of the data and communica- +tion costs before secure aggregation. As a result, they could achieve +better privacy-accuracy trade-offs in a more efficient manner. +HE-based defense. HE-based FL aims to combine traditional Ho- +momorphic Encryption (HE) and FL in a more suitable way. By +applying HE, FL is able to aggregate client models without reveal- +ing the information of the concrete model parameters. Therefore, +it is impossible to infer user privacy from the model. Hardy et al. +[50] proposed to encrypt FL with the homomorphic scheme in the +field of privacy-preserving entity resolution and federated logistic +regression. They bounded the difference between the empirical loss +of their classifier on the true data and showed an improved conver- +gence speed. Besides, their experiments found that even rates for +generalization cannot be significantly affected by entity resolution. +Liu et al. [94] designed a secure FL framework through leveraging +the additive property of partial homomorphic encryption, which +effectively avoids the exposure of client models at the server side. +Besides, the authors introduced two optimization mechanisms to +further enhance efficiency. Zhang et al. [165] proposed BatchCrypt, +an efficient homomorphic encryption system for cross-Silo feder- +ated learning. As shown in Fig. 12, there exist five typical steps to +achieve a cross-silo FL system. In the beginning, the aggregator +needed to select a client to generate an HE key-pair and distribute +it to others. Then for each iteration, clients conducted local gradi- +ent updates and further encrypted them by the public key. These +encrypted parameters were uploaded to the server where aggre- +gation happened and the aggregated model is transferred to each +client. Finally, the client side decrypted the received information +and implemented the local training as the next round. BatchCrypt +proposed two novel schemes to further improve efficiency. First, a +feasible batch encryption scheme was presented to directly sum up +the ciphertexts of two batches. Second, an efficient analytical model +dACIQ was presented to choose optimal clipping thresholds with +the minimum cumulative error. As a result, BatchCrypt achieved +23×-93× training speedup while reducing the communication over- +head by 66×-101×. +TEE-based defense. The aforementioned secure FL approaches +provide security guarantee mainly from the perspective of software. +In real-world scenarios, hardware protection is also widely applied +by designing crafted architecture. Trusted Execution Environment +(TEE) is a trusted component that establishes an isolated region +on the main processor to ensure the confidentiality and integrity +of data and programs [5, 25]. Compared to traditional encryption +schemes such as homomorphic encryption, TEE is more efficient +with respect to the computation cost since it only requires some +simple operations to connect the trusted and untrusted part in OS. +Recently there have been a large number of works targeting at +applying TEE to deep/federated learning, in order to achieve pro- +tection from hardware level. For example, Mo et al. [107] proposed +DarkneTZ that enabled executing DNNs more secure with TEE in +an edge device. They partitioned DNNs into a set of non-sensitive +layers and sensitive layers, which are respectively processed by +TEE or normal OS. Here the partition choice is based on the under- +lying system’s CPU execution time, memory usage, and accurate +power consumption of different DNN layers. Besides, the authors +developed a threat model to validate DarkneTZ’s robustness un- +der the membership inference attack and the results showed that +DarkneTZ could defend against this type of attack with negligible +performance overhead. +Based on the combination of DNNs and TEE, Mo et al. [106] +further attempted to apply TEEs to federated learning. Specifically, +they proposed PPFL, a framework that limited privacy leakages +in federated learning via implementing local training in TEEs. As +shown in Fig. 13, to address the challenge of limited memory size of +TEEs, the authors designed a greedy layer-wise training to conduct +local updates until convergence. In this way, this approach could +support sophisticated settings such as training one or more layers +(block) each time, which potentially speed up the training process. +Zhang et al. [171] proposed TEESlice, a system to provide a strong +security guarantee while maintaining low inference latency with +the help of TEEs. Concretely, TEESlice executed the more private +model slices on TEEs and others on normal AI accelerators. As a + +Single Client +Aggregator +HE Public Key +Gradients +③Aggregation +Aggregated +HE Private Key +Gradients +ClientA +②Encryption +@Decryption +ClientNi +Client B +② Encryption +4 +@Decryption +Gradient +?Model +computation +update +① Gradient +(5 Model +computation +updateRecent Advances on Federated Learning: A Systematic Survey +Conference’17, July 2017, Washington, DC, USA +Figure 13: The illustration of PPFL [106]. +result, TEESlice can achieve more than 10× throughput promotion +with the same level of strong security guarantee. +3.5 +Fair federated learning +Existing works of federated learning pay more attention to improv- +ing learning performance based on the accuracy of the model and +the time of learning task completion. However, the interests of the +FL clients are often ignored and this may lead to unfairness. The +problem of fairness can occur in the whole FL training process, in- +cluding client selection, model optimization, incentive distribution, +and contribution evaluation. The unfairness can have a negative +impact on both the FL clients and the FL server, as clients are dis- +couraged to join FL training, and servers are less likely to attract +potentially high-quality clients. Recently, to achieve fairness from +different angles, various Fairness-Aware Federated Learning (FAFL) +approaches have been proposed. In this section, we will discuss +recent FAFL methods in detail. +3.5.1 +Fair client selection. Unfairness in FL Client Selection mainly +consists of three types, over-representation, under-representation, +and never-representation. Suppose an FL system prefers to se- +lect clients with high performance (such as a faster GPU), and +clients with the highest performance may be selected much more +than any other clients (i.e., over-representation), while clients with +poor performance may be selected just a few times (i.e., under- +representation). At the same time, the client with the lowest per- +formance may never be selected (i.e., never-representation). Addi- +tionally, due to the heterogeneity among clients, fairness does not +indicate giving everyone the same possibility to be selected. It is +important to balance the interests of the server and the interests +of the clients. If clients from specific groups are oversampled, the +global FL model will be partial to their data, so the model’s perfor- +mance will deteriorate [23]. Existing FAFL client selection methods +can be partitioned into two categories, considering fairness factors +and customization for each client. +1) Fairness factors. Fairness factors are designed to allow rarely +selected clients, such as clients with lower computational abilities +or smaller datasets, to join the FL training more frequently. Yang et +al. [152] proposed a client selection algorithm based on the Com- +binatorial Multi-Armed Bandit (CMAB) framework to reduce the +class imbalance effect. Inspired by [76], Huang et al. [59] converts +the original offline problem to an online Lyapunov optimization +problem and uses dynamic queues to quantify the long-term guar- +antee of the client participation rate. Moreover, Huang introduces +a long-term fairness constraint to make sure the average client’s +long-term chosen rate is above a constant. After [59], Huang et al. +[58] improves the performance by replacing dynamic queues to +the Exp3 algorithms [6], and the fairness parameter determining +the selection possibility in each round can be different. However, +these works all design the fairness factor without considering the +real-time contribution of individual clients. Song et al. [128] ad- +dresses this problem and proposed a client selection policy with +fairness constraints based on reputation, using a fairness parameter +to balance reputation and the number of successful transmissions. +2) Client customization. This approach pays attention to cus- +tomized model settings or customized model procedures. Clients +often receive the same initial models at the first training round +in most current FL paradigms. Therefore, clients with lower ca- +pabilities, such as bad network connections, require more time to +complete each training round and are likely to be kept out of sub- +sequent rounds, leading to under-presented and never-presented +problems. To alleviate this problem, dynamically adapting the FL +model framework or the training procedure based on client capa- +bilities is often used. +Caldas et al. [14] proposed Federated Dropout (FD), which dis- +tributes sub-models with sizes suitable for each client based on +their computational resources. The process of FD is shown in Fig +14. Although FD diminishes communication and local computation +costs largely, it uses dropout operations and treats the neural net- +works as black-box functions. Bouaciada et al. noticed this problem +and proposed Adaptive Federated Dropout (AFD) [13]. AFD keeps +an activation score map to generate the best-fit sub-model for each +client. FD and AFD both make sure clients with low capabilities +could participate in FL training, but they do not provide custom +pruned submodels to different clients. To address this limitation, +Horvath et al. [56] augmented FD to Ordered Dropout (OD). Dif- +ferent from FD, OD drops neighboring components of the model +despite random neurons. OD divides clients with comparable com- +putational capabilities into clusters, and clients in the same cluster +apply the same dropout rate. Moreover, OD applies the knowledge +distillation method [54] to enhance feature extraction for smaller +submodels. +Clients’ communication capabilities can also affect client selec- +tion. A poor network may cause too much retransmission and lead + +Server +Clients +Configuration +Move to next block of layers +Secure channel + after conyergence +TEE +Transferring knowledge if any @ +if transferring +TEE +Model initialization ② +Public +model broadcasting ③ +Know- +Reporting +ledge +Private +④ +Layer-wise local training @ +Class- +Dataset +model reporting @ +ifier +Secure aggregation @ +Class- +Class- +Class- +Class- +@? +ifier +ifier +ifier +ifier +≥Data transmission +?I +? +Public layers +Forward pass + Private layers +Backward passConference’17, July 2017, Washington, DC, USA +Liu et al. +Figure 14: The summary of the Federated Dropout (FD) train- +ing procedure [14]. +Figure 15: The illustration of ThrowRightAway (TRA) +scheme [178]. +to extra delays in FL model training, which makes clients with a +poorer network less likely to aggregate their model updates into +the final model and leads to model bias. To deal with this issue, +Zhou et al. [178] proposed ThrowRightAway (TRA), a loss-tolerant +FL framework that makes the FL training faster by ignoring few +lost packets. As is shown in Fig 15, at first every participating FL +client reports their network conditions to the FL server, and the +server divides the clients into two categories: sufficient type and +insufficient type. Only the clients in the sufficient type can get a +re-transmission request and then re-transmit their loss packets. +Apparently, the method can only be effective when the category is +accurate. +This method means assigning less work to clients with lower +capabilities to make them available to pass threshold-based FL +client selection. Li et al. [80] proposed FedProx which allowed each +client performed partial training based on its accessible resources. +FedProx allows various local epochs, and thus more clients are +encouraged to join the training process. +3.5.2 +Fair model optimization. In the optimization during FL model +training, the model may discriminate against definite preserved +groups, or overfit some clients at the expense of others. Recent +works dealing with this issue can be approximately divided into +two types: 1) objective function-based and 2) gradient-based. +1) Objective function-based methods: Objective function-based +methods focus on the global/local objectives of the FL model, such +as minimizing the loss function. Mohri et al. [108] proposed AFL, +which aims to prevent the model overfitting any specific client at +the expense of others. AFL just optimizes the global model for the +target distribution made up of a mixture of clients. However, this +method only works for a small number of clients. Zhou et al. [178] +proposed q-FFL to diminish the scalability limitation of AFL. q-FFL +adds parameter q to reweigh the aggregate loss. To improve the +model robustness and maintain good-intent fairness at the same +time, Hu et al. [57] proposed fedMGDA+ which optimizes each FL +client’s loss function respectively and simultaneously. Addressing +the same issue, Li et al. [79] proposed Ditto, which improves fairness +and robustness at the same time. +While the methods mentioned all pay attention to the accuracy +parity notion of fairness, there are also many kinds of research +focusing on group fairness. Du et al. [29] proposed AgnosticFair, +which incorporates an agnostic fairness constraint. Although it has +good accuracy and fairness on unknown testing data distribution, it +needs prior knowledge to design the re-weighting function, which +limits its application in dynamic systems. Cui et al. [26] proposed +FCFL, a multi-objective optimization framework that achieves good- +intent fairness and group fairness at the same time. Different from +AFL, it minimizes the loss of the client with the worst performance +and uses a smooth surrogate maximum function considering all +clients. A fairness constraint is also added to calculate the disparities +among all clients. +2) Gradient-based approaches: Here, gradient means the local +updated gradient of each client in every local iteration. Wang et +al. [144] proposed the federated fair averaging (FedFV) algorithm, +which aims to average clients’ gradients after mitigating potential +conflicts among clients. FedFV detects gradient conflicts through the +cosine similarity and modifies both the direction and magnitude of +the gradients by iteratively eliminating such conflicts. However, the +estimated gradients may be incompatible with the latest updates. +3.5.3 +Fair contribution evaluation. Contribution evaluation in FL +learning indicates that an FL system can evaluate the contribution +of different clients without accessing data from the clients. Many +methods designed for non-privacy machine learning environments +cannot be applied to FL scenarios directly. A general method is +to evaluate each client’s model contribution to the aggregated FL +model, and a fair evaluation is critical. Unfairness in contribution +evaluation may lead to the free-rider issue [49], which implies that +clients contribute little but can get similar benefits as the clients who +contribute more. In this part we will introduce five types of existing +FL contribution evaluation methods with their typical works. +1) Self-reported information: This method of evaluation contribu- +tion is based on clients reporting their information actively. Most +works based on this method believe their clients are reliable, which +is not always correct in practice. Proposed by Zhang et al. [166], +Hierarchically fair federated learning (HFFL) follows the idea of +’contribute more, get more reward’, which is proved effective in so- +cial psychology [137], game theory [117] and bandwidth allocation +[77]. Hence, it’s critical to figure out how to evaluate a client’s con- +tribution and how much proportion of reward a client should get to +ensure fairness. Data Shapley can be used to evaluate contribution +in machine learning, but Shapley value is model-dependent [41] +and incompatible with FL tasks. As a result, Zhang proposes evalu- +ating contributions based on publicly verifiable factors of clients, +such as cost of data collection, data volume, and data quality, to +avoid the inconsistency of model-dependent methods. To distribute +proportional rewards to clients, Zhang introduces hierarchically +fair federated learning (HFFL), as is shown in Figure 16. The pub- +licly verifiable factors determined by the clients’ consensus about +each client are reported to the FL server, and the FL server then + +Server +Server +Clientnsufficient Sufficient speed +retransfer(Loss) +Adaptive Aggregation +setZero(Loss)Recent Advances on Federated Learning: A Systematic Survey +Conference’17, July 2017, Washington, DC, USA +Figure 16: The illustration of hierarchically fair federated +learning (HFFL) [166]. +uses the information to rate each client, which at the same level +are supposed to contribute to the model equally and will get the +equal reward. +2) Individual evaluation: Individual evaluation implies evaluating +contribution through performance on specific tasks and pays more +attention to individual performance instead of global performance. +The method often adopts two assumptions that both the server and +the client are reliable and clients with a similar model to others are +regarded to supply more contribution, which is not always feasible. +To achieve fairness without sacrificing the model performance, Lyu +et al. [98] proposed a Collaborative Fair Federated Learning (CFFL) +framework based on reputation, which uses a reputation mecha- +nism to achieve collaborative fairness. Lyu definites collaborative +fairness as the reward is proportional to the client’s contribution. +Different standard FL process, CFFL allows clients to receive only +the allocated aggregated updates according to their reputations, +and the server is in charge of a reputation list which is updated in +each communication round relying on the quality of the uploaded +gradients of each participant. +3) Utility game: The utility game [45] refers to a game where +each player chooses an available team to maximize their payoffs, +while the universal social welfare is the total utility produced by +all the teams cumulatively. FL contribution evaluation methods +based on utility games have a deep connection with profit-sharing +schemes, and there are three diffusely used profit-sharing schemes: +(1) Egalitarian: any part of the utility produced by a team is sepa- +rated equally between the members. (2) Marginal gain: the payoff +of a player in a team is equal to the team gained when the player +joined. (3) Marginal loss: the payoff of a player in a team is equal +to the team will lose if the player leaves. +Among the three types above, the marginal loss scheme is the +most commonly adopted. Wang et al. [140] proposed a deletion +method to evaluate contributions in horizontal federated learning. +This evaluation method consists of removing the instances supplied +from one definite party, retraining the model, calculating the differ- +ence between the original model and the new model, and using this +difference to define the contribution of this party. Wang formulates +the influence measure as follows, +𝐼𝑛𝑓 𝑙𝑢𝑒𝑛𝑐𝑒−𝑖 = 1 +𝑛 +𝑛 +∑︁ +𝑗=1 +���ˆ𝑦𝑗 − ˆ𝑦−𝑖 +𝑗 +���, +(11) +where n is the size of the dataset, ˆ𝑦𝑗 is the model trained on all data +prediction on jth instance, and ˆ𝑦−𝑖 +𝑗 +s the model trained without the +ith instance prediction on jth instance. +Then Huang defines a party’s contribution as the total influence +of all instances it possesses. +𝐼𝑛𝑓 𝑙𝑢𝑒𝑛𝑐𝑒−𝐷 = +∑︁ +𝑖 ∈𝐷 +𝐼𝑛𝑓 𝑙𝑢𝑒𝑛𝑐𝑒−𝑖, +(12) +For vertical horizontal learning, Huang uses shapley value which +will be introduced in the next part. +4) Shapley value: Shapley value (SV) was first introduced in coop- +erative game theory [122]. Different from marginal loss, SV-based +FL contribution evaluation approaches can reflect the contribu- +tion of a client’s own data, in spite of its joining order, and can +produce a fairer evaluation. However, SV’s computational complex- +ity is 𝑂(2𝑛), so many approaches have been proposed to improve +efficiency. +4 +PREVALENT FRAMEWORKS OF +FEDERATED LEARNING +In this section, we will introduce several prevalent frameworks of +federated learning, including FedLab, Flower, FedML, FATE, and +FedScale. +FedLab. Since most FL schemes follow the same basic steps and +just a few changes in some steps are needed in different scenar- +ios, Zeng et al. [164] proposed FedLab [126], which is designed +flexible and customizable, offers essential functional modules, and +has highly customizable interfaces. Two main roles in FL settings +are provided: Server and Client, and both of them are made up +of two components, NetworkManager and ParameterServerHan- +dler/Trainer. The design focuses more on communication efficiency +and FL algorithm effectiveness. To support methods improving +Communication Efficiency, FedLab uses tensor-based communi- +cation, supports customizable communication agreement, and im- +plements both Synchronous and Asynchronous communication +patterns according to Federated Optimization algorithms. For Opti- +mization Effectiveness, FedLab applies a "high-cohesion and low- +coupling" optimization module which provides aggregation and +data partition methods. Additionally, FedLab can be used in various +scenarios, such as Standalone, Cross-process and Hierarchical FL +simulation. +Flower. Due to the lack of frameworks that are able to support +scalably executing FL methods on mobile and edge devices, Beutel +et al. [9] proposed Flower [2], which can run large-scale FL experi- +ments on different FL device scenarios. Flower makes it possible to +smoothly transition from experimental research to system research +on a large group of real edge devices. Designed to be scalable, client- +agnostic, communication-agnostic, privacy-agnostic, and flexible, +Flower has extensive implementations, such as communication +stack, serialization, ClientProxy, and Virtual Client Engine(VCE). +FedML. Proposed by He et al. [52], FedML [35] aims to solve +the lack of support for diverse FL computing paradigms, support +of diverse FL configurations, and standardized FL algorithm im- +plementations and benchmarks. FedML library is mainly made up +of high-level API FedML-API and low-level API FedML-core. To + +Central server +f2 +f3 +f3 +m3 +Data volume +Level 1 +m2 +f2 +Level 2 +m1 +f1 +Level 3 +Level 1 +Level 2 Level 3Conference’17, July 2017, Washington, DC, USA +Liu et al. +support FL on real-world hardware platforms, FedML offers on- +device FL testbeds called FedML-Mobile and FedML-IoT which are +built upon real-world hardware platforms. FedML programming +interface allows worker/client-oriented programming, message def- +inition beyond gradient and model, topology management, trainer +and coordinator, privacy, security, and robustness, so users can +just pay attention to algorithms implementations and ignore the +backend details. +FATE. Since most open-sourced frameworks are research-oriented +and lack the implementation on industry, Liu et al. [95] proposed +FATE(Federated AI Technology Enabler) [145], which is the first +production-oriented platform. Built on FederatedML, FATE pro- +vides Private Set Intersection(PSI), and uses distributed computa- +tion framework Eggroll to improve computation efficiency. FATE +provides three main components, scheduling system FATE-Flow, +visualization tool FATE-Board, and high-performance inference +platform FATE-Serving. In addition, kinds of deployments are sup- +ported, including building FATE on top of Kubernetes in data cen- +ters through KubeFATE, manual or docker deployments on Mac +and Linux, and cross-cloud deployment and management through +FATE-cloud. +FedScale. Lai et al. [70] proposed FedScale [131], which con- +tains many realistic FL datasets for different tasks, and FedScale +Runtime which is an automated evaluation platform aiming to +simplify and standardize FL evaluation in more realistic environ- +ments. The raw data of FedScale datasets are collected from various +sources, processed into consistent formats, sorted into different FL +use cases and packed into standardized APIs for users to easily use +in other frameworks. The evaluation platform, FedScale Runtime, +is equipped with both mobile and cluster backends to enable both +on-device FL evaluation on smartphones, and FL evaluations in real +deployments and in-cluster simulations. +5 +DISCUSSION +This section summarizes some limitations of current FL approaches +and discusses possible future directions. +Dynamic federated learning. Current federated learning ap- +proaches assume that data in each client are stable and unchanged. +However, in real-world scenarios, clients may be in an ever-changing +environment, where the local data are continuously observed and +processed by sensors. Under this condition, directly conducting +conventional training and aggregation will suffer from the cata- +strophic forgetting problem, which indicates that the prior knowl- +edge learned by the model might be forgotten as new data arrive. +Incremental learning [15, 96, 148] is a hot research topic to address +the issue, targeting at learning new knowledge while maintain- +ing the ability to recognize previous ones. In the future, how to +effectively combine federated learning and incremental learning is +worth exploring. +Decentralized federated learning. A central server is of vi- +tal importance to traditional federated learning since aggregation +needs to be conducted in this side. Considering that the third-party +server may not be honest, uploading parameters or gradients to +it potentially exists security risks. Therefore, it is necessary to +achieve federated learning without a server involved. Although He +et al. [53] has made a preliminary attempt to decentralized FL, they +only target logistic regression and the experiments are insufficient. +How to accomplish general decentralized FL still remains an open +problem. +Scalability of federated learning. Recent FL papers paid more +attention to designing new algorithms to improve FL performance +under different conditions. However, they ignore the scalability +property, which determines whether we could operate large-scale +FL. In many cooperation scenarios, there might be a huge number +of parties and we should provide guidance to the cooperation im- +provement as the number of participants increases. In a word, FL +scalability deserves future investigation. +Unified benchmark. Although a large number of datasets have +been used for evaluating the performance of FL, there is still a lack +of a unified benchmark to align the results for a fair comparison. +On one hand, in order to achieve different federated goals (e.g., +personalization, robustness), researchers use different datasets to +test the performance. On the other hand, two typical types of FL, +horizontal FL and vertical FL, also apply distinctive datasets to +demonstrate the performance of different FL types. Thus a unified +benchmark will definitely benefit the FL community. +6 +CONCLUSION +Federated learning has gained more and more attention due to its +ability of collaboratively generating a global model without leaking +sensitive information. Recent surveys have summarized many re- +lated works devoted to offering a comprehensive understanding to +developers and readers in this community. 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Neural architecture search with reinforcement +learning. arXiv preprint arXiv:1611.01578 (2016). + diff --git a/NtAzT4oBgHgl3EQfWPy8/content/tmp_files/load_file.txt b/NtAzT4oBgHgl3EQfWPy8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc8781ff2de7f2b7153b163c12aee53f8eba94aa --- /dev/null +++ b/NtAzT4oBgHgl3EQfWPy8/content/tmp_files/load_file.txt @@ -0,0 +1,1608 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf,len=1607 +page_content='Recent Advances on Federated Learning: A Systematic Survey Bingyan Liu bingyanliu@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='cn Beijing University of Posts and Telecommunications Beijing, China Nuoyan Lv lvnuoyan@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='cn Beijing University of Posts and Telecommunications Beijing, China Yuanchun Guo gyc2001@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='cn Beijing University of Posts and Telecommunications Beijing, China Yawen Li warmly0716@126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='com Beijing University of Posts and Telecommunications Beijing, China ABSTRACT Federated learning has emerged as an effective paradigm to achieve privacy-preserving collaborative learning among different parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Compared to traditional centralized learning that requires collecting data from each party, in federated learning, only the locally trained models or computed gradients are exchanged, without exposing any data information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, it is able to protect privacy to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In recent years, federated learning has become more and more prevalent and there have been many surveys for summarizing related methods in this hot research topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, most of them focus on a specific perspective or lack the latest research progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this paper, we provide a systematic survey on federated learn- ing, aiming to review the recent advanced federated methods and applications from different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Specifically, this paper includes four major contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' First, we present a new taxonomy of fed- erated learning in terms of the pipeline and challenges in federated scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Second, we summarize federated learning methods into several categories and briefly introduce the state-of-the-art meth- ods under these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Third, we overview some prevalent federated learning frameworks and introduce their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Finally, some potential deficiencies of current methods and several future directions are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' KEYWORDS Decentralized AI, Federated Learning, Neural Networks, Survey ACM Reference Format: Bingyan Liu, Nuoyan Lv, Yuanchun Guo, and Yawen Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Recent Advances on Federated Learning: A Systematic Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In Proceedings of ACM Conference (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' ACM, New York, NY, USA, 18 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION Over the past few years, deep neural networks (DNNs) have re- ceived a lot of attention due to their remarkable performance on Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='nnnnnnn various tasks such as Computer Vision (CV) [68, 89, 91, 125], Nat- ural Language Processing (NLP) [27, 139, 151], Recommendation Systems (RS) [16–18] and Data Mining (DM) [85, 93, 121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, the superiority of DNNs depends on the support of big data, which is hard to access in a certain party considering the limitation of the storage space and the difficulty of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Gathering data from different parties to a central server for training is a direct solu- tion to the issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Nevertheless, data in each party may be sensitive or include some user privacy information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, medical images in a hospital are prohibited from outsourcing due to their privacy property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, policies such as General Data Protection Regulation (GDPR) [4] also highlight the importance of protecting privacy when sharing information among different organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Thus, how to aggregate the data knowledge from different parties while ensuring privacy is an important and practical problem in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Federated learning (FL) [104], which enables multiple parties to collaboratively train a DNN with the help of a central server, can be regarded as an effective solution to the aforementioned problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Different from the traditional centralized learning that needs to collect data from each party, in FL, data do not need to upload for a joint training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Instead, the local trained models are exchanged with a central server, which are used to aggregate the knowledge from all of the uploaded models and then distribute the global model to each party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, each party is able to benefit from other parties, improving the model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In recent years, there have been many applications based on FL in practice, such as loan status prediction, health situation assessment, and next-word prediction [48, 153, 154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' We take Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 1 as an example to illustrate a typical FL pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' First, each hospital (party) trains the local model distributed from a central cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The training process is usually implemented based on SGD with local data and then generates corresponding local updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Second, the local updates rather than local data are transferred to the cloud, where the updates are sampled in terms of some heuristic rules to ensure the overhead and some aggregation algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', FedAvg [104]) are conducted to achieve effective knowledge integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this way, the cloud can get an improved new global model and distributes it to each hospital for further tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' These steps may repeat several times until the healthcare service can be satisfied (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', the accuracy of the learned model is acceptable for practical deployment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='01299v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='LG] 3 Jan 2023 Conference’17, July 2017, Washington, DC, USA Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' There have been other surveys on FL over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For instance, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [78] summarized related FL methods from the system perspective, where the authors provided the definition of federated learning systems and analyzed the system components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [87] focused on the FL application in mobile edge net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [99] paid more attention to the security and privacy issues existed in current FL schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, these surveys only review a specific aspect of federated learning, failing to give read- ers a comprehensive understanding on FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Towards the general FL overviews, most of them are out of date and cannot catch the latest trend in FL research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [153] divided FL meth- ods into three categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', horizontal federated learning, vertical federated learning and federated transfer learning) and described their features respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Kairouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [65] gave a comprehensive introduction of federated learning theory and application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Notice that both of the surveys mainly cited papers published before 2020, which is impossible to track the latest research progress on FL considering the rapid development in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 2, we can clearly see that the number of accepted FL papers in top-tier conferences increases dramatically after 2020, which calls for a timely survey to summarize the advances in the FL commu- nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, the rapid update of FL frameworks also requires us to highlight their latest features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this paper, we attempt to provide a systematic survey on feder- ated learning, targeting at reviewing the recent advanced federated methods and applications from different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Specifically, the key contributions of this survey are as follows: (1) we present a new taxonomy based on the federated learning pipeline and chal- lenges, which includes four typical aspects: aggregation optimiza- tion, heterogeneity, privacy protection, fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' We will give detailed explanation in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' (2) we summarize different federated learning methods into the proposed categories and briefly describe the state-of-the-art methods under these categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' (3) we overview the latest federated learning frameworks and introduce their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' (4) we discuss some potential deficiencies of current methods and several future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The remainder of this survey is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In Section 2, we first introduce preliminaries of federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In Section 3, we propose the taxonomy of federated learning according to different aspects, in which various federated learning approaches are discussed and categorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Then, in Section 4, we introduce some prevalent frameworks to show the practical deployment of federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Finally, Section 5 and Section 6 discuss the future work and concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 2 PRELIMINARIES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='1 Problem formulation In this section, we first introduce some notations and symbols used in this survey to formally define federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In general, there are two ends participated in the round of federated learning: client end and server end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The client end holds a series of local private data D = {D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', D𝑁 }, which are then used to train the model in each client and generate local models M = {𝑀1, 𝑀2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', 𝑀𝑁 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Here 𝑁 denotes the number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' After the local training process, the local models M, rather than the data D, are uploaded to the server end, where aggregation algorithms are implemented to obtain a global model 𝑀𝑔𝑙𝑜𝑏𝑎𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The process can be defined as 𝑀𝑔𝑙𝑜𝑏𝑎𝑙 = 𝐴𝐺𝐺(𝑀1, 𝑀2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', 𝑀𝑁 ), (1) where 𝐴𝐺𝐺 represents the aggregation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this way, we finish one round of federated learning and distribute the global model to each client side for further local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The concrete number of round is usually determined by the model performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', we stop the process until the model can achieve desirable ac- curacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In addition, to provide a more rigorous privacy protection, each client may enforce some encryption techniques to the models before uploading them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Differential privacy (DP) [30] and homo- morphic encryption (HE) [39] are widely used to conduct such protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Based on the aforementioned statement, we can see that the per- formance of federated learning largely depends on the aggregation algorithm in the server end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Formally, the goal of federated learning is to optimize the following objective function min 𝑤 𝐿(𝑤), 𝑤ℎ𝑒𝑟𝑒 𝐿(𝑤) = 𝑁 ∑︁ 𝑖=1 𝑓𝑖𝐿𝑖 (𝑤), (2) where 𝑤 is the weights of DNNs, 𝐿(𝑤) is the global loss function and 𝐿𝑖 (𝑤) is the local loss function in the 𝑖𝑡ℎ client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 𝑓𝑖 represents the importance of the 𝑖𝑡ℎ client and �𝑁 𝑘=1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In federated learning, the aggregation algorithm determines the value allocation for 𝑓𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Many research papers that try to improve the accuracy performance of federated learning are focused on this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='2 Key challenges Different from traditional centralized learning or distributed learn- ing, federated learning faces the following key challenges: Heterogeneity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In federated learning, the hetero- geneity comes from three aspects:(1) Data heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Considering that each participator collects data from its lo- cal end, the overall data distribution inevitably conforms to the non-independent identically distribution (non-iid) situ- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, the same object image collected from different environments, or the same activity coming from dif- ferent people, can lead to different data distributions, which will further affect the performance of federated aggrega- tion [174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' (2) Model heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In real-world scenarios, it is hard to limit the federated clients to use an identical model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Instead, each client may prefer a dis- tinctive model architecture for improved task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, how to aggregate these heterogeneous models is challenging in practical federated learning conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' (3) System heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Because of the variability in hardware, different parties may have different storage space, computa- tion power, and communication capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, the server end needs to decide whether to wait for all parties to upload their models for better accuracy or remove strag- glers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', the parties with weak hardware performance) for accelerating the federation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Privacy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The key idea of federated learning is to achieve collaborative learning in a privacy-preserving man- ner, which differs from the traditional paradigm that ex- changes data or other sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Keeping data Recent Advances on Federated Learning: A Systematic Survey Conference’17, July 2017, Washington, DC, USA Figure 1: An example of the FL pipeline [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' AAAI AISTATS ICLR ICML KDD NeurIPS 0 10 20 30 40 50 0 0 0 3 0 0 6 3 4 6 2 17 14 8 10 18 6 26 19 19 21 37 8 57 year 2019 2020 2021 2022 Figure 2: The number of pulished FL papers in top-tie con- ference from 2019-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' in the local end and transferring corresponding models is the original privacy protection design in federated learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, the parameters of the uploaded models may also be exploited by attackers to infer the user privacy in- formation [180].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' So we require more rigorous encryption or obfuscation methods to ensure privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In traditional centralized learning or distributed learning, the unfairness problem does not exist since the participants belong to a same organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, the participants in federated learning come from various parties with different data resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' According to a previous work [31], if individuals with similar preferences and character- istics receive substantially different outcomes, then we say that the model violates individual fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Thus, it is neces- sary to generate federated models that go beyond average accuracy to further consider the fairness performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3 APPROACHES OF FEDERATED LEARNING In this section, we first present a taxonomy of federated learning and allocate different federated approaches into different categories according to the taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Then for each category, we describe in detail how various methods achieve their goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='1 Taxonomy In this survey, we propose a new taxonomy to classify the existing federated learning methods (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Our taxonomy is motivated by the pipeline and challenges in federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As stated in the previous section, the key step in the federated learning pipeline is the aggregation algorithm and the key challenges come from three different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, in our taxonomy, federated learn- ing approaches can be summarized into four cases: aggregation optimization, heterogeneous federated learning, secure federated learning and fair federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Aggregation optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Considering that the number of participants in a federated learning system is usually large, it is essential to implement an effective aggregation optimiza- tion for outputting a better global model compared to the ones with local training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' This survey investigates various aggregation methods such as FedAvg [104, 109, 174], FedMA [142] and FedProx [81], with a focus on how to combine local models into an improved global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Heterogeneous federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In real-world scenar- ios, federated clients may come from different environments local data local data + local local +!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' updates /new global learntmodel: model personalhealthcare + local data local dataConference’17, July 2017, Washington, DC, USA Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Federated learning Aggregation optimization Weight-level aggregation Feature-level aggregation Other aggregation Heterogeneous federated learning Data heterogeneity Model heterogeneity System heterogeneity Secure federated learning Attack methods Defense methods Fair federated learning Fair client selection Fair model optimization Fair contribution evaluation Figure 3: Our taxonomy of different federated learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' or equip with various hardware, leading to the heterogeneity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In the following sections, we respectively explore how related research efforts address the issue of data hetero- geneity, model heterogeneity and system heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In particular, techniques such as meta-learning [1, 19, 33, 63, 66, 176], multi-task learning [21, 24, 51, 60, 79, 103, 127, 138, 163, 177], transfer learning [112, 116, 143, 159] and cluster- ing [42, 43, 97, 119, 120, 167] are incorporated to achieve our goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Secure federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Although traditional federated learning has attempted to protect data privacy by only ex- changing parameters of the local trained models, malicious attackers can still design some scheme to infer the properties of raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In our survey, we first summarize a series of attacks targeting federated learning, where we describe how backdoor attack [7, 111, 130, 141, 149, 150, 172], gradients attack [38, 55, 72, 86, 156, 173, 179, 180] and poison attack [10, 114, 129, 147] are applied to compromise federated learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Then we introduce how to combine federated learning, differential privacy (DP) [3, 40, 44, 64, 105, 142, 146, 168, 176], homomorphic encryption (HE) [50, 165], trusted execution environment (TEE) [106, 107] and other algorithms [8, 61, 133, 149] to defend aforementioned attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Fair federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' During federated learning, it is possible that the performance of the global model varies sig- nificantly across the devices, resulting in the fairness prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' This survey reviews literature about how to ensure fair federated learning, such as designing minimax optimization strategies [123, 134] and sample reweighting approaches [32, 175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='2 Aggregation optimization The goal of aggregation optimization is to improve the performance of the final global model, which is the core output in federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' There have been a large number of aggregation algorithms proposed to combine these local models to a better global one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In the following parts, we will describe in detail how different types of aggregation methods work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='1 Weight-level aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' A typical and prevalent weight- level aggregation method called FedAvg [104] is mostly adopted by developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The key idea of FedAvg is to aggregate these local models in a coordinate-based weight averaging manner, which can be denoted as 𝑊 𝑟 𝑔 = 1 𝑁 𝑁 ∑︁ 𝑘=1 𝑤𝑟 𝑘, (3) where N is the number of federated clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 𝑤𝑘 denotes the weight parameters of the 𝑘𝑡ℎ client and𝑊 𝑟𝑔 is the final aggregated model at the𝑟𝑡ℎ round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Researchers have shown the remarkable performance of FedAvg on a variety of public datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', MNIST [73] and CIFAR-10 [67]) and provided some theoretical analyses to prove why FedAvg works well [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Despite being widely applied, FedAvg still suffers from the weight divergence problem [174]: the weight in the same coordinates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', same layer or same filter) may have a large mismatching due to Recent Advances on Federated Learning: A Systematic Survey Conference’17, July 2017, Washington, DC, USA Figure 4: The illustration of PFNM [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' the highly skewed data distribution in each distinctive client/party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, directly averaging them will degrade the accuracy of the generated global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To solve the issue, researchers leverage a particular DNN principle, weight permutation invariance, which has been mentioned and discussed by recent works [142, 161, 162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The key idea of this principle is that the weights in a DNN can be spe- cially shuffled without incurring much accuracy drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Concretely, suppose 𝑙𝑗 and 𝑙𝑗+1 are the weight of two continuous layers in a DNN model, where the output function can be denoted as 𝑂𝑗+1 = 𝑙𝑗+1𝑙𝑗𝐼, (4) where 𝐼 is the input and 𝑂𝑗+1 is the output of the 𝑗 + 1𝑡ℎ layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Note that for each weight matrix 𝑙, it can be further decomposed as follows 𝑙 = 𝑙1 = 𝑙ΠΠ𝑇, (5) where Π represents the permutation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In terms of this equa- tion, we can transform Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 4 to the following form 𝑂𝑗+1 = (𝑙𝑗+1Π𝑗+1Π𝑇 𝑗+1)𝑙𝑗𝐼 = (𝑙𝑗+1Π𝑗+1)(Π𝑇 𝑗+1𝑙𝑗)𝐼, (6) Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 6, we can clearly see that the original layer weight can be losslessly transformed with a pair of well-designed permutation matrices, which we call it weight permutation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In federated learning, traditional aggregation methods fuse lo- cal models according to their weight location, which may be sub- optimal since the weight permutation invariance principle indicates that we can change the weight value in a specific location while ensuring the same performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Thus, the location-based aggre- gation cannot achieve accurate knowledge fusion, leading to the weight mismatching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To address this problem, a large number of federated optima- tion works attempt to achieve weight-level alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, Yurochkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [162] developed Probabilistic Federated Neural Matching (PFNM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 4, the key idea is to identify subsets of neurons in each local model that matches neurons in other local models and then combine the matched neurons to an improved global model by leveraging Bayesian nonparametric ma- chinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For single-layer neural matching, they presented a Beta Bernoulli Process [169] based model of MLP weight parameters, Figure 5: Comparison between FedAvg and 𝐹𝑒𝑑2 [158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' where the corresponding neurons in the output layer are used to convert the neurons in each batch and form a cost matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Then the matched neurons can be aggregated to generate the final global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For multilayer neural matching, they extended the single strategy by defining a generative model of deep neural network weights from outputs back to inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this way, they could adopt a greedy inference procedure that first infers the matching of the top layer and then proceeds down the layers of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Unfortunately, PFNM only performs well on simple architec- tures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' fully connected feedforward networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For more com- plex CNNs and LSTMs, it just receives minor improvements over location-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', FedAvg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To further achieve the weight alignment goal, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [142] proposed Federated Matched Av- eraging (FedMA) to effectively align advanced CNNs and LSTMs in a layer-wise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The key idea is to search for the best permu- tation matrices by addressing the following optimization problem min � 𝜋 𝑗 𝑙𝑖 � 𝐿 ∑︁ 𝑖=1 ∑︁ 𝑗,𝑙 min 𝜃𝑖 𝜋 𝑗 𝑙𝑖𝑐 � 𝑤𝑗𝑙,𝜃𝑖 � (7) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' ∑︁ 𝑖 𝜋 𝑗 𝑙𝑖 = 1∀𝑗,𝑙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' ∑︁ 𝑙 𝜋 𝑗 𝑙𝑖 = 1∀𝑖, 𝑗, (8) where 𝜃𝑖 is the 𝑖𝑡ℎ neuron in the current global model, 𝑤𝑗𝑙 is the output weights processed by permutation matrix 𝜋 𝑗 𝑙𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 𝑐() is the dis- tance metric served as determining the similarity between neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To solve this optimization problem, unlike PFNM that used heuristic choices, FedMA addressed it by the Hungarian matching algorithm [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='2 Feature-level aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Despite effectiveness, the perfor- mance of weight-level aggregation/alignment largely depends on the selection of distance metric, which may not fully reflect the inherent feature information embedded in the neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In addition, the computation cost of the matching process is significantly heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To address these limitations, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [158] designed a feature-level alignment method, named 𝐹𝑒𝑑2 which is composed of a feature- oriented structure adaptation and a model fusion algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 5, compared with traditional weight alignment, 𝐹𝑒𝑑2 paid more attention to the neuron features and then aggregated the corresponding neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, similar knowledge can be fused to achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Concretely, the authors developed two schemes to accomplish feature-based federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 6 shows the pipeline of the Server 1 Server 2 Server 3 Outputs Hidden layers Input Match and merge neurons to form aggregate layer Outputs Global hidden layer InputNeuron Coordinates Neuron Group Coordinates Collaborative Nodes Model Average Model Average (a) FedAvg (IID) (b) Our Framework (IID)Conference’17, July 2017, Washington, DC, USA Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Figure 6: The illustration of 𝐹𝑒𝑑2 [158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' proposed 𝐹𝑒𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The first scheme is model structure adaptation, where 𝐹𝑒𝑑2 takes advantage of the group-convolution technique to allocate and learn the distinctive neuron features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Next, a feature paired averaging policy is presented to aggregate different neu- rons according to the partitioned group features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this way, 𝐹𝑒𝑑2 enables more accurate feature alignment as well as avoiding the expensive distance-based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='3 Other aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The aforementioned works mainly focus on alignment, in fact, there are also many other literatures target- ing federated aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [155] proposed a robust aggregation method for distributed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In the begin- ning, this work mainly analyzed two robust distributed gradient descent (GD) algorithms, including the coordinate-wise median and the coordinate-wise trimmed mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They proved statistical error rates for three kinds of population loss functions: strongly convex, non-strongly convex, and smooth non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Furthermore, to re- duce the communication cost, the authors designed a median-based distributed algorithm and demonstrate its effectiveness by exten- sive experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [22] further considered the federated learning scenario, and found that heterogeneous data in different nodes will harm the training convergence to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Based on this observation, they developed a novel gradient correction mechanism that can perturb the local gradients with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The main advantage of the proposed scheme is that it offers a provable convergence guarantee even when data are non-iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, Yurochkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [160] leveraged Bayesian nonpara- metrics to design a meta-model that can potentially capture the global structure through statistical parameter matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The au- thors pointed out that their approach is model-independent and is applicable to a wide range of model types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [20] proposed FEDBE, a novel method to apply bayesian model ensemble into conventional federated learning, aiming at making the aggregation more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Motivated by prior work [100], the authors utilized bayesian inference to construct an improved global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In addi- tion, stochastic weight average (SWA) [62] is also used to further boost the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='3 Heterogeneous federated learning Heterogeneous federated learning aims to effectively aggregate models generated from heterogeneous environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Here the heterogeneous property could be reflected from data, models or device systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' We will dive into each aspect in the next parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='1 Data heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Data heterogeneity indicates that collab- orative clients might be in different situations, resulting in various data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, the dog images collected from in- doors and outdoors display highly heterogeneous data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To address the issue, the research community borrows the idea from other AI techniques to alleviate the heterogeneity influence, which we list as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Multi-task learning based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Multi-task learning enables learning models for multiple related tasks at the same time [11, 102, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The core design principle is to capture the relationship among tasks and leverage the relationship to facilitate the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In federated learning, clients with different data distributions could also be considered as a type of multi-task learning, where each task has a distinctive statistical representation [46, 47, 60, 132, 138, 163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For instance, Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [127] first proposed to combine federated learning and multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' By a series of concept formula- tions and theoretical analyses, they suggested multi-task learning is a natural choice to handle the statistical problem in the federated setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Based on the combination, they further developed a novel approach MOCHA, in order to accomplish their goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Specifically, the authors formulated the problem as a dual optimization problem as follows min 𝜶 � D(𝜶) := 𝑚 ∑︁ 𝑡=1 𝑛𝑡 ∑︁ 𝑖=1 ℓ∗ 𝑡 � −𝜶𝑖 𝑡 � + R∗(X𝜶) � , (9) where 𝑙∗ 𝑡 and R∗ are the conjugate dual functions of 𝑙𝑡 and R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To solve 9, they carefully designed the quadratic ap- proximation of the dual problem to separate computation across the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Despite federated multi-task learning being demonstrated effec- tive, it has been applied only on convex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To address the limitation, Corinzia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [24] proposed a more general approach, named VIRTUAL, to achieve federation on non-convex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The key idea is to construct a hierarchical Bayesian network in terms of the central server and the clients, such that the inference could be performed with variational methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this way, each client can obtain a task-specific model that benefits from the server model in a transfer learning manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Feature-to-Structure 国C Allocation Feature Paired Averaging AoA Different Data Distribution Collaboration Shared Decoupled (a) Model Structure Adaptation (b) Feature Paired AveragingRecent Advances on Federated Learning: A Systematic Survey Conference’17, July 2017, Washington, DC, USA Marfoq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [103] further proposed to study federated multi-task learning under the flexible assumption that each local data distri- bution is a mixture of unknown underlying distributions, which is a more challenging and practical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In the beginning, the au- thors showed the fact that t federated learning is impossible without assumptions on local data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Then they made the flexible assumption and developed Federated Expectation-Maximization to accomplish their objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, the proposed approach is proven generalizable to unseen clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Meta-learning based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Meta-learning is commonly consid- ered as learning to learn [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Compared with conventional deep learning algorithms that learn specific feature knowledge, meta- learning focus more on learning the learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In the field of federated learning, meta-learning techniques can also be applied to generate a more personalized federation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [63] first proposed to combine them, where they believed meta-learning had a number of similarities with the objective of addressing the statistical challenge in FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Concretely, they developed a novel algo- rithm to further combine FedAvg [104] and Reptile [110], with two modifications: the first one is to decrease the local learning rate to make training more stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' another is to design a fine-tuning stage based on Reptile with smaller K and Adam as the server optimizer, which could improve the initial model as well as preserving and stabilizing the personalized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Khodak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [66] built a theoretical framework to further char- acterize meta-learning methods and apply them into federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They introduced Average Regret-Upper-Bound Analysis (ARUBA), which enables meta-learning to leverage more sophis- ticated structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' With ARUBA, researchers could improve the results of many ML tasks, including adapting to the task-similarity, adapting to dynamic environments, adapting to the inter-task geom- etry and statistical learning-to-Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Towards FL, they improved meta-test-time performance on few-shot learning and effectively added user-personalization to FedAvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Fallah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [33] aims to find an initial shared model that can be easily fitted to their local data with one or a few steps of gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They achieved their objective by incorporating Model- Agnostic Meta-Learning (MAML) [36, 37] into current FL pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Specifically, the authors proposed a personalized variant of the FedAvg algorithm, named Per-FedAvg, which can be formulated as optimizing the following equation min 𝑤∈R𝑑 𝐹 (𝑤) := 1 𝑛 𝑛 ∑︁ 𝑖=1 𝑓𝑖 (𝑤 − 𝛼∇𝑓𝑖 (𝑤)) , (10) where 𝑛 is the number of clients and 𝛼 is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The detailed solution for the optimization problem can be seen in the paper if readers have an interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Acar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [1] further modified meta-learning to benefit federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 7, they proposed PFL, a gradient correc- tion method based on prior works, which explicitly de-biased the meta-model in the distributed heterogeneous data setting to learn a more personalized device model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' During the process, convergence guarantees of PFL for strongly convex, convex and nonconvex meta objectives are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Transfer learning based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Transfer learning aims to trans- fer the information learned from a source task to a target task [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Figure 7: The illustration of PFL [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' A large number of research works have been proposed to advance this promising field [82, 84, 157, 170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In federated learning, trans- ferring the knowledge of the federated model to each client model will significantly facilitate the personalization performance under the data heterogeneity environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [143] proposed to use fine-tuning, a typical transfer learning algorithm to achieve personalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They first conducted traditional FL to obtain a global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Then the federated model is regarded as the source model and further retrained using individual client’s training cache data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this way, each client model can acquire and benefit the transferred knowledge, outputting an improved customized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Based on the aforementioned work, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [159] extended the simple fine-tuning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They investigated how three adapta- tion mechanisms: fine-tuning, multi-task learning, and knowledge distillation affect the personalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The authors characterized these mechanisms as local adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In addition, different model protection techniques such as differential privacy and robust aggregation were applied to further validate the effec- tiveness of local adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Finally, they used both CV and NLP datasets to demonstrate the superiority and necessity to conduct local adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [115] considered a new FL+TL scenario beyond fine- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Instead, they paid more attention to domain shift, which means that the labeled data collected by source nodes statistically differ from the target node’s unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Based on this setting, they proposed the problem of federated domain adaptation and address it by Federated Adversarial Domain Adaptation (FADA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The key idea is to apply adversarial adaptation and representation disentanglement to FL settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Ozkara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [112] introduced a quantized and personalized FL algorithm to deal with the data issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The quantized training process is conducted via knowledge distillation (KD) among clients who 2 0 VfioTi(w) unbiased 6 Device 1 8 Device 2 debiasing Device 3 10 Global Server w 8 6 4 2 0 2 4 6 8Conference’17, July 2017, Washington, DC, USA Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Figure 8: The illustration of IFCA [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' have access to heterogeneous data and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, they developed an alternating proximal gradient update to address this compressed personalization challenge and analyzed its convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Clustering-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Clustering-based FL attempts to tackle the data heterogeneity issue via partitioning clients into different clusters, each of which conforms to a similar distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In terms of this key idea, much research effort is made to explore cluster- based FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Sattler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [120] proposed Clustered Federated Learning (CFL), to utilize geometric properties of the FL loss surface, in order to group the client population into clusters with jointly trainable data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' It is worth noting that CFL is orthogonal to the current FL communication protocol and can be applied to general non-convex objectives beyond DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Ghosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [42] proposed the Iterative Federated Clustering Al- gorithm (IFCA), which alternately estimated the cluster identities of the users and optimized model parameters for the user clusters via gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 8, the server broadcasted models and the workers dynamically identified their cluster memberships and run local updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' This process will continue to operate until the clusters become stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To train high-quality cluster models, Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [119] suggested FedSoft, which uses proximal updates to restrict client burden by asking a subset of clients to complete just one optimization task per communication round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [92] proposed a framework to accomplish privacy- preserving federated adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The key idea is to group the clients with similar distribution to collaboratively adapt the federated model, rather than just adapting it with the data in a single device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' PFA leveraged the sparsity property of neural networks to generate privacy-preserving representations and used them to efficiently identify clients with similar data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this way, PFA can conduct an FL process in a group-wise way on the federated model to achieve adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, in order to achieve clustering without uploading any extra information, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [90] further proposed DistFL, targeting at finishing accurate, automated and efficient cluster-based FL in terms of distribution feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Specifically, they extracted the distri- bution knowledge from the uploaded model via existing synthesis techniques [101] and then compared them to obtain the clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Finally, they aggregated models in each cluster, getting rid of the influence of heterogeneous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Figure 9: The illustration of HeteroFL [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Figure 10: The illustration of Oort [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='2 Model heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Model heterogeneity means that the federated model might not be identical due to the different hard- ware and data distributions of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, in order to fit various computation capabilities of clients, we require deploying different model architectures to match each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' On the other hand, NAS techniques [181] have been widely used to search a crafted architecture based on the data in each device, thus leading to the model heterogeneity situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To tackle the problem, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [75] used transfer learning and knowledge distillation to develop a universal framework, which enabled federated learning with uniquely designed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [88] further proposed a distillation framework for robust feder- ated model fusion and leveraged entropy-reduction to accelerate convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Diao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [28] designed HeteroFL to address hetero- geneous clients equipped with highly different computation and communication capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 9, the federation is achieved by aggregating parameters on the same location while unlearning the other non-overlapping area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='3 System heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' System heterogeneity is a practical property in FL scenarios because different clients/parties naturally own heterogeneous hardware and memory limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, how to accomplish FL under the condition of system heterogeneity is worth exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' A key design for system acceleration is to develop different client selection strategies for avoiding the influence of latency stragglers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Coordinator ① Job Info update Oort Execution submission Metastore Selector Driver Selection 3 Execution Aggregation Participants Participants Client Pool(01,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='02) (a) (c) 02 (d) 2Global model parameters W Local model parameters W3 Local model parameters W?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Local model parameters W,Recent Advances on Federated Learning: A Systematic Survey Conference’17, July 2017, Washington, DC, USA Figure 11: The illustration of model replacement [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Here stragglers refer to the clients with weak computing power and thus could slow down the overall FL process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [71] proposed Oort, a system to improve the performance of federated training and testing with guided participant selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 10, Oort cherry-picked participants according to the tradeoff between statistical and system efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Specifically, they defined "Client Statistical Utility" to measure the importance of each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [124] developed FedBalancer, a framework to actively select clients’ training samples in terms of the more “informative" data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, they introduced an adaptive deadline control scheme to predict the optimal deadline for each round, in order to further speed up global training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [74] observed that current client selection was coarse-grained due to their under-exploitation on the clients’ data and system heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Based on this finding, they proposed PyramidFL, a fine-grained client selection framework to speed up the FL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The key idea is to not only focus on the divergence of those selected participants but also fully exploited the data and system heterogeneity within selected clients to profile their utility more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, PyramidFL is able to achieve better performance compared to other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='4 Secure federated learning The original design of federated learning considers the security problem via exchanging parameters while keeping raw data in their own devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, recent studies have proved that attackers might steal the privacy information from the uploaded models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, more rigorous secure FL should be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In the following parts, we will introduce the attack methods and defense methods in FL scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='1 Attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The goal of backdoor at- tacks is to manipulate a subset of training data by injecting adversar- ial triggers such that DNN models will output incorrect prediction on the test set when the same trigger occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In federated learning, directly applying current backdoor attacks is unsuitable since the aggregation process might destroy the triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Bagdasaryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [7] is the first to backdoor federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They achieved their objective by proposing model replacement, which means the backdoor is injected to the joint model rather than raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 11, the attacker trained a model on the backdoor data using the constrain-and-scale technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this way, the av- eraging function is largely affected by this attack model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [141] proposed edge-case backdoors, which forced a model to misclassify on seemingly easy inputs that are unlikely to be part of the training or testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, they may exist on the tail of the input distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, it is extremely hard to detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [150] further developed distributed backdoor attack (DBA) to compromise FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They mainly took advantage of the distributed nature of FL, decomposing a global trigger pattern into separate local patterns and introducing them into the training set of different adversarial parties respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, DBA is more persistent and stealthy compared to centralized ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In FL models, backdoors can be inserted, but these backdoors are often not durable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', they do not remain in the model after poisoned updates stop being uploaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Since training occurs gradually in FL systems, an inserted backdoor may not survive until deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [172] proposed Neurotoxin, which is a simple modifi- cation to existing backdoor attacks that target parameters that are not changed in magnitude as much during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Gradients attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Gradients attack targets at reverse some pri- vacy information from gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In federated learning, exchanging gradients is a typical step for knowledge update and aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, gradient attack poses a high risk to the federal partic- ipants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [180] found since training occurs gradually in FL systems, an inserted backdoor may not survive until deploy- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' that it is possible to obtain the private training data from the publicly shared gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They first randomly generated a pair of “dummy” inputs and labels and used them to compute correspond- ing gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Then the gradients were compared to the shared ones and continually optimize the dummy inputs and labels to minimize the distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, the dummy data are close to the original ones and can peek into user privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Lam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [72] further realized gradients attack from the aggregated model up- dates/gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The authors leveraged the summary information from device analytics and reconstructed the user participant matrix, which invalided the current secure aggregation protocols [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [179] proposed Recursive Gradient Attack on Privacy (R-GAP), an approach to analyze how and when the target gradients can lead to the unique recovery of original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Concretely, the authors designed a recursive, depth-wise algorithm for recovering training data from the gradient information, which is the first closed-form algorithm that works on both CNN layers and FC layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [86] found that under certain defense settings, generative gradient leakage can still leak private training information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Model poison attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The goal of poison attacks is to induce the FL model to output the target label specified by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, Tolpegin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [136] implemented data poison attack by flipping the labels of training data from one class to another class in the local training epoch to mislead the global model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Although the aggregation process in FL can mitigate the attack to some extent, when the number of malicious clients becomes large, FL is inevitably poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [34] conducted the first systematic study on local model poisoning attacks to federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Based on this study, they proposed local model poisoning attacks to Byzantine robust federated learning via manipulating the local model parameters on compromised worker devices during the benign participants user C user B userA Diocal train Federated Gt Averaging User M constrain It+1 and Dbackdoor scaleConference’17, July 2017, Washington, DC, USA Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Figure 12: The illustration of BatchCrypt [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, the authors further stated two defense strategies and test their performance on the proposed attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='2 Defense methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' DP-based defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Differential privacy (DP) [30] has been widely used to prevent information leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The key idea is to add some noises to obfuscate the original information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, attackers are hard to infer the privacy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Federated learning also requires this type of protection since the uploaded model parameters can be easily exploited to extract sensitive infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [146] proposed NbAFL, a framework that applied DP into FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Specifically, they added noises to parameters of the local model at the client side before aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, the authors theoretically analyzed the convergence property of differentially private FL algorithms and proved the effectiveness of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Kairouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [64] presented a comprehensive end-to-end sys- tem, where they discretized the data and added discrete Gaussian noise before conducting secure aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In addition, the authors provided a novel privacy analysis for sums of discrete Gaussians and carefully analyzed the effects of data quantization and mod- ular summation arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Experiments demonstrated that their method can achieve comparable performance with 16 bits of pre- cision per value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [3] proposed a multi-dimensional Skellam mechanism, where two independent poisson random vari- ables are used to measure the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The authors applied their mechanism to FL and provided a novel algorithm that appropriately discretized the data and used the Skellam mechanism along with modular arithmetic to bound the range of the data and communica- tion costs before secure aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, they could achieve better privacy-accuracy trade-offs in a more efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' HE-based defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' HE-based FL aims to combine traditional Ho- momorphic Encryption (HE) and FL in a more suitable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' By applying HE, FL is able to aggregate client models without reveal- ing the information of the concrete model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, it is impossible to infer user privacy from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Hardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [50] proposed to encrypt FL with the homomorphic scheme in the field of privacy-preserving entity resolution and federated logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They bounded the difference between the empirical loss of their classifier on the true data and showed an improved conver- gence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, their experiments found that even rates for generalization cannot be significantly affected by entity resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [94] designed a secure FL framework through leveraging the additive property of partial homomorphic encryption, which effectively avoids the exposure of client models at the server side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, the authors introduced two optimization mechanisms to further enhance efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [165] proposed BatchCrypt, an efficient homomorphic encryption system for cross-Silo feder- ated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 12, there exist five typical steps to achieve a cross-silo FL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In the beginning, the aggregator needed to select a client to generate an HE key-pair and distribute it to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Then for each iteration, clients conducted local gradi- ent updates and further encrypted them by the public key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' These encrypted parameters were uploaded to the server where aggre- gation happened and the aggregated model is transferred to each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Finally, the client side decrypted the received information and implemented the local training as the next round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' BatchCrypt proposed two novel schemes to further improve efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' First, a feasible batch encryption scheme was presented to directly sum up the ciphertexts of two batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Second, an efficient analytical model dACIQ was presented to choose optimal clipping thresholds with the minimum cumulative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, BatchCrypt achieved 23×-93× training speedup while reducing the communication over- head by 66×-101×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' TEE-based defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The aforementioned secure FL approaches provide security guarantee mainly from the perspective of software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In real-world scenarios, hardware protection is also widely applied by designing crafted architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Trusted Execution Environment (TEE) is a trusted component that establishes an isolated region on the main processor to ensure the confidentiality and integrity of data and programs [5, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Compared to traditional encryption schemes such as homomorphic encryption, TEE is more efficient with respect to the computation cost since it only requires some simple operations to connect the trusted and untrusted part in OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Recently there have been a large number of works targeting at applying TEE to deep/federated learning, in order to achieve pro- tection from hardware level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For example, Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [107] proposed DarkneTZ that enabled executing DNNs more secure with TEE in an edge device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' They partitioned DNNs into a set of non-sensitive layers and sensitive layers, which are respectively processed by TEE or normal OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Here the partition choice is based on the under- lying system’s CPU execution time, memory usage, and accurate power consumption of different DNN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Besides, the authors developed a threat model to validate DarkneTZ’s robustness un- der the membership inference attack and the results showed that DarkneTZ could defend against this type of attack with negligible performance overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Based on the combination of DNNs and TEE, Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [106] further attempted to apply TEEs to federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Specifically, they proposed PPFL, a framework that limited privacy leakages in federated learning via implementing local training in TEEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 13, to address the challenge of limited memory size of TEEs, the authors designed a greedy layer-wise training to conduct local updates until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this way, this approach could support sophisticated settings such as training one or more layers (block) each time, which potentially speed up the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [171] proposed TEESlice, a system to provide a strong security guarantee while maintaining low inference latency with the help of TEEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Concretely, TEESlice executed the more private model slices on TEEs and others on normal AI accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a Single Client Aggregator HE Public Key Gradients ③Aggregation Aggregated HE Private Key Gradients ClientA ②Encryption @Decryption ClientNi Client B ② Encryption 4 @Decryption Gradient ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='Model computation update ① Gradient (5 Model computation updateRecent Advances on Federated Learning: A Systematic Survey Conference’17, July 2017, Washington, DC, USA Figure 13: The illustration of PPFL [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' result, TEESlice can achieve more than 10× throughput promotion with the same level of strong security guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='5 Fair federated learning Existing works of federated learning pay more attention to improv- ing learning performance based on the accuracy of the model and the time of learning task completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, the interests of the FL clients are often ignored and this may lead to unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The problem of fairness can occur in the whole FL training process, in- cluding client selection, model optimization, incentive distribution, and contribution evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The unfairness can have a negative impact on both the FL clients and the FL server, as clients are dis- couraged to join FL training, and servers are less likely to attract potentially high-quality clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Recently, to achieve fairness from different angles, various Fairness-Aware Federated Learning (FAFL) approaches have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this section, we will discuss recent FAFL methods in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='1 Fair client selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Unfairness in FL Client Selection mainly consists of three types, over-representation, under-representation, and never-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Suppose an FL system prefers to se- lect clients with high performance (such as a faster GPU), and clients with the highest performance may be selected much more than any other clients (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', over-representation), while clients with poor performance may be selected just a few times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', under- representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' At the same time, the client with the lowest per- formance may never be selected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', never-representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Addi- tionally, due to the heterogeneity among clients, fairness does not indicate giving everyone the same possibility to be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' It is important to balance the interests of the server and the interests of the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' If clients from specific groups are oversampled, the global FL model will be partial to their data, so the model’s perfor- mance will deteriorate [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Existing FAFL client selection methods can be partitioned into two categories, considering fairness factors and customization for each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 1) Fairness factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Fairness factors are designed to allow rarely selected clients, such as clients with lower computational abilities or smaller datasets, to join the FL training more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [152] proposed a client selection algorithm based on the Com- binatorial Multi-Armed Bandit (CMAB) framework to reduce the class imbalance effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Inspired by [76], Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [59] converts the original offline problem to an online Lyapunov optimization problem and uses dynamic queues to quantify the long-term guar- antee of the client participation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Moreover, Huang introduces a long-term fairness constraint to make sure the average client’s long-term chosen rate is above a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' After [59], Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [58] improves the performance by replacing dynamic queues to the Exp3 algorithms [6], and the fairness parameter determining the selection possibility in each round can be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, these works all design the fairness factor without considering the real-time contribution of individual clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [128] ad- dresses this problem and proposed a client selection policy with fairness constraints based on reputation, using a fairness parameter to balance reputation and the number of successful transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 2) Client customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' This approach pays attention to cus- tomized model settings or customized model procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Clients often receive the same initial models at the first training round in most current FL paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, clients with lower ca- pabilities, such as bad network connections, require more time to complete each training round and are likely to be kept out of sub- sequent rounds, leading to under-presented and never-presented problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To alleviate this problem, dynamically adapting the FL model framework or the training procedure based on client capa- bilities is often used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Caldas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [14] proposed Federated Dropout (FD), which dis- tributes sub-models with sizes suitable for each client based on their computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The process of FD is shown in Fig 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Although FD diminishes communication and local computation costs largely, it uses dropout operations and treats the neural net- works as black-box functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Bouaciada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' noticed this problem and proposed Adaptive Federated Dropout (AFD) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' AFD keeps an activation score map to generate the best-fit sub-model for each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FD and AFD both make sure clients with low capabilities could participate in FL training, but they do not provide custom pruned submodels to different clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To address this limitation, Horvath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [56] augmented FD to Ordered Dropout (OD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Dif- ferent from FD, OD drops neighboring components of the model despite random neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' OD divides clients with comparable com- putational capabilities into clusters, and clients in the same cluster apply the same dropout rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Moreover, OD applies the knowledge distillation method [54] to enhance feature extraction for smaller submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Clients’ communication capabilities can also affect client selec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' A poor network may cause too much retransmission and lead Server Clients Configuration Move to next block of layers Secure channel after conyergence TEE Transferring knowledge if any @ if transferring TEE Model initialization ② Public model broadcasting ③ Know- Reporting ledge Private ④ Layer-wise local training @ Class- Dataset model reporting @ ifier Secure aggregation @ Class- Class- Class- Class- @?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' ifier ifier ifier ifier ≥Data transmission ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='I ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Public layers Forward pass Private layers Backward passConference’17, July 2017, Washington, DC, USA Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Figure 14: The summary of the Federated Dropout (FD) train- ing procedure [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Figure 15: The illustration of ThrowRightAway (TRA) scheme [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' to extra delays in FL model training, which makes clients with a poorer network less likely to aggregate their model updates into the final model and leads to model bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To deal with this issue, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [178] proposed ThrowRightAway (TRA), a loss-tolerant FL framework that makes the FL training faster by ignoring few lost packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As is shown in Fig 15, at first every participating FL client reports their network conditions to the FL server, and the server divides the clients into two categories: sufficient type and insufficient type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Only the clients in the sufficient type can get a re-transmission request and then re-transmit their loss packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Apparently, the method can only be effective when the category is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' This method means assigning less work to clients with lower capabilities to make them available to pass threshold-based FL client selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [80] proposed FedProx which allowed each client performed partial training based on its accessible resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FedProx allows various local epochs, and thus more clients are encouraged to join the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='2 Fair model optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In the optimization during FL model training, the model may discriminate against definite preserved groups, or overfit some clients at the expense of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Recent works dealing with this issue can be approximately divided into two types: 1) objective function-based and 2) gradient-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 1) Objective function-based methods: Objective function-based methods focus on the global/local objectives of the FL model, such as minimizing the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Mohri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [108] proposed AFL, which aims to prevent the model overfitting any specific client at the expense of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' AFL just optimizes the global model for the target distribution made up of a mixture of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, this method only works for a small number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [178] proposed q-FFL to diminish the scalability limitation of AFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' q-FFL adds parameter q to reweigh the aggregate loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To improve the model robustness and maintain good-intent fairness at the same time, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [57] proposed fedMGDA+ which optimizes each FL client’s loss function respectively and simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Addressing the same issue, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [79] proposed Ditto, which improves fairness and robustness at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' While the methods mentioned all pay attention to the accuracy parity notion of fairness, there are also many kinds of research focusing on group fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [29] proposed AgnosticFair, which incorporates an agnostic fairness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Although it has good accuracy and fairness on unknown testing data distribution, it needs prior knowledge to design the re-weighting function, which limits its application in dynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [26] proposed FCFL, a multi-objective optimization framework that achieves good- intent fairness and group fairness at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Different from AFL, it minimizes the loss of the client with the worst performance and uses a smooth surrogate maximum function considering all clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' A fairness constraint is also added to calculate the disparities among all clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 2) Gradient-based approaches: Here, gradient means the local updated gradient of each client in every local iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [144] proposed the federated fair averaging (FedFV) algorithm, which aims to average clients’ gradients after mitigating potential conflicts among clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FedFV detects gradient conflicts through the cosine similarity and modifies both the direction and magnitude of the gradients by iteratively eliminating such conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, the estimated gradients may be incompatible with the latest updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='3 Fair contribution evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Contribution evaluation in FL learning indicates that an FL system can evaluate the contribution of different clients without accessing data from the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Many methods designed for non-privacy machine learning environments cannot be applied to FL scenarios directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' A general method is to evaluate each client’s model contribution to the aggregated FL model, and a fair evaluation is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Unfairness in contribution evaluation may lead to the free-rider issue [49], which implies that clients contribute little but can get similar benefits as the clients who contribute more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In this part we will introduce five types of existing FL contribution evaluation methods with their typical works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 1) Self-reported information: This method of evaluation contribu- tion is based on clients reporting their information actively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Most works based on this method believe their clients are reliable, which is not always correct in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Proposed by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [166], Hierarchically fair federated learning (HFFL) follows the idea of ’contribute more, get more reward’, which is proved effective in so- cial psychology [137], game theory [117] and bandwidth allocation [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Hence, it’s critical to figure out how to evaluate a client’s con- tribution and how much proportion of reward a client should get to ensure fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Data Shapley can be used to evaluate contribution in machine learning, but Shapley value is model-dependent [41] and incompatible with FL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' As a result, Zhang proposes evalu- ating contributions based on publicly verifiable factors of clients, such as cost of data collection, data volume, and data quality, to avoid the inconsistency of model-dependent methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To distribute proportional rewards to clients, Zhang introduces hierarchically fair federated learning (HFFL), as is shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The pub- licly verifiable factors determined by the clients’ consensus about each client are reported to the FL server, and the FL server then Server Server Clientnsufficient Sufficient speed retransfer(Loss) Adaptive Aggregation setZero(Loss)Recent Advances on Federated Learning: A Systematic Survey Conference’17, July 2017, Washington, DC, USA Figure 16: The illustration of hierarchically fair federated learning (HFFL) [166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' uses the information to rate each client, which at the same level are supposed to contribute to the model equally and will get the equal reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 2) Individual evaluation: Individual evaluation implies evaluating contribution through performance on specific tasks and pays more attention to individual performance instead of global performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The method often adopts two assumptions that both the server and the client are reliable and clients with a similar model to others are regarded to supply more contribution, which is not always feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To achieve fairness without sacrificing the model performance, Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [98] proposed a Collaborative Fair Federated Learning (CFFL) framework based on reputation, which uses a reputation mecha- nism to achieve collaborative fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Lyu definites collaborative fairness as the reward is proportional to the client’s contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Different standard FL process, CFFL allows clients to receive only the allocated aggregated updates according to their reputations, and the server is in charge of a reputation list which is updated in each communication round relying on the quality of the uploaded gradients of each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 3) Utility game: The utility game [45] refers to a game where each player chooses an available team to maximize their payoffs, while the universal social welfare is the total utility produced by all the teams cumulatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FL contribution evaluation methods based on utility games have a deep connection with profit-sharing schemes, and there are three diffusely used profit-sharing schemes: (1) Egalitarian: any part of the utility produced by a team is sepa- rated equally between the members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' (2) Marginal gain: the payoff of a player in a team is equal to the team gained when the player joined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' (3) Marginal loss: the payoff of a player in a team is equal to the team will lose if the player leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Among the three types above, the marginal loss scheme is the most commonly adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [140] proposed a deletion method to evaluate contributions in horizontal federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' This evaluation method consists of removing the instances supplied from one definite party, retraining the model, calculating the differ- ence between the original model and the new model, and using this difference to define the contribution of this party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Wang formulates the influence measure as follows, 𝐼𝑛𝑓 𝑙𝑢𝑒𝑛𝑐𝑒−𝑖 = 1 𝑛 𝑛 ∑︁ 𝑗=1 ���ˆ𝑦𝑗 − ˆ𝑦−𝑖 𝑗 ���, (11) where n is the size of the dataset, ˆ𝑦𝑗 is the model trained on all data prediction on jth instance, and ˆ𝑦−𝑖 𝑗 s the model trained without the ith instance prediction on jth instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Then Huang defines a party’s contribution as the total influence of all instances it possesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 𝐼𝑛𝑓 𝑙𝑢𝑒𝑛𝑐𝑒−𝐷 = ∑︁ 𝑖 ∈𝐷 𝐼𝑛𝑓 𝑙𝑢𝑒𝑛𝑐𝑒−𝑖, (12) For vertical horizontal learning, Huang uses shapley value which will be introduced in the next part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 4) Shapley value: Shapley value (SV) was first introduced in coop- erative game theory [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Different from marginal loss, SV-based FL contribution evaluation approaches can reflect the contribu- tion of a client’s own data, in spite of its joining order, and can produce a fairer evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, SV’s computational complex- ity is 𝑂(2𝑛), so many approaches have been proposed to improve efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 4 PREVALENT FRAMEWORKS OF FEDERATED LEARNING In this section, we will introduce several prevalent frameworks of federated learning, including FedLab, Flower, FedML, FATE, and FedScale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FedLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Since most FL schemes follow the same basic steps and just a few changes in some steps are needed in different scenar- ios, Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [164] proposed FedLab [126], which is designed flexible and customizable, offers essential functional modules, and has highly customizable interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Two main roles in FL settings are provided: Server and Client, and both of them are made up of two components, NetworkManager and ParameterServerHan- dler/Trainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The design focuses more on communication efficiency and FL algorithm effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To support methods improving Communication Efficiency, FedLab uses tensor-based communi- cation, supports customizable communication agreement, and im- plements both Synchronous and Asynchronous communication patterns according to Federated Optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' For Opti- mization Effectiveness, FedLab applies a "high-cohesion and low- coupling" optimization module which provides aggregation and data partition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Additionally, FedLab can be used in various scenarios, such as Standalone, Cross-process and Hierarchical FL simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Flower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Due to the lack of frameworks that are able to support scalably executing FL methods on mobile and edge devices, Beutel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [9] proposed Flower [2], which can run large-scale FL experi- ments on different FL device scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Flower makes it possible to smoothly transition from experimental research to system research on a large group of real edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Designed to be scalable, client- agnostic, communication-agnostic, privacy-agnostic, and flexible, Flower has extensive implementations, such as communication stack, serialization, ClientProxy, and Virtual Client Engine(VCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FedML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Proposed by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [52], FedML [35] aims to solve the lack of support for diverse FL computing paradigms, support of diverse FL configurations, and standardized FL algorithm im- plementations and benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FedML library is mainly made up of high-level API FedML-API and low-level API FedML-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' To Central server f2 f3 f3 m3 Data volume Level 1 m2 f2 Level 2 m1 f1 Level 3 Level 1 Level 2 Level 3Conference’17, July 2017, Washington, DC, USA Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' support FL on real-world hardware platforms, FedML offers on- device FL testbeds called FedML-Mobile and FedML-IoT which are built upon real-world hardware platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FedML programming interface allows worker/client-oriented programming, message def- inition beyond gradient and model, topology management, trainer and coordinator, privacy, security, and robustness, so users can just pay attention to algorithms implementations and ignore the backend details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Since most open-sourced frameworks are research-oriented and lack the implementation on industry, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [95] proposed FATE(Federated AI Technology Enabler) [145], which is the first production-oriented platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Built on FederatedML, FATE pro- vides Private Set Intersection(PSI), and uses distributed computa- tion framework Eggroll to improve computation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FATE provides three main components, scheduling system FATE-Flow, visualization tool FATE-Board, and high-performance inference platform FATE-Serving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In addition, kinds of deployments are sup- ported, including building FATE on top of Kubernetes in data cen- ters through KubeFATE, manual or docker deployments on Mac and Linux, and cross-cloud deployment and management through FATE-cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' FedScale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [70] proposed FedScale [131], which con- tains many realistic FL datasets for different tasks, and FedScale Runtime which is an automated evaluation platform aiming to simplify and standardize FL evaluation in more realistic environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The raw data of FedScale datasets are collected from various sources, processed into consistent formats, sorted into different FL use cases and packed into standardized APIs for users to easily use in other frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' The evaluation platform, FedScale Runtime, is equipped with both mobile and cluster backends to enable both on-device FL evaluation on smartphones, and FL evaluations in real deployments and in-cluster simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 5 DISCUSSION This section summarizes some limitations of current FL approaches and discusses possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Dynamic federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Current federated learning ap- proaches assume that data in each client are stable and unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, in real-world scenarios, clients may be in an ever-changing environment, where the local data are continuously observed and processed by sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Under this condition, directly conducting conventional training and aggregation will suffer from the cata- strophic forgetting problem, which indicates that the prior knowl- edge learned by the model might be forgotten as new data arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Incremental learning [15, 96, 148] is a hot research topic to address the issue, targeting at learning new knowledge while maintain- ing the ability to recognize previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In the future, how to effectively combine federated learning and incremental learning is worth exploring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Decentralized federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' A central server is of vi- tal importance to traditional federated learning since aggregation needs to be conducted in this side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Considering that the third-party server may not be honest, uploading parameters or gradients to it potentially exists security risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Therefore, it is necessary to achieve federated learning without a server involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Although He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' [53] has made a preliminary attempt to decentralized FL, they only target logistic regression and the experiments are insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' How to accomplish general decentralized FL still remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Scalability of federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Recent FL papers paid more attention to designing new algorithms to improve FL performance under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, they ignore the scalability property, which determines whether we could operate large-scale FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In many cooperation scenarios, there might be a huge number of parties and we should provide guidance to the cooperation im- provement as the number of participants increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In a word, FL scalability deserves future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Unified benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Although a large number of datasets have been used for evaluating the performance of FL, there is still a lack of a unified benchmark to align the results for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' On one hand, in order to achieve different federated goals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=', personalization, robustness), researchers use different datasets to test the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' On the other hand, two typical types of FL, horizontal FL and vertical FL, also apply distinctive datasets to demonstrate the performance of different FL types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Thus a unified benchmark will definitely benefit the FL community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 6 CONCLUSION Federated learning has gained more and more attention due to its ability of collaboratively generating a global model without leaking sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Recent surveys have summarized many re- lated works devoted to offering a comprehensive understanding to developers and readers in this community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' However, most of them focus on a specific aspect of FL or fail to catch the latest progress of this hot research topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' This paper provides a systematic survey, which investigates recent development on federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' By analyzing the pipeline and challenges of FL, we propose a taxonomy with different FL aspects involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' In addition, we also explore some practical FL frameworks and characterize their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' Finally, some limitations and future direction are concluded in order to promote the evolution of the FL community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' REFERENCES [1] Durmus Alp Emre Acar, Yue Zhao, Ruizhao Zhu, Ramon Matas, Matthew Mat- tina, Paul Whatmough, and Venkatesh Saligrama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' 2021.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content=' arXiv preprint arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} +page_content='01578 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfWPy8/content/2301.01299v1.pdf'} diff --git a/PdFJT4oBgHgl3EQf1y2a/content/tmp_files/2301.11653v1.pdf.txt b/PdFJT4oBgHgl3EQf1y2a/content/tmp_files/2301.11653v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bf819618b81fc6f5a8c6cb9d88e0c06e04fb67b --- /dev/null +++ b/PdFJT4oBgHgl3EQf1y2a/content/tmp_files/2301.11653v1.pdf.txt @@ -0,0 +1,1378 @@ +Exact linear reductions of dynamical models +Alexander Demin∗, Elizaveta Demitraki† , and Gleb Pogudin‡ +Abstract. Dynamical models described by ordinary differential equations (ODEs) are a fundamental tool in +the sciences and engineering. +Exact reduction aims at producing a lower-dimensional model in +which each macro-variable can be directly related to the original variables, and it is thus a natural +step towards the model’s formal analysis and mechanistic understanding. We present an algorithm +which, given a polynomial ODE model, computes a longest possible chain of exact linear reductions +of the model such that each reduction refines the previous one, thus giving a user control of the +level of detail preserved by the reduction. This significantly generalizes over the existing approaches +which compute only the reduction of the lowest dimension subject to an approach-specific constraint. +The algorithm reduces finding exact linear reductions to a question about representations of finite- +dimensional algebras. We provide an implementation of the algorithm, demonstrate its performance +on a set of benchmarks, and illustrate the applicability via case studies. Our implementation is freely +available at https://github.com/x3042/ExactODEReduction.jl. +Key words. ordinary differential equations, exact reduction, lumping, dimensionality reduction, matrix algebras +MSC codes. 34C20, 34-04, 16G10 +1. Introduction. Ordinary Differential Equations (ODEs) provide a powerful and expres- +sive language for describing systems evolving in real-time and, thus, are widely used both +in the sciences and engineering. This motivates development of formal methods to analyse +the structure of models defined using ODEs. One important problem which has been studied +actively in the past decade from this angle is model reduction [16, 17, 6, 7, 28]. +In general, model reduction refers to a variety of techniques aiming at replacing the model +of interest with a simpler one while preserving, at least approximately, some of the important +features of the original model. Traditionally, approximate methods such as, e.g., balanced +truncation [1] have been employed. Exact model reduction is a complementary approach in +which one lowers the dimension of the model without introducing approximation errors. Such +reductions are of particular interest in the context of performing formal analysis or deriving +mechanistic insights. +In this paper, we will focus on an important class of such reductions, exact linear lumpings, +which correspond to finding a self-consistent system of differential equations for a set of macro- +variables in which each macro-variable is a linear combination of the original variables. The +case of the macro-variables being sums of the original variables has been extensively studied, +see e.g. [16, 8, 17, 7]. In particular, ERODE software has been developed [6] which efficiently +finds the optimal partition of the original variables into macro-variables. A recent software +CLUE [28] was a step towards lifting these restrictions on the macro-variables. Unlike the prior +∗Corresponding author. National Research University, Higher School of Economics, Moscow, Russia +alexander.demin.eternal@gmail.com +†National Research University, Higher School of Economics, Moscow, Russia egdemitraki@edu.hse.ru +‡LIX, +CNRS, +´Ecole +Polytechnique, +Institute +Polytechnique +de +Paris, +Palaiseau, +France +gleb.pogudin@polytechnique.edu +1 +arXiv:2301.11653v1 [eess.SY] 27 Jan 2023 + +2 +A. DEMIN, E. DEMITRAKI, AND G. POGUDIN +approaches, the macro-variables produced by CLUE are allowed to involve any coefficients +(not just zeroes and ones as before), and, thus, the dimension of the reduced model could +be significantly lower [28, Table 1]. However, the input of the algorithm consisted not only +of a model but also of linear forms in the state variables to be preserved (the observables). +Such a set of observables may or may not be available, and guessing it correctly is crucial +for finding low-dimensional reductions. In this paper, we aim at taking the best from both +worlds: requiring only a model as input (as ERODE) and allowing the macro-variables to be +any linear combinations of the original variables (as CLUE). +The main result of the paper is an algorithm for finding arbitrary exact linear reductions +when given only a polynomial ODE model with rational coefficients. Note that the question +of finding such an arbitrary linear lumping of the lowest possible dimension may not be +the most meaningful one since any linear first integral yields a reduction of dimension one +with constant dynamics. Instead, our algorithm finds a longest possible chain of lumpings +in which each reduction refines the next one (for details, see Section 2) so that a user can +choose the desired level of details to be preserved by reduction by moving along the chain and +may find reductions which would likely be missed by ERODE and CLUE (e.g., see Example 2.4 +and subsection 5.2). Such generality comes with a price: our software is typically slower than +CLUE and ERODE. +Our algorithm is based on combining the connection of the linear lumping problem to the +problem of finding a common invariant subspace of a set of matrices [25, 28] with the structure +theory of finite-dimensional algebras. We use the general framework of existing algorithms +over finite [33] and algebraically closed [9, 36] fields and achieve desired efficiency by +• sparsity-aware algorithm for finding a basis of an algebra (Subsection 3.2); +• exploiting the structure of the input to compute mostly with rational numbers and +postponing passing to algebraic number fields as much as possible; +• using sparse linear algebra and modular computation to avoid large matrices and +expression swell, respectively. +We implemented our algorithm, and the implementation is publicly available at https: +//github.com/x3042/ExactODEReduction.jl. We evaluate its performance on a set of bench- +marks from the BioModels database [27], a large collection of models from life sciences, and +demonstrate the produced reduction for two case studies. +The rest of the paper is organized as follows. In Section 2, we give precise definition of +exact linear reduction, and formulate explicitly the algorithmic problem we solve in the paper. +Section 3 contains detailed description of the algorithm and its justification. We describe our +implementation and report its performance in Section 4. Section 5 contains the case studies +describing the reductions produced by our software. +2. Problem statement. In the paper, the transpose of a matrix M is denoted by MT . +For a vector x = (x1, . . . , xn) of indeterminates, by C[x] (resp., R[x], Q[x]) we will denote the +set of polynomials in x with complex (resp., real, rational) coefficients. +Definition 2.1 (Lumping). Consider a system of ODEs with polynomial right-hand side of +the form +(2.1) +x′ = f(x), + +EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS +3 +where x = (x1, . . . , xn), f = (f1, . . . , fn), and f1, . . . , fn ∈ C[x]. We say that a linear trans- +formation y = xL with y = (y1, . . . , ym), L ∈ Cn×m, and rank L = m is a lumping of (2.1) if +there exist g = (g1, . . . , gm) with g1, . . . , gm ∈ C[y] such that +y′ = g(y) +for every solution x of (2.1). We say that m is the dimension of the lumping. The variables +y in the reduced system are called macro-variables. +Throughout this section, we will work with the following running example [30, Example 1]. +We will consider a chemical reaction network consisting of +• A chemical species X. +• Species AUU, AUX, AXU, and AXX. Each of them is one of the states of a molecule +A with two identical binding sites, which can be either unbound (U in the subscript) +or bound (X in the subscript) to X. +For simplicity, we will assume that all the reaction rates are equal to one. The dynamics of +the network is defined by the following reactions: +(2.2) +X + AU∗ +AX∗ +X + A∗U +A∗X +where ∗ ∈ {X, U}. Under the laws of the mass-action kinetics [14, Ch. 7], the reactions (2.2) +yield the following ODE system (where [S] denotes the concentration of the species S): +(2.3) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +[X]′ = [AXU] + [AUX] + 2[AXX] − [X]([AXU] + [AUX] + 2[AUU]), +[AUU]′ = [AXU] + [AUX] − 2[X][AUU], +[AUX]′ = [AXX] + [X][AUU] − [X][AUX] − [AUX], +[AXU]′ = [AXX] + [X][AUU] − [X][AXU] − [AXU], +[AXX]′ = [X][AXU] + [X][AUX] − 2[AXX]. +Example 2.2 (Conservation laws as lumpings). We show that matrix L = +� +0 +1 +1 +1 +1 +�T +yields a lumping of (2.3). We have +y = +� +[X] +[AUU] +[AUX] +[AXU] +[AXX] +� +· L = [AUU] + [AUX] + [AXU] + [AXX]. +Using (2.3) one can check that y′ = 0, so we can take g(y) = 0. +Indeed, y is the total +concentration of A and must be constant. Furthermore, any linear conservation law yields a +lumping of dimension one. +Example 2.3 (More informative lumping). +Another lumping for the same system (2.3) is +given by the matrix +L = +� +� +� +� +� +� +1 +0 +0 +0 +2 +0 +0 +1 +1 +0 +1 +1 +0 +0 +2 +� +� +� +� +� +� +=⇒ +� +� +� +� +� +y1 = [X], +y2 = 2[AUU] + [AUX] + [AXU], +y3 = [AUX] + [AXU] + 2[AXX]. + +4 +A. DEMIN, E. DEMITRAKI, AND G. POGUDIN +The macro-variables will satisfy a self-contained system +y′ +1 = y3 − y1y2, +y′ +2 = y3 − y1y2, +y′ +3 = −y3 + y1y2. +The rationale behind this reduction is that y2 and y3 are “concentrations” of unbound and +binded sites, respectively. +The above examples demonstrate that one system can have several lumpings (in fact, (2.3) +has more), so a natural question is how to find useful lumpings. The state-of-the-art software +tools CLUE [28] and ERODE [6] approach this question by finding the lumping of the smallest +dimension satisfying certain constraints: +• preserving some quantities of interest unlumped (for CLUE [28]); +• or coming from a partition of the state variables (for ERODE [6]). +Both constraints may be too restrictive: not all interesting lumpings come from a partition +(like the ones in Examples 2.2 and 2.3; see also [28, Table 1]), and it may be complicated to +guess in advance meaningful quantities to preserve. +Example 2.4 (Example hard for CLUE and ERODE, see also Subsection 5.2). Consider another +chemical reaction network [15, Eq. (19.20)] (originally due to Daniel Knight): +E∗ +E +E + S +ES +E + P +E∗ + S +As in the case of (2.2), we transform the reactions into an ODE system using the law of +mass-action kinetics taking all the rates to be one1: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +[E]′ = 2[ES] + [E∗] − [E][S] − [E][P] − [E], +[S]′ = 2[ES] − [E][S], +[P]′ = [ES] − [E][P], +[ES]′ = [E][S] + [E][P] − 3[ES], +[E∗]′ = [E] + [ES] − [E∗] +One meaningful linear reduction is y = [E] + [ES] − [E∗] with the equation y′ = −2y. The +macro-variable y can be understood as a potential between the amount of E (typically en- +zyme), both in the free form E and as a part of the complex ES, and E∗ (typically inactivated +enzyme).This reduction does not come from a subdivision of the species, so it cannot be found +by ERODE. Furthermore, finding it using CLUE would require knowing this macro-variable in +advance. +An alternative approach would be to find all the lumpings and let the user to choose +which ones to use. The problem is that there may be easily an inifinite number of lumpings, +for example, similarly to Example 2.2, one can show that the matrix +L = +� +α +1 +1 + α +1 + α +1 + 2α +�T +1For the reduction in this example, it is sufficient that the rates of ES → E+ + S and E → E+ coincide + +EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS +5 +yields a lumping of (2.3) for every number α. +Furthermore, as we will explain later, the +lumpings are in a bijection with the invariant subspace of certain matrices coming from the +Jacobian of f(x) and (at least for arbitrary matrices) the invariant subspaces can form an +arbitrary algebraic variety [32]. +The approach we take in this paper is to find a sequence of reductions refining each other +with the guarantee that this sequence is of maximal possible length. +Definition 2.5 (Chain of lumpings). For an ODE system of the form x′ = f(x), a sequence +of linear transformations +y1 = xL1, y2 = xL2, . . . , yℓ = xLℓ, +where L1 ∈ Cn×m1, . . . , Lℓ ∈ Cn×mℓ, is called a chain of lumpings if +1. 0 < m1 < . . . < mℓ < n; +2. yi = xLi is a lumping of (2.1) for every 1 ⩽ i ⩽ ℓ; +3. for every 1 < i ⩽ ℓ, there exists a matrix Ai such that Li−1 = LiAi. +The latter means that the reductions given by L1, . . . , Lℓ refine each other. +Such a chain +(L1, . . . , Lℓ) will be called maximal if it is not contained as a subsequence in any longer chain. +We will show (see Corollary 3.6) that all maximal chains are of the same length, so they +are also the longest possible chains. Given a maximal chain of lumpings, a user can “zoom +in/out” by going left/right along the chain depending on the desired tradeoff between the size +of the reduced model and the amount of information retained. Thus, we can now formally +state the main problem studied in this paper. +Main problem 2.6. +Given a system x′ = f(x) with f being a vector of polynomials over Q; +Compute a maximal chain of lumpings for the system. +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +[X]′ = [AXU] + [AUX] + 2[AXX] − [X]([AXU] + [AUX] + 2[AUU]), +[AUU]′ = [AXU] + [AUX] − 2[X][AUU], +[AUX]′ = [AXX] + [X][AUU] − [X][AUX] − [AUX], +[AXU]′ = [AXX] + [X][AUU] − [X][AXU] − [AXU], +[AXX]′ = [X][AXU] + [X][AUX] − 2[AXX]. +� +� +� +� +� +� +� +� +� +y′ +4,1 = y4,3 + 2y4,4 − y4,1(y4,3 + 2y4,2), +y′ +4,2 = y4,3 − 2y4,1y4,2, +y′ +4,3 = 2y4,4 − y4,3y4,1 − y4,3 + 2y4,2y4,1, +y′ +4,4 = y4,1y4,3 − 2y4,4 +� +� +� +� +� +y′ +3,1 = y3,3 − y3,1y3,2, +y′ +3,2 = y3,3 − y3,1y3,2, +y′ +3,3 = −y3,3 + y3,1y3,2. +� +y′ +2,1 = 0, +y′ +2,2 = 0. +� +y′ +1,1 = 0. +� +y4,1 = [X], y4,2 = [AUU], +y4,3 = [AUX] + [AXU], y4,4 = [AXX] +� +� +� +� +� +y3,1 = y4,1, +y3,2 = y4,3 + 2y4,2, +y3,3 = y4,3 + 2y4,4 +� +y2,1 = y3,2 + y3,3, +y2,2 = y3,1 + y3,3 +y1,1 = y2,1 +Figure 1: Maximal chain of lumpings for (2.3) and the corresponding reductions + +6 +A. DEMIN, E. DEMITRAKI, AND G. POGUDIN +Example 2.7 (Maximal chain of lumpings for (2.3)). Figure 1 shows a chain of lumpings +and the corresponding reductions for our example system (2.3). +The blocks contain the +reduced systems and the arrows are labeled with the transformations between the consecutive +reductions (matrices Ai in the terms of Definition 2.5). This chain of reductions includes our +preceding Examples 2.2 and 2.3 as y1 and y3, respectively. +In this example we have the original dimension n = 5 and the dimensions of the reductions +m1 = 1, m2 = 2 , m3 = 3, m4 = 4, so this chain is clearly maximal. +3. Algorithm. For finding a maximal chain of lumpings, we first use theory developed +in [28] to reduce the problem to a problem about common invariant subspaces of a set of +matrices (Subsection 3.1) and then solve the new problem using the structure theory of +finite-dimensional algebras (Subsections 3.3 to 3.5). +The overall algorithm is summarized +in Subsection 3.6. +3.1. Reduction to the search for common invariant subspaces. Let x′ = f(x) be an ODE +system in variables x = (x1, . . . , xn) and f being a row vector of polynomials f1, . . . , fn ∈ C[x]. +Let J(x) be the Jacobian matrix of f with respect to x. +We denote the monomials in x +appearing in J(x) by m1(x), . . . , mN(x). Then J(x) can be written uniquely as +(3.1) +J(x) = +� +∇f1 +. . . +∇fn +� += +N +� +i=1 +Ji · mi(x), +where ∇g := +� +∂g +∂x1 +. . . +∂g +∂xn +�T +and J1, . . . , JN are constant matrices. +Example 3.1. Consider the system +x′ +1 = x1 − 2x2 +2, +x′ +2 = −x2 + x2 +2. +In this case the decomposition (3.1) will be +J(x1, x2) = +� +1 +0 +−4x2 +−1 + 2x2 +� += +�1 +0 +0 +−1 +� ++ +� 0 +0 +−4 +2 +� +x2. +Lemma 3.2. Using the notation above, linear transformation y = xL, where L ∈ Cn×m, +is a lumping of x′ = f(x) if and only if the column space of L is invariant with respect to +J1, . . . , JN. +Proof. For the case L ∈ Rn×m, the statement follows from [28, Lemmas S.I.1 and S.II.1]. +The proof of [28, Lemmas S.I.1] remains correct after replacing R with C, and the proof of [28, +Lemmas S.II.1] will be correct for the case of C if the real inner products are replaced with +the complex ones. +Corollary 3.3. A sequence of linear transformations y1 = xL1, . . . , yℓ = xLℓ is a chain of +lumpings if and only if the column spaces V1, V2, . . . , Vℓ of L1, . . . , Lℓ satisfy +• Vi is invariant with respect to J1, . . . , JN for every 1 ⩽ i ⩽ ℓ; +• {0} ⊊ V1 ⊊ . . . ⊊ Vℓ ⊊ Cn. +Furthermore, the chain of lumpings is maximal if and only if the chain V1, . . . , Vℓ is not a +subsequence of a chain of subspaces satisfying the two properties above. + +EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS +7 +In order to search for such chains of invariant subspaces, we will use theory of finite +dimensional matrix algebras. +Definition 3.4 (Matrix algebra). Let k be a field (e.g., k = Q, R, C). A subspace A ⊆ kn×n +of matrices is called an algebra if it is closed under multiplication and contains the identity +matrix. +For a finite set A1, . . . , Am ∈ kn×n, we denote the smallest algebra containing A1, . . . , Am +by ⟨A1, . . . , Am⟩. This algebra is equal to the span of all possible products these matrices. +For an ODE system x′ = f(x) with x = (x1, . . . , xn) and f1, f2, . . . , fn ∈ C[x], we consider +the coefficients J1, . . . , JN of the Jacobian matrix of f(x) written as a polynomial in x as +in (3.1). We will call the algebra ⟨In, J1, . . . , JN⟩ (where In is the identity n × n-matrix) the +Jacobian algebra of the system x′ = f(x). +Example 3.5. +• Let Tn be the set of all upper-triangular matrices in kn×n. Since the product of two +upper-triangular matrices is upper-triangular again, Tn is an algebra. +• Consider the system from Example 3.1. Its Jacobian algebra is +��1 +0 +0 +1 +� +, +�1 +0 +0 +−1 +� +, +� 0 +0 +−4 +2 +�� += {MT | M ∈ T2}. +Since a subspace V ⊂ Cn is invariant with respect to J1, . . . , JN if and only if it is invariant +with respect to ⟨In, J1, . . . , JN⟩, that is, invariant w.r.t. any element of the algebra, we will +further focus on finding invariant subspaces of this Jacobian algebra. An immediate benefit +is that we can use the Jordan-H¨older theorem [13, Theorem 1.5.1] to clarify our notion of +the maximal chain of lumpings (Definition 2.5): the definition only requires that a maximal +chain cannot be further refined, and this, in general, does not preclude the existence of longer +chains. The following direct consequence of [13, Theorem 1.5.1] guarantees that a maximal +chain indeed has the maximal possible length. +Corollary 3.6. For a given ODE system x′ = f(x), all maximal chains of lumpings have +the same length. +3.2. Generating the algebra. For performing explicit computation with the Jacobain +algebra ⟨In, J1, . . . , JN⟩ (Definition 3.4), we will compute its basis. +Algorithm 3.1 gives a +simplified version of our approach, which is essentially [28, Algorithm 2] applied to matrices +instead of vectors. Similarly to [28], we employ modular computation (cf. [28, Algorithm 3]) +to avoid the intermediate expression swell and use sparse linear algebra. +Building upon this straightforward adaptation of the approach from [28], we significantly +improve the performance by taking further advantage of the sparsity of the input and output. +In applications, only a couple of J1, . . . , JN (typically, the constant term of (3.1)) are not so +sparse, and the rest are extremely sparse; furthermore, the basis of the Jacobian algebra also +can be often chosen to be very sparse. Hence, many of matrices C from (Step 2)(b)i will +be sparse as well. However, some of the matrices computed at the intermediate steps may be +still quite dense slowing down the whole algorithm. We deal with the issue by temporarily +deferring (Step 2)(b)ii for relatively dense matrices C and then, once the outer loop exits +signaling that P is empty, we add each of the deferred matrices to P and restart the iteration. + +8 +A. DEMIN, E. DEMITRAKI, AND G. POGUDIN +This way we ensure that we have generated enough sparse matrices in the algebra so that +the reductions of the dense matrices will be more sparse now. Thanks to this optimization, +Algorithm 3.1 is not a bottleneck in our computation which it was when we used the approach +from [28] directly. +Algorithm 3.1 Finding a basis of matrix algebra (basic version) +Input a set of square matrices A1, . . . , Aℓ ∈ kn×n; +Output a basis S of the smallest linear subspace A ⊆ kn×n containing all possible products +of A1, . . . , Aℓ; +(Step 1) Set S to be any basis of the linear span of A1, . . . , Aℓ and let P := S. +(Step 2) While P ̸= ∅ do +(a) Take B to be an element of P and remove it from P. +(b) For every A in {A1, . . . , Aℓ} do +i. +Compute C := AB and reduce C w.r.t. S via Gaussian reduction. +ii. +If C ̸= 0, set S := S ∪ {C} and P := P ∪ {C}. +(Step 3) Return S. +3.3. Search for invariant subspaces: algebraic preliminaries. For this section, we fix +a ground field k of characteristic zero. The cases we are mostly interested in are rational +numbers Q, algebraic numbers Q, and complex numbers C. +Definition 3.7 (Radical of an algebra). +Let A ⊆ kn×n be an algebra. +• A subspace I ⊆ A is called an ideal (resp., left ideal) if AB, BA ∈ I (resp., AB ∈ I) +for every A ∈ A and B ∈ I. +• An ideal (resp., left ideal) I ⊆ A is nilpotent if there exists N such that the product +of any N elements of I is zero. +• The set of all elements A ∈ A such that the left ideal A · A is nilpotent is called the +radical of A. It is a nilpotent ideal of A by [13, Theorems 3.1.6, 3.1.10]. +Example 3.8. Let Tn be the set of all upper-triangluar matrices in kn×n. Consider a subset +Un ⊂ Tn of strictly upper-triangular matrices. One can easily verify that Un is an ideal and +the product of any n elements of Un is zero. Since, for every A ∈ Un, we have Tn · A ⊆ Un, +we deduce that Un is the radical of Tn. +Dixon’s theorem [5, Theorem 11] implies that the radical of an algebra A ⊆ kn×n can be +computed by finding the kernel of a square matrix of order dim A ⩽ n2. The relevance of the +notion of radical to our problem is demonstrated by the following lemma. +Lemma 3.9. Let A ⊆ kn×n be an algebra, and let R ⊂ A be its radical. If R ̸= {0}, then +the intersection +� +R∈R +Ker R is nontrivial and is invariant w.r.t. A. +Proof. Let N be the smallest integer such that the product of any N elements of R is +zero. Then there exists 0 ̸= M ∈ kn×n which is a product of N − 1 elements of R. Then, we +have RM = 0 for every R ∈ R, so V := � +R∈R +Ker R ⊇ Im M is nontrivial. + +EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS +9 +Consider v ∈ V , A ∈ A, and R ∈ R. Since RA ∈ R, we have RAv = 0, so Av ∈ Ker R. +Thus, V is invariant w.r.t. A. +Example 3.10. Consider the system from Example 3.1. +In Example 3.5, it was shown +that the Jacobian algebra of this system is the set of lower triangular matrices. Similarly +to Example 3.8 we find that the radical of this algebra is +�0 +0 +λ +0 +� +. The common kernel of +the radical is spanned by the second basis vector yielding the reduction y′ = −y + y2 (with +y = x2). +Definition 3.11 (Semisimple algebra). An algebra A ⊆ kn×n is called semisimple if its rad- +ical is zero. +We will use the following characterization of semisimple algebras. +Theorem 3.12 (Wedderburn-Artin, [13, Theorems 2.4.3 and 2.6.2]). +Let A ⊆ kn×n be a +semisimple algebra. Then there exist +1. algebras A1 ⊆ kn1×n1, . . . , Aℓ ⊆ knℓ×nℓ such that Ai does not have a nontrivial proper +invariant subspace in kni for every 1 ⩽ i ⩽ ℓ, +2. integers m1, . . . , mℓ such that n1m1 + . . . + nℓmℓ = n, +3. a basis in kn +such that, in this basis, we have +(3.2) +A = +� +� +�Diag(A1, . . . , A1 +� +�� +� +m1 times +, . . . , Aℓ, . . . , Aℓ +� +�� +� +mℓ times +) | A1 ∈ A1, . . . , Aℓ ∈ Aℓ +� +� +� , +where Diag(B1, . . . , BN) denotes the block-diagonal matrix with blocks B1, . . . , BN. +Example 3.13. Consider the set of all matrices of the form +� +� +� +� +a +b +0 +0 +−b +a +0 +0 +0 +0 +c +0 +0 +0 +0 +c +� +� +� +� , +where a, b, c ∈ Q. +This is a semisimple algebra in the form (3.2) with ℓ = 2, m1 = 1, and m2 = 2. +In the case ℓ = 1 and m1 = 1 in the decomposition (3.2) from Theorem 3.12, there are no +invariant subspaces in kn but there still may be invariant subspaces in k +n if k ̸= k, where k is +the algebraic closure of field k. These subspaces can be found using the center of the algebra. +Definition 3.14 (Center/Centralizer). +Let A ⊆ kn×n be an algebra. +• The center of A is the set of all M ∈ A such that MA = AM for every A ∈ A. +• The centralizer of A is the set of all M ∈ kn×n such that MA = AM for every A ∈ A. +Since, for every fixed A, AM = MA is a system of linear equations in the entries, the +center and centralizer can be computed by solving a system of linear equations. +Lemma 3.15. Let A ⊆ kn×n be an algebra without nontrivial proper invariant subspaces in +kn. Let C be the center of A. For every C ∈ C, every eigenspace of C is an invariant subspace +of A in k +n. + +10 +A. DEMIN, E. DEMITRAKI, AND G. POGUDIN +Proof. Let V be an eigenspace of C corresponding to the eigenvalue λ. Let A ∈ A. Then +C(Av) = (CA)v = (AC)v = λAv. +Lemma 3.16. Let A ⊆ Qn×n be a semisimple algebra. Let M ∈ A be a matrix such that +the characteristic polynomial of M is of the form p(t)d, where p(t) is Q-irreducibe. Let Z +and C be the center and centralizer of A, respectively. Then the equality dim C = d2 dim Z is +equivalent to the fact that, in the Wedderburn-Artin decomposition (3.2) of A, we have ℓ = 1 +and m1 = d. +Proof. We consider the Wedderburn-Artin decomposition (3.2) of A. For every 1 ⩽ i ⩽ ℓ, +we denote the center of Ai by Zi. Then dim Z = dim Z1 +. . .+Zℓ. The number of irreducible +factors of a characteristic polynomial of any element of A will be at least m1 + . . . + mℓ, so +d ⩾ m1 +. . .+mℓ. A direct computation using the Schur’s lemma [13, Theorem 2.1.1] implies +that the centralizer C of A is isomorphic to Matm1(Z1) × . . . × Matmℓ(Zℓ), where Matmi(Zi) +denotes the space of mi × mi-block matrices with each block being an element of Zi (cf. [13, +Theorem 2.6.4]). Therefore +dim C = m2 +1 dim Z1 + m2 +2 dim Z2 + . . . + m2 +ℓ dim Zℓ. +Bounding the right-hand side, we can write +dim C ⩽ (m1 + . . . + mℓ)2(dim Z1 + . . . + dim Zℓ) ⩽ d2 dim Z +Both inequalities will be equalities if and only if ℓ = 1 and d = m1, and this proves the +lemma. +3.4. Search for invariant subspaces: how to find one. In this subsection, we present Al- +gorithm 3.2 for finding an invariant subspace if there is any. The rest of the subsection is +devoted to justifying its correctness and termination, see Proposition 3.20. +Proposition 3.17. Let A ⊆ Qn×n be a semisimple algebra such that there are no nontriv- +ial proper A-invariant subspaces in Qn. Let M1, . . . , MN be a linear basis of A. Then the +polynomial +det(x1M1 + . . . + xNMN) ∈ Q[x1, . . . , xN] +is of the form P d, where P is irreducible over Q. +Remark 3.18 (On the importance of being a basis). While the statement of Proposition 3.17 +may sound quite natural, there situation is in fact quite subtle: if one replaces linear basis +with a set of generators of A in the statement of the proposition, it will not longer be true [24, +Theorem 1.2]. +Proof of Proposition 3.17. By performing a change of coordinates over Q, we will assume +that M1 is the identity matrix. +Let A be the complexification of A. By the Wedderburn-Artin theorem [13, Corollary +2.4.4], there exist n1, . . . , nℓ such that N = n2 +1 + . . . + n2 +ℓ and +(3.3) +A ∼= Matn1(C) × . . . × Matnℓ(C). + +EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS +11 +Algorithm 3.2 Finding a nontrivial invariant subspace of an algebra +Input +a basis B1, . . . , BN ∈ Qn×n of an algebra A ⊆ Qn×n; +Output +One of the following: +• NO if there is no subspace in Q +n invariant w.r.t. A; +• nontrivial proper subspace in Qn invariant w.r.t. A; +• a maximal chain of subspaces in Q +n invariant w.r.t. A. +Considering corner cases: +(Step 1) If N = n2, return NO. +(Step 2) For an arbitrary nonzero vector v, consider a space V spanned by B1v, . . . , BNv. If +dim V < n, return V . +Examining the radical: +(Step 3) Find a basis of the radical R of A (Definition 3.7) using Dixon’s theorem [5, Theo- +rem 11]. +(Step 4) If dim R > 0 compute the common kernel of the basis elements of R and return it +(see Lemma 3.9). +Semisimple case: +(Step 5) Set D := 1. +(Step 6) Compute M := �N +i=1 aiBi, where a1, . . . , aN are sampled independently and uni- +formly at random from {1, 2, . . . , D}. +(Step 7) If the characteristic polynomial of M has at least two distinct Q-irreducible factors +(say, p1(t) and p2(t)): +(a) Check the invariance of Ker p1(M) w.r.t. B1, . . . , BN. +(b) If it is invariant, return Ker p1(M). +Otherwise, set D := 2D and go +to (Step 6). +(Step 8) Write the characteristic polynomial of M as p(t)d, where p(t) is Q-irreducible. +(Step 9) Compute the center Z and centralizer C of A (Definition 3.14). +(Step 10) If dim C < d2 dim Z, set D := 2D and go to (Step 6). +(Step 11) Let C1, . . . , Cs be a basis of C. Set C := �s +i=1 biCi, where b1, . . . , bs are sampled +independently and uniformly at random from {1, 2, . . . , D}. +(Step 12) Compute q(t), the minimal polynomial of C. If q is Q-reducible or deg q < d dim Z, +set D := 2D and go to (Step 6). +(Step 13) Let V1, . . . , Vℓ (where ℓ = d dim Z) be the eigenspaces of C. +(Step 14) Return V1 ⊂ V1 ⊕ V2 ⊂ . . . ⊂ V1 ⊕ V2 ⊕ . . . ⊕ Vℓ−1. +Then the complexification Cn of the original representation Qn of A can be decom- +posed [13, Theorem 2.6.2] as +(3.4) +Cn = k1V1 ⊕ k2V2 ⊕ . . . ⊕ kℓVℓ, + +12 +A. DEMIN, E. DEMITRAKI, AND G. POGUDIN +where Vi ∼= Cni is the unique irreducible representation of Matni(C). We denote the base +change corresponding to (3.4) by C ∈ Cn×n. Then CMC−1, where M := x1M1+. . .+xNMN, +is block diagonal with the dimensions of blocks as in (3.4). Furthermore, there exists and +invertible matrix B ∈ CN×N such that, for y := Bx, one has +CMC−1 = diag(Y1, . . . , Y1 +� +�� +� +k1times +, . . . , Yℓ, . . . , Yℓ +� +�� +� +kℓtimes +), +where Yi is a matrix with entries yn1+...+n2 +i−1+1, . . . , yn2 +1+...+n2 +i for every 1 ⩽ i ⩽ ℓ. +Then we have +det(M) = det(CMC−1) = det(Y1)k1 . . . det(Yℓ)kℓ. +Furthermore, since M1 is the identity, det(M)|x1=x1+t as a polynomial in t is the characteristic +polynomial of −M. Let Q(x) := det Y1 . . . det Yℓ ∈ Q[x]. Then Q|x1=x1+t as a polynomial in +t is the minimal polynomial of −M. +Since det Yi is a determinant of a matrix with independent entries, it is irreducible over +C. Let p(x) be a Q-irreducible divisor of det M. Then p divides Q, so, by reordering Yi’s +if necessary, we can assume that p(x) = det Y1 . . . det Yr for r ⩽ ℓ. Assume that r < ℓ. Set +p0(t) := p(x1 − t, x2, . . . , xN) and consider p0(M). We will have +Cp0(M)C−1 = diag( +0, . . . , 0 +� �� � +k1+...+kr times +, p0(Yr+1), . . . , p0(Yr+1) +� +�� +� +kr+1 times +, . . . , p0(Yr+1), . . . , p0(Yr+1) +� +�� +� +kℓ times +). +Since p0 is coprime with the charpolys of Yr+1, . . . , Yℓ, the matrices p0(Yr+1), . . . , p0(Yℓ) are +nonsingular. Therefore, the kernel of Cp0(M)C−1 is exactly the span of the first k1 + . . . + kr +basis vectors. Therefore, the kernel of p0(M) is the span of this many first columns of C−1. +Therefore, the kernel of p0(M) is A-invariant and is defined over C. On the other hand, the +entries of p0(M) belong to Q(x), so the kernel of p0(M) in fact is defined over C ∩ Q(x) = Q. +Therefore, the kernel of p0(M) yields a nontrivial A-invariant subspace of Qn contradicting +with the irreducibility of this representation. Therefore p must be equal to Q and, thus, det M +must be a power of p. +The proof of the proposition provides a way to find the degree of deg P. +Corollary 3.19. In the notation of the proof (see (3.3)) of Proposition 3.17, deg P = n1 + +n2 + . . . + nℓ. +Proposition 3.20. Algorithm 3.2 is correct and terminates with probability one. +Proof. We will first prove the correctness. If the algorithm returned on (Step 1), then +A is the full matrix algebra, and does not have any nontrivial proper invariant subspace. If +the algorithm returned on (Step 2), then the returned subspace is invariant by construction. +If the algorithm returned on (Step 4), the returned subspace is nonzero and invariant due +to Lemma 3.9. +It remains to consider the case when the algorithm returns after (Step 5). If the algo- +rithm returned on (Step 7)ii., then the returned subspace is invariant by construction and is + +EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS +13 +nonzero because p1(t) divides the charpoly of M, so p1(M) is a singular matrix. Finally, con- +sider the case when the algorithm returned on (Step 14). Consider the decomposition (3.2) +from Theorem 3.12 for A. If the algorithm reached (Step 11), it contains a matrix with +the charpoly being p(t)d with Q-irreducible p(t) such that dim C = d2 dim Z. Lemma 3.16 +implies that, in the decomposition (3.2), we have ℓ = 1 and m1 = d. Thus, the whole space +Qn can be written as U1 ⊕ U2 ⊕ . . . ⊕ Ud such that each of Ui’s is A-invariant without proper +nontrivial A-invariant subspaces. [13, Corollary 2.2.4] implies that, over Q, each of Ui’s can +be decomposed as a direct sum of at most dim Z A-invariant subspaces. Therefore, the whole +Q +n can be decomposed into at most d dim Z A-invariant subspaces by [13, Theorem 2.6.2]. +Lemma 3.15 implies that each of Vi’s from (Step 13) is an invariant subspace w.r.t. A. +Since there are d dim Z of them, each of Vi’s does not contain nontrivial proper A-invariant +subspaces. Therefore, the chain V1 ⊂ V1⊕V2 ⊂ . . . ⊂ V1⊕V2⊕. . .⊕Vℓ−1 returned at (Step 14) +is maximal. This finished the proof of the correctness of the algorithm. +We will now prove that the algorithm terminates with probability one. +Consider the +decomposition (3.2) of A from Theorem 3.12. +Consider variables z1, . . . , zN and a ma- +trix M0 := �N +i=1 ziBi. +Then M at (Step 6) is a specialization of M0 at zi = ai. +Let +P(z1, . . . , zN, t) be the charpoly of M0. +Consider the decomposition (3.2) for A. +For ev- +ery 1 ⩽ i ⩽ ℓ, we apply Proposition 3.17 to the block corresponding to Ai and obtain a +Q-irreducible Pi and its power di. Thus, we obtain the following factorization for M0 +P = P d1m1 +1 +P d2m2 +2 +. . . P dℓmℓ +ℓ +. +The characteristic polynomial of M computed at (Step 6) is equal to P(a1, . . . , aN, t). Assume +that Pi(a1, . . . , aN, t) is Q-reducible for every 1 ⩽ i ⩽ s and these polynomials are distinct. +• Assume that ℓ > 1. Then p1(t) from (Step 7) will be equal to Pi(a1, . . . , aN, t) for +some i. Then Ker p1(M) will be the subspace corresponding to the i-th block in the +decomposition (3.2). The subspace is invariant, so it will be returned on (Step 7)ii.. +• Assume that ℓ = 1. We will study matrix C similarly to the way we studied M above. +Let y1, . . . , ys be independent variables, and we define C0 := y1C1 +. . .+ysCs. By the +same argument as in the proof of Lemma 3.16, we have C ∼= Matr(Z) for some integer +r. By [13, Proposition 2.3.4], algebra C is simple and every C-module (in particular, +our ambient space Qn) is a direct sum of isomorphic copies of the same C-module. We +apply Proposition 3.17 to this module and deduce that the characteristic polynomial +of C0 is of the form Q(y1, . . . , ys, t)h for some integer h and Q-irreducible polynomial +Q. Furthermore, deg Q = d dim Z by Corollary 3.19. Assume that Q(b1, . . . , bs, t) +is Q-irreducible. Then Q(b1, . . . , bs, t) will be the minimal polynomial of C, so this +polynomial will not satisfy the condition of (Step 12) and, thus, the algorithm will +terminate without going back to (Step 6). +Combining the two underlined assumptions in the text above, we see that the algorithm +will return for a fixed value of D if the following conditions hold: +1. Pi(a1, . . . , aN, t) is Q-reducible for every 1 ⩽ i ⩽ s and these polynomials are all +distinct; +2. Q(b1, . . . , bs, t) is Q-irreducible. +[11, Theorem 2.1] implies that there exists constants C0, C1 such that the probability of any +of Pi(a1, . . . , aN, t)’s and Q(b1, . . . , bs, t) being Q-reducible is less that +C1 +3√ +D if D > C0. Fur- + +14 +A. DEMIN, E. DEMITRAKI, AND G. POGUDIN +thermore, the Schwartz-Zippel lemma [38, Proposition 98] implies that there exists a constant +C2 such that the probability of any of Pi(a1, . . . , aN, t)’s being equal does not exceed C2 +D . +Therefore, for D > C0, the probability that D will be updated is at most +C1 +3√ +D + C2 +D . This +number will eventually become less than 0.99, so the probability of non-termination will be +bounded by 0.99 · 0.99 · . . . = 0. +3.5. Search for invariant subspaces: how to find a chain. In this section, we describe +how to use Algorithm 3.2 in a recursive manner to find a maximal chain of invariant subspaces +in Q w.r.t. the Jacobian algebra A ⊂ Qn×n of an ODE system. We will denote a basis of A +by B1, . . . , BN +In the cases when Algorithm 3.2 applied to B1, . . . , BN returned NO or a maximal chain of +invariant subspaces, we are done. Therefore, we consider the case when Algorithm 3.2 returns +a single invariant subspace V ⊂ Qn. In this case, we consider two subproblems: +1. Restriction. Since V is invariant w.r.t. B1, . . . , BN, there are well-defined restrictions +B1|V , . . . , BN|V . We fix a basis in V and will denote the matrix representations for +these restricted operators also by B∗ +1, . . . , B∗ +N. +2. Quotients. Consider the quotient space Qn/V and the quotient map π: Qn → Qn/V +(see [3, 3.83, 3.88]). +Since V is invariant w.r.t. +B1, . . . , BN, we can consider the +quotient operators [3, 5.14] B1/V, . . . , BN/V , we denote their matrix representations +by B◦ +1, . . . , B◦ +N. Note that, for every their common invariant subspace U ⊂ Qn/V , the +subspace π−1(U) ⊂ Qn is invariant w.r.t. B1, . . . , BN. +Note that the aforementioned matrix representations can be computed solving linear sys- +tems in n variables. Thus, we can work recursively with algebras ⟨B∗ +1, . . . , B∗ +N⟩ on V and +⟨B◦ +1, . . . , B◦ +N⟩ on Qn/V . If the resulting maximal chains of invariant subspaces are +0 ⊊ V1 ⊊ . . . ⊊ Vs ⊊ V +and +0 ⊊ U1 ⊊ . . . ⊊ Ur ⊊ Qn/V, +then we can return the following maximal chain of invariant subspaces for B1, . . . , BN +0 ⊊ V1 ⊊ . . . ⊊ Vs ⊊ V ⊊ π−1(U1) ⊊ . . . ⊊ π−1(Ur) ⊊ Qn. +3.6. Putting everything together. In this section we collect the subroutines from the +preceding sections into the complete algorithm for finding a maximal chain of lumpings. +Algorithm 3.3 Finding a maximal chain of lumpings +Input +an ODE system x′ = f(x) with x = (x1, . . . , xn) and f = (f1, . . . , fn) ∈ Q[x]n; +Output +a maximal chain of lumpings (see Definition 2.5 and Example 2.7); +(Step 1) Compute the Jacobian J(x) of f and the matrices J1, . . . , Jℓ ∈ Qn×n from its de- +composition as in (3.1). +(Step 2) Use Algorithm 3.1 to compute the basis B1, . . . , BN of the Jacobian algebra A = +⟨In, J1, . . . , Jℓ⟩ of the system. +(Step 3) Apply Algorithm 3.2 in a recursive way as decribed in Subsection 3.5 to compute a +maximal chain V1 ⊊ . . . ⊊ Vs of subspaces in Q +n invariant w.r.t. A. +(Step 4) For each 1 ⩽ i ⩽ s, find a matrix Li with the columns being a basis of Vi. +(Step 5) Return L1, . . . , Ls. + +EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS +15 +4. Implementation and performance. We have implemented Algorithm 3.3 (and all the +algorithms it relies on) in Julia language [4] as a part of ExactODEReduction.jl package. +The package together with relevant resources to replicate our results is freely available at +https://github.com/x3042/ExactODEReduction.jl +We use libraries AbstractAlgebra.jl and Nemo.jl [18]. Internally, this results in us- +ing FLINT [21] and Calcium [23] (for complex number arithmetic). We use a version of the +code from [30] to improve interpretability of the computed lumpings. Additionaly, during the +development stage, various components of the package were profiled on collections of sparse +matrices from the SuiteSparse dataset [12]. Our implementation accepts models typed man- +ually or from the files in the ERODE *.ode format [6, Section 3.2]. We provide documentation, +installation instructions, and usage examples. +We will demonstrate the performance of our implementation on a set of benchmarks2. +We use benchmarks from the BioModels database [27] collected in [29] of dimensions ranging +from 4 to 133. We run Algorithm 3.3 over rationals on each of the models. Table 1 contains +benchmark results aggregated by models’ dimension. For each range, we report: +• the number of models considered; +• the (average) length of a chain of reductions found; +• the (average) number of nonequivalent reductions, where equivalence is taken up to +adding states with constant dynamics. We have chosen to report this because we think +is it a reasonable first approximation to the number of “interesting” reductions; +• the (minimum, average, maximum) elapsed runtime of our implementation; +Models info +Reductions +Runtime (sec.) +Dimension # Models # Total # Non-equivalent +Min. +Average +Max. +2 - 9 +44 +4.02 +1.39 +0.0 s +0.6 s +0.66 s +10 - 19 +41 +8.15 +2.61 +0.01 s +0.21 s +1.46 s +20 - 29 +46 +9.65 +2.13 +0.08 s +0.44 s +1.48 s +30 - 39 +17 +19.41 +2.71 +0.33 s +1.74 s +5.91 s +40 - 59 +25 +29.08 +6.08 +0.78 s +4.58 s +26.71 s +60 - 79 +20 +37.25 +6.95 +7.7 s +34.57 s +102.92 s +80 - 99 +11 +42.91 +7.09 +24.46 s +96.38 s +497.26 s +100 - 133 +4 +89.0 +21.5 +75.15 s 202.52 s 312.02 s +Table 1: Benchmark results aggregated by model dimension +The timings were produced on a laptop with 2 cores 1.60GHz each and 8 Gb RAM3. We +would like to note that out of the 208 models considered, at least one reduction was found in +202 models, and 154 of them admit a non-constant reduction. +The timings in the table do not include the cost of the positivization step [30], which is +2Models are available at https://github.com/x3042/ExactODEReduction.jl/tree/main/data/ODEs, com- +mit hash 678d32c5bbc8beedc9e22b673238cde0ec673a46. +3For +the +overall +table, +we +refer +to +https://github.com/x3042/ExactODEReduction.jl/blob/main/ +benchmark/biomodels benchmark results.md, commit hash 23c9f532aa316cbef59a8e3e6be04156a3d9c3eb. + +16 +A. DEMIN, E. DEMITRAKI, AND G. POGUDIN +optional. Here, our algorithm uses the Polymake [2] library. With the positivization step, +the running time increases no more than by a factor of two in most instances, and usually +the increase is indistinguishable at all4. In the earlier versions of the implementation of Al- +gorithm 3.3, computing the algebra basis on (Step 2) had often been a clear bottleneck on +our dataset. With the modifications to the Algorithm 3.1 as described in Subsection 3.2, +currently, the most time-consuming steps are the restriction and quotienting procedures ap- +plied on (Step 3) of Algorithm 3.3. Solving a number of linear systems to find the matrix +representations of restricted and quotient operators is a clear bottleneck here. +5. Case studies. +5.1. Inactivation of factor Va. We will consider a model from [22] which appears in the +BioModels database [27] as BIOMD0000000365. Factor V is a protein involved in the process of +coagulation (transforming blood from liquid to gel), and thus is closely related to blood vessel +repair and thrombosis. In particular, it can assist in activating protein anticoagulant protein +C. The activated factor V, factor Va, can no longer do this. A model describing deactivation +of Va by means of activated protein C (APC) was proposed and studied in [22]. +Factor Va consists of the heavy chain (HC) and light chain (LC), and the binding of APC +happens through the light chain. The model consists of the following species +• Factor Va and its versions Va3, Va5, Va6, Va53, Va56, Va36, and Va536; +• LC, HC, and the versions of the latter (HC3, HC5, etc) corresponding to the versions +of Va; +• the A1 domain of factor Va, VaLC·A1 and versions of the A2 domain such as VaA3, +VaA53, etc. +• APC, complexes formed by it and LC/Va (such as APC·Va3). +In total, the model contains 30 variables and 9 parameters. Our code finds a maximal chain +of lumpings of length 14 in under 5 second on a laptop. The smallest reduction with nonzero +dynamics has dimension three and involves two parameters (similar to the one in Example 2.7): +(5.1) +� +� +� +� +� +y′ +1 = −k1y1y2 + k2y3, +y′ +2 = −k1y1y2 + k2y3, +y′ +3 = k1y1y2 − k2y3. +The macro-variables are +y1 = [APC], +y2 = [LC] + [Va] + [Va3] + [Va36] + [Va5] + [Va53] + [Va536] + [Va56] + [VaLC ·A1], +y3 = [LC · APC] + [Va · APC] + [Va3 · APC] + [Va36 · APC] + [Va5 · APC] ++ [Va53 · APC] + [Va536 · APC] + [Va56 · APC] + [VaLC ·A1 · APC]. +Variable y2 (resp., y3) can be described as the total concentration of the light chains without +(resp., with) bound APC. Therefore, the reduction (5.1) focuses on the process of bind- +ing/unbinding of APC to the light chains, and it turns out that the other processes such as +4One notable exception are models that admit large reductions with large coefficients. For example, model +BIOMD0000000153 of dimension 76 has 22 nontrivial reductions of dimensions 55 and more, and applying the +positivization routine increases the total runtime from 40 s to 1240 s. + +EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS +17 +reactions between the heavy and light chains become irrelevant and, in particular, the HCn +species do not appear in the macro-variables at all. +(a) States of the original model appearing in y2 +(b) Macro-variable y2 of the reduced model +(c) Macro-variables y1, y3 of the reduced model +Figure 2: Numerical simulation for the model from [22] and its reduction using the initial +conditions and parameter values from [22] +From numerical perspective5, the reduction (5.1) can be interpreted as “exact timescale +separation” since the dynamics of the macro-variables turns out to be transient compared to +the dynamics of the original system. More precisely, the original system was studied in [22] +and has nontrivial dynamics on the timespan of 1200 second. In particular, this is the case +for the variables contributing to the macro-variable y2, see Figure 2a. On the other hand, as +Figures 2b and 2c show, the macro-variables y1, y2, y3 have much faster dynamics and reach +the steady state after less than one second. +5All +numerical +simulations +in +this +paper +have +been +done +using +ModelingToolkit +[26] +and +DifferentialEquations.jl [31] + +2.00×10 +1.50×10 +LC +Va +Va3 +Va36 +1.00×10-7 +Va5 +Va53 +Va536 +Va56 +VaLCA1 +5.00×10-8 +0 +0 +250 +500 +750 +1000 +Time (s)2.0000×10-7 +1.9750×10-7 +1.9500×10-7 +y2 +1.9250×10-7 +1.9000×10-7 +0.00 +0.25 +0.50 +0.75 +1.00 +Time (s)1.00×10-8 +7.50×10-9 +y1 +5.00×10-9 +y3 +2.50×10-9 +0.00 +0.25 +0.50 +0.75 +1.00 +Time (s)18 +A. DEMIN, E. DEMITRAKI, AND G. POGUDIN +42 other species, +including the ligand- +receptor complex +NFkB +A20 mRNA +A20 +FLIP mRNA +FLIP +Figure 3: The relevant chemical species and dependencies between them +5.2. Model of cell death. In this subsection, we consider a model designed in [34] in +order to study the sensitivity of the apoptosis (programmed cell death) to the TNF (tumor +necrosis factor) stimulation. The overall model involves 47 chemical species and numerous +interactions between them schematically described in [34, Figure 1]. Our code produces a +maximal chain of lumpings of length 23 (16 out of them with nonconstant dynamics). +We will consider the nonconstant reduction of the smallest dimension. It involves two pro- +teins, A20 and FLIP, whose concentrations depend on the concentrations of the corresponding +mRNAs, A20 mRNA and FLIP mRNA. The concentrations of these mRNAs are governed by +the concentrations of nuclear NF-κB (NFkB N). The latter depends (directly or indirectly) +on many other species including the aforementioned protein A20. +These species and relations between them are summarized on Figure 3, and the corre- +sponding differential equations are: +[A20]′ = k1[A20 mRNA] + k2, +[A20 mRNA]′ = k5[NFκB N], +[FLIP]′ = k3[FLIPmRNA] + k4, +[FLIP mRNA]′ = k6[NFκB N], +where k1, . . . , k6 are numeric parameters. +Our code finds a three-dimensional reduction +which can be straightforwardly simplified further a two-dimensional with the following macro- +variables y1, y2 and the reduced system: +� +y1 = k6 +k1 [A20] − k5 +k3 [FLIP], +y2 = k6[A20 mRNA] − k5[FLIP mRNA] +=⇒ +� +y′ +1 = y2 + k2k6 +k1 − k4k5 +k3 , +y′ +2 = 0 +So the idea is that, although both A20 and FLIP are involved in a complex reaction network, +one can, by eliminating the dependence on NFκB, find a linear combination of them satisfying +a simple system of differential equations which can be explicitly solved. Such explicit relations +on the states can be, for example, combined with the differential inequalities method in order +to obtain tighter reachability bounds [35]. +By going further along the chain of the reductions one can include gradually more species +into the reduced model, for example, a combination of the RIP protein and the transitional +receptor can be included in a similar fashion. +6. Conclusions. We have presented a new algorithm which takes as input a system of +ODEs and produces a longest possible chain of exact linear reductions of the system such that +each reduction in the chain is a refinement of the previous one. This specification is more + +EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS +19 +general compared to the existing tools as it does not put any restriction on the new variables +other than being the linear combinations of the original ones and it does not require any initial +observable/guess. +We provided a publicly available implementation in Julia. Our code is able to analyze +models of dimension over a hundred in a couple of minutes using commodity hardware. We +have also demonstrated its applicability to models arising in life sciences. +Since the produced reductions are exact, our tool can be used for formal verification and as +a preprocessing for approximate reduction techniques. While exactness is thus an important +feature, it can also be viewed as a limitation since some models have only a few exact reductions +(if any). +Therefore, one intriguing direction for future research is to produce a “relaxed” +version of our algorithm to find approximate lumpings together with rigorous error bounds. +For existing results on approximate lumping, see [37, 19] and references therein. 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I. Barton, Bounds on the reachable sets of nonlinear control systems, Automatica, +49 (2013), pp. 93–100, https://doi.org/10.1016/j.automatica.2012.09.020. +[36] D. +Speyer, +Response +to +“Is +there +a +clean +way +to +extract +the +subspaces +invariant +un- +der +a +list +of +matrices?”, +https://mathematica.stackexchange.com/questions/6519/ +is-there-a-clean-way-to-extract-the-subspaces-invariant-under-a-list-of-matrices/9442#9442. +[37] M. Tschaikowski and M. Tribastone, Approximate reduction of heterogenous nonlinear models with +differential hulls, IEEE Transactions on Automatic Control, 61 (2016), pp. 1099–1104, https://doi. +org/10.1109/TAC.2015.2457172. +[38] R. +Zippel, +Effective +Polynomial +Computation, +Springer, +1993, +http://dx.doi.org/10.1007/ +978-1-4615-3188-3. + diff --git a/PdFJT4oBgHgl3EQf1y2a/content/tmp_files/load_file.txt b/PdFJT4oBgHgl3EQf1y2a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..32dda7f8c6edcd76bcfcc6628b883953674d5f59 --- /dev/null +++ b/PdFJT4oBgHgl3EQf1y2a/content/tmp_files/load_file.txt @@ -0,0 +1,1594 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf,len=1593 +page_content='Exact linear reductions of dynamical models Alexander Demin∗, Elizaveta Demitraki† , and Gleb Pogudin‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Dynamical models described by ordinary differential equations (ODEs) are a fundamental tool in the sciences and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Exact reduction aims at producing a lower-dimensional model in which each macro-variable can be directly related to the original variables, and it is thus a natural step towards the model’s formal analysis and mechanistic understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We present an algorithm which, given a polynomial ODE model, computes a longest possible chain of exact linear reductions of the model such that each reduction refines the previous one, thus giving a user control of the level of detail preserved by the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' This significantly generalizes over the existing approaches which compute only the reduction of the lowest dimension subject to an approach-specific constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The algorithm reduces finding exact linear reductions to a question about representations of finite- dimensional algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We provide an implementation of the algorithm, demonstrate its performance on a set of benchmarks, and illustrate the applicability via case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Our implementation is freely available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='com/x3042/ExactODEReduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ordinary differential equations, exact reduction, lumping, dimensionality reduction, matrix algebras MSC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 34C20, 34-04, 16G10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Ordinary Differential Equations (ODEs) provide a powerful and expres- sive language for describing systems evolving in real-time and, thus, are widely used both in the sciences and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' This motivates development of formal methods to analyse the structure of models defined using ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' One important problem which has been studied actively in the past decade from this angle is model reduction [16, 17, 6, 7, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In general, model reduction refers to a variety of techniques aiming at replacing the model of interest with a simpler one while preserving, at least approximately, some of the important features of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Traditionally, approximate methods such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', balanced truncation [1] have been employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Exact model reduction is a complementary approach in which one lowers the dimension of the model without introducing approximation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Such reductions are of particular interest in the context of performing formal analysis or deriving mechanistic insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In this paper, we will focus on an important class of such reductions, exact linear lumpings, which correspond to finding a self-consistent system of differential equations for a set of macro- variables in which each macro-variable is a linear combination of the original variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The case of the macro-variables being sums of the original variables has been extensively studied, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' [16, 8, 17, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In particular, ERODE software has been developed [6] which efficiently finds the optimal partition of the original variables into macro-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A recent software CLUE [28] was a step towards lifting these restrictions on the macro-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Unlike the prior ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' National Research University, Higher School of Economics, Moscow, Russia alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='demin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='eternal@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='com †National Research University, Higher School of Economics, Moscow, Russia egdemitraki@edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='hse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='ru ‡LIX, CNRS, ´Ecole Polytechnique, Institute Polytechnique de Paris, Palaiseau, France gleb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='pogudin@polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='11653v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='SY] 27 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMIN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMITRAKI, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' POGUDIN approaches, the macro-variables produced by CLUE are allowed to involve any coefficients (not just zeroes and ones as before), and, thus, the dimension of the reduced model could be significantly lower [28, Table 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' However, the input of the algorithm consisted not only of a model but also of linear forms in the state variables to be preserved (the observables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Such a set of observables may or may not be available, and guessing it correctly is crucial for finding low-dimensional reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In this paper, we aim at taking the best from both worlds: requiring only a model as input (as ERODE) and allowing the macro-variables to be any linear combinations of the original variables (as CLUE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The main result of the paper is an algorithm for finding arbitrary exact linear reductions when given only a polynomial ODE model with rational coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Note that the question of finding such an arbitrary linear lumping of the lowest possible dimension may not be the most meaningful one since any linear first integral yields a reduction of dimension one with constant dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Instead, our algorithm finds a longest possible chain of lumpings in which each reduction refines the next one (for details, see Section 2) so that a user can choose the desired level of details to be preserved by reduction by moving along the chain and may find reductions which would likely be missed by ERODE and CLUE (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4 and subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Such generality comes with a price: our software is typically slower than CLUE and ERODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Our algorithm is based on combining the connection of the linear lumping problem to the problem of finding a common invariant subspace of a set of matrices [25, 28] with the structure theory of finite-dimensional algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We use the general framework of existing algorithms over finite [33] and algebraically closed [9, 36] fields and achieve desired efficiency by sparsity-aware algorithm for finding a basis of an algebra (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' exploiting the structure of the input to compute mostly with rational numbers and postponing passing to algebraic number fields as much as possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' using sparse linear algebra and modular computation to avoid large matrices and expression swell, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We implemented our algorithm, and the implementation is publicly available at https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='com/x3042/ExactODEReduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We evaluate its performance on a set of bench- marks from the BioModels database [27], a large collection of models from life sciences, and demonstrate the produced reduction for two case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In Section 2, we give precise definition of exact linear reduction, and formulate explicitly the algorithmic problem we solve in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Section 3 contains detailed description of the algorithm and its justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We describe our implementation and report its performance in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Section 5 contains the case studies describing the reductions produced by our software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In the paper, the transpose of a matrix M is denoted by MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For a vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , xn) of indeterminates, by C[x] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', R[x], Q[x]) we will denote the set of polynomials in x with complex (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', real, rational) coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1 (Lumping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider a system of ODEs with polynomial right-hand side of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1) x′ = f(x), EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS 3 where x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , xn), f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , fn), and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , fn ∈ C[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We say that a linear trans- formation y = xL with y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , ym), L ∈ Cn×m, and rank L = m is a lumping of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1) if there exist g = (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , gm) with g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , gm ∈ C[y] such that y′ = g(y) for every solution x of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We say that m is the dimension of the lumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The variables y in the reduced system are called macro-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Throughout this section, we will work with the following running example [30, Example 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will consider a chemical reaction network consisting of A chemical species X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Species AUU, AUX, AXU, and AXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Each of them is one of the states of a molecule A with two identical binding sites, which can be either unbound (U in the subscript) or bound (X in the subscript) to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For simplicity, we will assume that all the reaction rates are equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The dynamics of the network is defined by the following reactions: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) X + AU∗ AX∗ X + A∗U A∗X where ∗ ∈ {X, U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Under the laws of the mass-action kinetics [14, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 7], the reactions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) yield the following ODE system (where [S] denotes the concentration of the species S): (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3) � � � � � � � � � � � � � � � [X]′ = [AXU] + [AUX] + 2[AXX] − [X]([AXU] + [AUX] + 2[AUU]), [AUU]′ = [AXU] + [AUX] − 2[X][AUU], [AUX]′ = [AXX] + [X][AUU] − [X][AUX] − [AUX], [AXU]′ = [AXX] + [X][AUU] − [X][AXU] − [AXU], [AXX]′ = [X][AXU] + [X][AUX] − 2[AXX].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 (Conservation laws as lumpings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We show that matrix L = � 0 1 1 1 1 �T yields a lumping of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We have y = � [X] [AUU] [AUX] [AXU] [AXX] � L = [AUU] + [AUX] + [AXU] + [AXX].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3) one can check that y′ = 0, so we can take g(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Indeed, y is the total concentration of A and must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Furthermore, any linear conservation law yields a lumping of dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3 (More informative lumping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Another lumping for the same system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3) is given by the matrix L = � � � � � � 1 0 0 0 2 0 0 1 1 0 1 1 0 0 2 � � � � � � =⇒ � � � � � y1 = [X], y2 = 2[AUU] + [AUX] + [AXU], y3 = [AUX] + [AXU] + 2[AXX].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMIN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMITRAKI, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' POGUDIN The macro-variables will satisfy a self-contained system y′ 1 = y3 − y1y2, y′ 2 = y3 − y1y2, y′ 3 = −y3 + y1y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The rationale behind this reduction is that y2 and y3 are “concentrations” of unbound and binded sites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The above examples demonstrate that one system can have several lumpings (in fact, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3) has more), so a natural question is how to find useful lumpings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The state-of-the-art software tools CLUE [28] and ERODE [6] approach this question by finding the lumping of the smallest dimension satisfying certain constraints: preserving some quantities of interest unlumped (for CLUE [28]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' or coming from a partition of the state variables (for ERODE [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Both constraints may be too restrictive: not all interesting lumpings come from a partition (like the ones in Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' see also [28, Table 1]), and it may be complicated to guess in advance meaningful quantities to preserve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4 (Example hard for CLUE and ERODE, see also Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider another chemical reaction network [15, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='20)] (originally due to Daniel Knight): E∗ E E + S ES E + P E∗ + S As in the case of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2), we transform the reactions into an ODE system using the law of mass-action kinetics taking all the rates to be one1: � � � � � � � � � � � � � � � [E]′ = 2[ES] + [E∗] − [E][S] − [E][P] − [E], [S]′ = 2[ES] − [E][S], [P]′ = [ES] − [E][P], [ES]′ = [E][S] + [E][P] − 3[ES], [E∗]′ = [E] + [ES] − [E∗] One meaningful linear reduction is y = [E] + [ES] − [E∗] with the equation y′ = −2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The macro-variable y can be understood as a potential between the amount of E (typically en- zyme), both in the free form E and as a part of the complex ES, and E∗ (typically inactivated enzyme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='This reduction does not come from a subdivision of the species, so it cannot be found by ERODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Furthermore, finding it using CLUE would require knowing this macro-variable in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' An alternative approach would be to find all the lumpings and let the user to choose which ones to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The problem is that there may be easily an inifinite number of lumpings, for example, similarly to Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2, one can show that the matrix L = � α 1 1 + α 1 + α 1 + 2α �T 1For the reduction in this example, it is sufficient that the rates of ES → E+ + S and E → E+ coincide EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS 5 yields a lumping of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3) for every number α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Furthermore, as we will explain later, the lumpings are in a bijection with the invariant subspace of certain matrices coming from the Jacobian of f(x) and (at least for arbitrary matrices) the invariant subspaces can form an arbitrary algebraic variety [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The approach we take in this paper is to find a sequence of reductions refining each other with the guarantee that this sequence is of maximal possible length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5 (Chain of lumpings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For an ODE system of the form x′ = f(x), a sequence of linear transformations y1 = xL1, y2 = xL2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , yℓ = xLℓ, where L1 ∈ Cn×m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Lℓ ∈ Cn×mℓ, is called a chain of lumpings if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 0 < m1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' < mℓ < n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' yi = xLi is a lumping of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1) for every 1 ⩽ i ⩽ ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' for every 1 < i ⩽ ℓ, there exists a matrix Ai such that Li−1 = LiAi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The latter means that the reductions given by L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Lℓ refine each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Such a chain (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Lℓ) will be called maximal if it is not contained as a subsequence in any longer chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will show (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6) that all maximal chains are of the same length, so they are also the longest possible chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Given a maximal chain of lumpings, a user can “zoom in/out” by going left/right along the chain depending on the desired tradeoff between the size of the reduced model and the amount of information retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Thus, we can now formally state the main problem studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Main problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Given a system x′ = f(x) with f being a vector of polynomials over Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Compute a maximal chain of lumpings for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' � � � � � � � � � � � � � � � [X]′ = [AXU] + [AUX] + 2[AXX] − [X]([AXU] + [AUX] + 2[AUU]), [AUU]′ = [AXU] + [AUX] − 2[X][AUU], [AUX]′ = [AXX] + [X][AUU] − [X][AUX] − [AUX], [AXU]′ = [AXX] + [X][AUU] − [X][AXU] − [AXU], [AXX]′ = [X][AXU] + [X][AUX] − 2[AXX].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' � � � � � � � � � y′ 4,1 = y4,3 + 2y4,4 − y4,1(y4,3 + 2y4,2), y′ 4,2 = y4,3 − 2y4,1y4,2, y′ 4,3 = 2y4,4 − y4,3y4,1 − y4,3 + 2y4,2y4,1, y′ 4,4 = y4,1y4,3 − 2y4,4 � � � � � y′ 3,1 = y3,3 − y3,1y3,2, y′ 3,2 = y3,3 − y3,1y3,2, y′ 3,3 = −y3,3 + y3,1y3,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' � y′ 2,1 = 0, y′ 2,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' � y′ 1,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' � y4,1 = [X], y4,2 = [AUU], y4,3 = [AUX] + [AXU], y4,4 = [AXX] � � � � � y3,1 = y4,1, y3,2 = y4,3 + 2y4,2, y3,3 = y4,3 + 2y4,4 � y2,1 = y3,2 + y3,3, y2,2 = y3,1 + y3,3 y1,1 = y2,1 Figure 1: Maximal chain of lumpings for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3) and the corresponding reductions 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMIN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMITRAKI, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' POGUDIN Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='7 (Maximal chain of lumpings for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Figure 1 shows a chain of lumpings and the corresponding reductions for our example system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The blocks contain the reduced systems and the arrows are labeled with the transformations between the consecutive reductions (matrices Ai in the terms of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' This chain of reductions includes our preceding Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3 as y1 and y3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In this example we have the original dimension n = 5 and the dimensions of the reductions m1 = 1, m2 = 2 , m3 = 3, m4 = 4, so this chain is clearly maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For finding a maximal chain of lumpings, we first use theory developed in [28] to reduce the problem to a problem about common invariant subspaces of a set of matrices (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1) and then solve the new problem using the structure theory of finite-dimensional algebras (Subsections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The overall algorithm is summarized in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Reduction to the search for common invariant subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let x′ = f(x) be an ODE system in variables x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , xn) and f being a row vector of polynomials f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , fn ∈ C[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let J(x) be the Jacobian matrix of f with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We denote the monomials in x appearing in J(x) by m1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , mN(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then J(x) can be written uniquely as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1) J(x) = � ∇f1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ∇fn � = N � i=1 Ji · mi(x), where ∇g := � ∂g ∂x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ∂g ∂xn �T and J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , JN are constant matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider the system x′ 1 = x1 − 2x2 2, x′ 2 = −x2 + x2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In this case the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1) will be J(x1, x2) = � 1 0 −4x2 −1 + 2x2 � = �1 0 0 −1 � + � 0 0 −4 2 � x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Using the notation above, linear transformation y = xL, where L ∈ Cn×m, is a lumping of x′ = f(x) if and only if the column space of L is invariant with respect to J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , JN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For the case L ∈ Rn×m, the statement follows from [28, Lemmas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The proof of [28, Lemmas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1] remains correct after replacing R with C, and the proof of [28, Lemmas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1] will be correct for the case of C if the real inner products are replaced with the complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A sequence of linear transformations y1 = xL1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , yℓ = xLℓ is a chain of lumpings if and only if the column spaces V1, V2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Vℓ of L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Lℓ satisfy Vi is invariant with respect to J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , JN for every 1 ⩽ i ⩽ ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' {0} ⊊ V1 ⊊ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊊ Vℓ ⊊ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Furthermore, the chain of lumpings is maximal if and only if the chain V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Vℓ is not a subsequence of a chain of subspaces satisfying the two properties above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS 7 In order to search for such chains of invariant subspaces, we will use theory of finite dimensional matrix algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4 (Matrix algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let k be a field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', k = Q, R, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A subspace A ⊆ kn×n of matrices is called an algebra if it is closed under multiplication and contains the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For a finite set A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Am ∈ kn×n, we denote the smallest algebra containing A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Am by ⟨A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Am⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' This algebra is equal to the span of all possible products these matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For an ODE system x′ = f(x) with x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , xn) and f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , fn ∈ C[x], we consider the coefficients J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , JN of the Jacobian matrix of f(x) written as a polynomial in x as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will call the algebra ⟨In, J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , JN⟩ (where In is the identity n × n-matrix) the Jacobian algebra of the system x′ = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let Tn be the set of all upper-triangular matrices in kn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since the product of two upper-triangular matrices is upper-triangular again, Tn is an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider the system from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Its Jacobian algebra is ��1 0 0 1 � , �1 0 0 −1 � , � 0 0 −4 2 �� = {MT | M ∈ T2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since a subspace V ⊂ Cn is invariant with respect to J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , JN if and only if it is invariant with respect to ⟨In, J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , JN⟩, that is, invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' any element of the algebra, we will further focus on finding invariant subspaces of this Jacobian algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' An immediate benefit is that we can use the Jordan-H¨older theorem [13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1] to clarify our notion of the maximal chain of lumpings (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5): the definition only requires that a maximal chain cannot be further refined, and this, in general, does not preclude the existence of longer chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The following direct consequence of [13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1] guarantees that a maximal chain indeed has the maximal possible length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For a given ODE system x′ = f(x), all maximal chains of lumpings have the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Generating the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For performing explicit computation with the Jacobain algebra ⟨In, J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , JN⟩ (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4), we will compute its basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1 gives a simplified version of our approach, which is essentially [28, Algorithm 2] applied to matrices instead of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Similarly to [28], we employ modular computation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' [28, Algorithm 3]) to avoid the intermediate expression swell and use sparse linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Building upon this straightforward adaptation of the approach from [28], we significantly improve the performance by taking further advantage of the sparsity of the input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In applications, only a couple of J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , JN (typically, the constant term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1)) are not so sparse, and the rest are extremely sparse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' furthermore, the basis of the Jacobian algebra also can be often chosen to be very sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Hence, many of matrices C from (Step 2)(b)i will be sparse as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' However, some of the matrices computed at the intermediate steps may be still quite dense slowing down the whole algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We deal with the issue by temporarily deferring (Step 2)(b)ii for relatively dense matrices C and then, once the outer loop exits signaling that P is empty, we add each of the deferred matrices to P and restart the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMIN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMITRAKI, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' POGUDIN This way we ensure that we have generated enough sparse matrices in the algebra so that the reductions of the dense matrices will be more sparse now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Thanks to this optimization, Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1 is not a bottleneck in our computation which it was when we used the approach from [28] directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1 Finding a basis of matrix algebra (basic version) Input a set of square matrices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Aℓ ∈ kn×n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Output a basis S of the smallest linear subspace A ⊆ kn×n containing all possible products of A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Aℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 1) Set S to be any basis of the linear span of A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Aℓ and let P := S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 2) While P ̸= ∅ do (a) Take B to be an element of P and remove it from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (b) For every A in {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Aℓ} do i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Compute C := AB and reduce C w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' S via Gaussian reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If C ̸= 0, set S := S ∪ {C} and P := P ∪ {C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 3) Return S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Search for invariant subspaces: algebraic preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For this section, we fix a ground field k of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The cases we are mostly interested in are rational numbers Q, algebraic numbers Q, and complex numbers C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='7 (Radical of an algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let A ⊆ kn×n be an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A subspace I ⊆ A is called an ideal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', left ideal) if AB, BA ∈ I (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', AB ∈ I) for every A ∈ A and B ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' An ideal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', left ideal) I ⊆ A is nilpotent if there exists N such that the product of any N elements of I is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The set of all elements A ∈ A such that the left ideal A · A is nilpotent is called the radical of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' It is a nilpotent ideal of A by [13, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let Tn be the set of all upper-triangluar matrices in kn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider a subset Un ⊂ Tn of strictly upper-triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' One can easily verify that Un is an ideal and the product of any n elements of Un is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since, for every A ∈ Un, we have Tn · A ⊆ Un, we deduce that Un is the radical of Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Dixon’s theorem [5, Theorem 11] implies that the radical of an algebra A ⊆ kn×n can be computed by finding the kernel of a square matrix of order dim A ⩽ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The relevance of the notion of radical to our problem is demonstrated by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let A ⊆ kn×n be an algebra, and let R ⊂ A be its radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If R ̸= {0}, then the intersection � R∈R Ker R is nontrivial and is invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let N be the smallest integer such that the product of any N elements of R is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then there exists 0 ̸= M ∈ kn×n which is a product of N − 1 elements of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then, we have RM = 0 for every R ∈ R, so V := � R∈R Ker R ⊇ Im M is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS 9 Consider v ∈ V , A ∈ A, and R ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since RA ∈ R, we have RAv = 0, so Av ∈ Ker R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Thus, V is invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider the system from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5, it was shown that the Jacobian algebra of this system is the set of lower triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Similarly to Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='8 we find that the radical of this algebra is �0 0 λ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The common kernel of the radical is spanned by the second basis vector yielding the reduction y′ = −y + y2 (with y = x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='11 (Semisimple algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' An algebra A ⊆ kn×n is called semisimple if its rad- ical is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will use the following characterization of semisimple algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='12 (Wedderburn-Artin, [13, Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let A ⊆ kn×n be a semisimple algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then there exist 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' algebras A1 ⊆ kn1×n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Aℓ ⊆ knℓ×nℓ such that Ai does not have a nontrivial proper invariant subspace in kni for every 1 ⩽ i ⩽ ℓ, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' integers m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , mℓ such that n1m1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' + nℓmℓ = n, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' a basis in kn such that, in this basis, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) A = � � �Diag(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , A1 � �� � m1 times , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Aℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Aℓ � �� � mℓ times ) | A1 ∈ A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Aℓ ∈ Aℓ � � � , where Diag(B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN) denotes the block-diagonal matrix with blocks B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider the set of all matrices of the form � � � � a b 0 0 −b a 0 0 0 0 c 0 0 0 0 c � � � � , where a, b, c ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' This is a semisimple algebra in the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) with ℓ = 2, m1 = 1, and m2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In the case ℓ = 1 and m1 = 1 in the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='12, there are no invariant subspaces in kn but there still may be invariant subspaces in k n if k ̸= k, where k is the algebraic closure of field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' These subspaces can be found using the center of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='14 (Center/Centralizer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let A ⊆ kn×n be an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The center of A is the set of all M ∈ A such that MA = AM for every A ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The centralizer of A is the set of all M ∈ kn×n such that MA = AM for every A ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since, for every fixed A, AM = MA is a system of linear equations in the entries, the center and centralizer can be computed by solving a system of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let A ⊆ kn×n be an algebra without nontrivial proper invariant subspaces in kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let C be the center of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For every C ∈ C, every eigenspace of C is an invariant subspace of A in k n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMIN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMITRAKI, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' POGUDIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let V be an eigenspace of C corresponding to the eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let A ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then C(Av) = (CA)v = (AC)v = λAv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let A ⊆ Qn×n be a semisimple algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let M ∈ A be a matrix such that the characteristic polynomial of M is of the form p(t)d, where p(t) is Q-irreducibe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let Z and C be the center and centralizer of A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then the equality dim C = d2 dim Z is equivalent to the fact that, in the Wedderburn-Artin decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) of A, we have ℓ = 1 and m1 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We consider the Wedderburn-Artin decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For every 1 ⩽ i ⩽ ℓ, we denote the center of Ai by Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then dim Z = dim Z1 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='+Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The number of irreducible factors of a characteristic polynomial of any element of A will be at least m1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' + mℓ, so d ⩾ m1 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='+mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A direct computation using the Schur’s lemma [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1] implies that the centralizer C of A is isomorphic to Matm1(Z1) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' × Matmℓ(Zℓ), where Matmi(Zi) denotes the space of mi × mi-block matrices with each block being an element of Zi (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore dim C = m2 1 dim Z1 + m2 2 dim Z2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' + m2 ℓ dim Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Bounding the right-hand side, we can write dim C ⩽ (m1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' + mℓ)2(dim Z1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' + dim Zℓ) ⩽ d2 dim Z Both inequalities will be equalities if and only if ℓ = 1 and d = m1, and this proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Search for invariant subspaces: how to find one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In this subsection, we present Al- gorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 for finding an invariant subspace if there is any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The rest of the subsection is devoted to justifying its correctness and termination, see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let A ⊆ Qn×n be a semisimple algebra such that there are no nontriv- ial proper A-invariant subspaces in Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , MN be a linear basis of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then the polynomial det(x1M1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' + xNMN) ∈ Q[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , xN] is of the form P d, where P is irreducible over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='18 (On the importance of being a basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' While the statement of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='17 may sound quite natural, there situation is in fact quite subtle: if one replaces linear basis with a set of generators of A in the statement of the proposition, it will not longer be true [24, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' By performing a change of coordinates over Q, we will assume that M1 is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let A be the complexification of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' By the Wedderburn-Artin theorem [13, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4], there exist n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , nℓ such that N = n2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' + n2 ℓ and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3) A ∼= Matn1(C) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' × Matnℓ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS 11 Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 Finding a nontrivial invariant subspace of an algebra Input a basis B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN ∈ Qn×n of an algebra A ⊆ Qn×n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Output One of the following: NO if there is no subspace in Q n invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' nontrivial proper subspace in Qn invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' a maximal chain of subspaces in Q n invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Considering corner cases: (Step 1) If N = n2, return NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 2) For an arbitrary nonzero vector v, consider a space V spanned by B1v, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BNv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If dim V < n, return V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Examining the radical: (Step 3) Find a basis of the radical R of A (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='7) using Dixon’s theorem [5, Theo- rem 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 4) If dim R > 0 compute the common kernel of the basis elements of R and return it (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Semisimple case: (Step 5) Set D := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 6) Compute M := �N i=1 aiBi, where a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , aN are sampled independently and uni- formly at random from {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 7) If the characteristic polynomial of M has at least two distinct Q-irreducible factors (say, p1(t) and p2(t)): (a) Check the invariance of Ker p1(M) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (b) If it is invariant, return Ker p1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Otherwise, set D := 2D and go to (Step 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 8) Write the characteristic polynomial of M as p(t)d, where p(t) is Q-irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 9) Compute the center Z and centralizer C of A (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 10) If dim C < d2 dim Z, set D := 2D and go to (Step 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 11) Let C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Cs be a basis of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Set C := �s i=1 biCi, where b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , bs are sampled independently and uniformly at random from {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 12) Compute q(t), the minimal polynomial of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If q is Q-reducible or deg q < d dim Z, set D := 2D and go to (Step 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 13) Let V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Vℓ (where ℓ = d dim Z) be the eigenspaces of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 14) Return V1 ⊂ V1 ⊕ V2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊂ V1 ⊕ V2 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊕ Vℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then the complexification Cn of the original representation Qn of A can be decom- posed [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2] as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4) Cn = k1V1 ⊕ k2V2 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊕ kℓVℓ, 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMIN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMITRAKI, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' POGUDIN where Vi ∼= Cni is the unique irreducible representation of Matni(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We denote the base change corresponding to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4) by C ∈ Cn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then CMC−1, where M := x1M1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='+xNMN, is block diagonal with the dimensions of blocks as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Furthermore, there exists and invertible matrix B ∈ CN×N such that, for y := Bx, one has CMC−1 = diag(Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Y1 � �� � k1times , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Yℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Yℓ � �� � kℓtimes ), where Yi is a matrix with entries yn1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='+n2 i−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , yn2 1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='+n2 i for every 1 ⩽ i ⩽ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then we have det(M) = det(CMC−1) = det(Y1)k1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' det(Yℓ)kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Furthermore, since M1 is the identity, det(M)|x1=x1+t as a polynomial in t is the characteristic polynomial of −M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let Q(x) := det Y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' det Yℓ ∈ Q[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then Q|x1=x1+t as a polynomial in t is the minimal polynomial of −M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since det Yi is a determinant of a matrix with independent entries, it is irreducible over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let p(x) be a Q-irreducible divisor of det M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then p divides Q, so, by reordering Yi’s if necessary, we can assume that p(x) = det Y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' det Yr for r ⩽ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Assume that r < ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Set p0(t) := p(x1 − t, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , xN) and consider p0(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will have Cp0(M)C−1 = diag( 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , 0 � �� � k1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='+kr times , p0(Yr+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , p0(Yr+1) � �� � kr+1 times , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , p0(Yr+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , p0(Yr+1) � �� � kℓ times ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since p0 is coprime with the charpolys of Yr+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Yℓ, the matrices p0(Yr+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , p0(Yℓ) are nonsingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, the kernel of Cp0(M)C−1 is exactly the span of the first k1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' + kr basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, the kernel of p0(M) is the span of this many first columns of C−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, the kernel of p0(M) is A-invariant and is defined over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' On the other hand, the entries of p0(M) belong to Q(x), so the kernel of p0(M) in fact is defined over C ∩ Q(x) = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, the kernel of p0(M) yields a nontrivial A-invariant subspace of Qn contradicting with the irreducibility of this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore p must be equal to Q and, thus, det M must be a power of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The proof of the proposition provides a way to find the degree of deg P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In the notation of the proof (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3)) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='17, deg P = n1 + n2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' + nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 is correct and terminates with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will first prove the correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If the algorithm returned on (Step 1), then A is the full matrix algebra, and does not have any nontrivial proper invariant subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If the algorithm returned on (Step 2), then the returned subspace is invariant by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If the algorithm returned on (Step 4), the returned subspace is nonzero and invariant due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' It remains to consider the case when the algorithm returns after (Step 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If the algo- rithm returned on (Step 7)ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', then the returned subspace is invariant by construction and is EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS 13 nonzero because p1(t) divides the charpoly of M, so p1(M) is a singular matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Finally, con- sider the case when the algorithm returned on (Step 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='12 for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If the algorithm reached (Step 11), it contains a matrix with the charpoly being p(t)d with Q-irreducible p(t) such that dim C = d2 dim Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='16 implies that, in the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2), we have ℓ = 1 and m1 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Thus, the whole space Qn can be written as U1 ⊕ U2 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊕ Ud such that each of Ui’s is A-invariant without proper nontrivial A-invariant subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' [13, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4] implies that, over Q, each of Ui’s can be decomposed as a direct sum of at most dim Z A-invariant subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, the whole Q n can be decomposed into at most d dim Z A-invariant subspaces by [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='15 implies that each of Vi’s from (Step 13) is an invariant subspace w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since there are d dim Z of them, each of Vi’s does not contain nontrivial proper A-invariant subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, the chain V1 ⊂ V1⊕V2 ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊂ V1⊕V2⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='⊕Vℓ−1 returned at (Step 14) is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' This finished the proof of the correctness of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will now prove that the algorithm terminates with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) of A from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider variables z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , zN and a ma- trix M0 := �N i=1 ziBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then M at (Step 6) is a specialization of M0 at zi = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let P(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , zN, t) be the charpoly of M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2) for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For ev- ery 1 ⩽ i ⩽ ℓ, we apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='17 to the block corresponding to Ai and obtain a Q-irreducible Pi and its power di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Thus, we obtain the following factorization for M0 P = P d1m1 1 P d2m2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' P dℓmℓ ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The characteristic polynomial of M computed at (Step 6) is equal to P(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , aN, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Assume that Pi(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , aN, t) is Q-reducible for every 1 ⩽ i ⩽ s and these polynomials are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Assume that ℓ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then p1(t) from (Step 7) will be equal to Pi(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , aN, t) for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then Ker p1(M) will be the subspace corresponding to the i-th block in the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The subspace is invariant, so it will be returned on (Step 7)ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='. Assume that ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will study matrix C similarly to the way we studied M above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Let y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , ys be independent variables, and we define C0 := y1C1 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='+ysCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' By the same argument as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='16, we have C ∼= Matr(Z) for some integer r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' By [13, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='4], algebra C is simple and every C-module (in particular, our ambient space Qn) is a direct sum of isomorphic copies of the same C-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='17 to this module and deduce that the characteristic polynomial of C0 is of the form Q(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , ys, t)h for some integer h and Q-irreducible polynomial Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Furthermore, deg Q = d dim Z by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Assume that Q(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , bs, t) is Q-irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Then Q(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , bs, t) will be the minimal polynomial of C, so this polynomial will not satisfy the condition of (Step 12) and, thus, the algorithm will terminate without going back to (Step 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Combining the two underlined assumptions in the text above, we see that the algorithm will return for a fixed value of D if the following conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Pi(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , aN, t) is Q-reducible for every 1 ⩽ i ⩽ s and these polynomials are all distinct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Q(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , bs, t) is Q-irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' [11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1] implies that there exists constants C0, C1 such that the probability of any of Pi(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , aN, t)’s and Q(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , bs, t) being Q-reducible is less that C1 3√ D if D > C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Fur- 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMIN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMITRAKI, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' POGUDIN thermore, the Schwartz-Zippel lemma [38, Proposition 98] implies that there exists a constant C2 such that the probability of any of Pi(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , aN, t)’s being equal does not exceed C2 D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, for D > C0, the probability that D will be updated is at most C1 3√ D + C2 D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' This number will eventually become less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='99, so the probability of non-termination will be bounded by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='99 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='99 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Search for invariant subspaces: how to find a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In this section, we describe how to use Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 in a recursive manner to find a maximal chain of invariant subspaces in Q w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' the Jacobian algebra A ⊂ Qn×n of an ODE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will denote a basis of A by B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN In the cases when Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 applied to B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN returned NO or a maximal chain of invariant subspaces, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, we consider the case when Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 returns a single invariant subspace V ⊂ Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In this case, we consider two subproblems: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since V is invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN, there are well-defined restrictions B1|V , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN|V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We fix a basis in V and will denote the matrix representations for these restricted operators also by B∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , B∗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Consider the quotient space Qn/V and the quotient map π: Qn → Qn/V (see [3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='83, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='88]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since V is invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN, we can consider the quotient operators [3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='14] B1/V, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN/V , we denote their matrix representations by B◦ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , B◦ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Note that, for every their common invariant subspace U ⊂ Qn/V , the subspace π−1(U) ⊂ Qn is invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Note that the aforementioned matrix representations can be computed solving linear sys- tems in n variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Thus, we can work recursively with algebras ⟨B∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , B∗ N⟩ on V and ⟨B◦ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , B◦ N⟩ on Qn/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' If the resulting maximal chains of invariant subspaces are 0 ⊊ V1 ⊊ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊊ Vs ⊊ V and 0 ⊊ U1 ⊊ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊊ Ur ⊊ Qn/V, then we can return the following maximal chain of invariant subspaces for B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN 0 ⊊ V1 ⊊ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊊ Vs ⊊ V ⊊ π−1(U1) ⊊ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊊ π−1(Ur) ⊊ Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Putting everything together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In this section we collect the subroutines from the preceding sections into the complete algorithm for finding a maximal chain of lumpings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3 Finding a maximal chain of lumpings Input an ODE system x′ = f(x) with x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , xn) and f = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , fn) ∈ Q[x]n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Output a maximal chain of lumpings (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5 and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 1) Compute the Jacobian J(x) of f and the matrices J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Jℓ ∈ Qn×n from its de- composition as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 2) Use Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1 to compute the basis B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , BN of the Jacobian algebra A = ⟨In, J1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Jℓ⟩ of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 3) Apply Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2 in a recursive way as decribed in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5 to compute a maximal chain V1 ⊊ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' ⊊ Vs of subspaces in Q n invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 4) For each 1 ⩽ i ⩽ s, find a matrix Li with the columns being a basis of Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (Step 5) Return L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Implementation and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We have implemented Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3 (and all the algorithms it relies on) in Julia language [4] as a part of ExactODEReduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='jl package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The package together with relevant resources to replicate our results is freely available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='com/x3042/ExactODEReduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='jl We use libraries AbstractAlgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='jl and Nemo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='jl [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Internally, this results in us- ing FLINT [21] and Calcium [23] (for complex number arithmetic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We use a version of the code from [30] to improve interpretability of the computed lumpings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Additionaly, during the development stage, various components of the package were profiled on collections of sparse matrices from the SuiteSparse dataset [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Our implementation accepts models typed man- ually or from the files in the ERODE *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='ode format [6, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We provide documentation, installation instructions, and usage examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will demonstrate the performance of our implementation on a set of benchmarks2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We use benchmarks from the BioModels database [27] collected in [29] of dimensions ranging from 4 to 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We run Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3 over rationals on each of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Table 1 contains benchmark results aggregated by models’ dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For each range, we report: the number of models considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' the (average) length of a chain of reductions found;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' the (average) number of nonequivalent reductions, where equivalence is taken up to adding states with constant dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We have chosen to report this because we think is it a reasonable first approximation to the number of “interesting” reductions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' the (minimum, average, maximum) elapsed runtime of our implementation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Models info Reductions Runtime (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=') Dimension # Models # Total # Non-equivalent Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Average Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 2 - 9 44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='0 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='6 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='66 s 10 - 19 41 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='01 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='21 s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='46 s 20 - 29 46 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='08 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='44 s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='48 s 30 - 39 17 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='33 s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='74 s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='91 s 40 - 59 25 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='78 s 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='58 s 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='71 s 60 - 79 20 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='95 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='7 s 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='57 s 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='92 s 80 - 99 11 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='91 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='09 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='46 s 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='38 s 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='26 s 100 - 133 4 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='15 s 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='52 s 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='02 s Table 1: Benchmark results aggregated by model dimension The timings were produced on a laptop with 2 cores 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='60GHz each and 8 Gb RAM3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We would like to note that out of the 208 models considered, at least one reduction was found in 202 models, and 154 of them admit a non-constant reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The timings in the table do not include the cost of the positivization step [30], which is 2Models are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='com/x3042/ExactODEReduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='jl/tree/main/data/ODEs, com- mit hash 678d32c5bbc8beedc9e22b673238cde0ec673a46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 3For the overall table, we refer to https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='com/x3042/ExactODEReduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='jl/blob/main/ benchmark/biomodels benchmark results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='md, commit hash 23c9f532aa316cbef59a8e3e6be04156a3d9c3eb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMIN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMITRAKI, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' POGUDIN optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Here, our algorithm uses the Polymake [2] library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' With the positivization step, the running time increases no more than by a factor of two in most instances, and usually the increase is indistinguishable at all4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In the earlier versions of the implementation of Al- gorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3, computing the algebra basis on (Step 2) had often been a clear bottleneck on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' With the modifications to the Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1 as described in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2, currently, the most time-consuming steps are the restriction and quotienting procedures ap- plied on (Step 3) of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Solving a number of linear systems to find the matrix representations of restricted and quotient operators is a clear bottleneck here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Inactivation of factor Va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will consider a model from [22] which appears in the BioModels database [27] as BIOMD0000000365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Factor V is a protein involved in the process of coagulation (transforming blood from liquid to gel), and thus is closely related to blood vessel repair and thrombosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In particular, it can assist in activating protein anticoagulant protein C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The activated factor V, factor Va, can no longer do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' A model describing deactivation of Va by means of activated protein C (APC) was proposed and studied in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Factor Va consists of the heavy chain (HC) and light chain (LC), and the binding of APC happens through the light chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The model consists of the following species Factor Va and its versions Va3, Va5, Va6, Va53, Va56, Va36, and Va536;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' LC, HC, and the versions of the latter (HC3, HC5, etc) corresponding to the versions of Va;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' the A1 domain of factor Va, VaLC·A1 and versions of the A2 domain such as VaA3, VaA53, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' APC, complexes formed by it and LC/Va (such as APC·Va3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In total, the model contains 30 variables and 9 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Our code finds a maximal chain of lumpings of length 14 in under 5 second on a laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The smallest reduction with nonzero dynamics has dimension three and involves two parameters (similar to the one in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='7): (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1) � � � � � y′ 1 = −k1y1y2 + k2y3, y′ 2 = −k1y1y2 + k2y3, y′ 3 = k1y1y2 − k2y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The macro-variables are y1 = [APC], y2 = [LC] + [Va] + [Va3] + [Va36] + [Va5] + [Va53] + [Va536] + [Va56] + [VaLC ·A1], y3 = [LC · APC] + [Va · APC] + [Va3 · APC] + [Va36 · APC] + [Va5 · APC] + [Va53 · APC] + [Va536 · APC] + [Va56 · APC] + [VaLC ·A1 · APC].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Variable y2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', y3) can be described as the total concentration of the light chains without (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=', with) bound APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, the reduction (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1) focuses on the process of bind- ing/unbinding of APC to the light chains, and it turns out that the other processes such as 4One notable exception are models that admit large reductions with large coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For example, model BIOMD0000000153 of dimension 76 has 22 nontrivial reductions of dimensions 55 and more, and applying the positivization routine increases the total runtime from 40 s to 1240 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS 17 reactions between the heavy and light chains become irrelevant and, in particular, the HCn species do not appear in the macro-variables at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' (a) States of the original model appearing in y2 (b) Macro-variable y2 of the reduced model (c) Macro-variables y1, y3 of the reduced model Figure 2: Numerical simulation for the model from [22] and its reduction using the initial conditions and parameter values from [22] From numerical perspective5, the reduction (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1) can be interpreted as “exact timescale separation” since the dynamics of the macro-variables turns out to be transient compared to the dynamics of the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' More precisely, the original system was studied in [22] and has nontrivial dynamics on the timespan of 1200 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In particular, this is the case for the variables contributing to the macro-variable y2, see Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' On the other hand, as Figures 2b and 2c show, the macro-variables y1, y2, y3 have much faster dynamics and reach the steady state after less than one second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 5All numerical simulations in this paper have been done using ModelingToolkit [26] and DifferentialEquations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='jl [31] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='00×10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='50×10 LC Va Va3 Va36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='00×10-7 Va5 Va53 Va536 Va56 VaLCA1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='00×10-8 0 0 250 500 750 1000 Time (s)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='0000×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='9750×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='9500×10-7 y2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='9250×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='9000×10-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='00 Time (s)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='00×10-8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='50×10-9 y1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='00×10-9 y3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='50×10-9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='00 Time (s)18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMIN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' DEMITRAKI, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' POGUDIN 42 other species, including the ligand- receptor complex NFkB A20 mRNA A20 FLIP mRNA FLIP Figure 3: The relevant chemical species and dependencies between them 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Model of cell death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' In this subsection, we consider a model designed in [34] in order to study the sensitivity of the apoptosis (programmed cell death) to the TNF (tumor necrosis factor) stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The overall model involves 47 chemical species and numerous interactions between them schematically described in [34, Figure 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Our code produces a maximal chain of lumpings of length 23 (16 out of them with nonconstant dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We will consider the nonconstant reduction of the smallest dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' It involves two pro- teins, A20 and FLIP, whose concentrations depend on the concentrations of the corresponding mRNAs, A20 mRNA and FLIP mRNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The concentrations of these mRNAs are governed by the concentrations of nuclear NF-κB (NFkB N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' The latter depends (directly or indirectly) on many other species including the aforementioned protein A20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' These species and relations between them are summarized on Figure 3, and the corre- sponding differential equations are: [A20]′ = k1[A20 mRNA] + k2, [A20 mRNA]′ = k5[NFκB N], [FLIP]′ = k3[FLIPmRNA] + k4, [FLIP mRNA]′ = k6[NFκB N], where k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' , k6 are numeric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Our code finds a three-dimensional reduction which can be straightforwardly simplified further a two-dimensional with the following macro- variables y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' y2 and the reduced system: � y1 = k6 k1 [A20] − k5 k3 [FLIP],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' y2 = k6[A20 mRNA] − k5[FLIP mRNA] =⇒ � y′ 1 = y2 + k2k6 k1 − k4k5 k3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' y′ 2 = 0 So the idea is that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' although both A20 and FLIP are involved in a complex reaction network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' one can,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' by eliminating the dependence on NFκB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' find a linear combination of them satisfying a simple system of differential equations which can be explicitly solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Such explicit relations on the states can be, for example, combined with the differential inequalities method in order to obtain tighter reachability bounds [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' By going further along the chain of the reductions one can include gradually more species into the reduced model, for example, a combination of the RIP protein and the transitional receptor can be included in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We have presented a new algorithm which takes as input a system of ODEs and produces a longest possible chain of exact linear reductions of the system such that each reduction in the chain is a refinement of the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' This specification is more EXACT LINEAR REDUCTIONS OF DYNAMICAL MODELS 19 general compared to the existing tools as it does not put any restriction on the new variables other than being the linear combinations of the original ones and it does not require any initial observable/guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We provided a publicly available implementation in Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Our code is able to analyze models of dimension over a hundred in a couple of minutes using commodity hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We have also demonstrated its applicability to models arising in life sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Since the produced reductions are exact, our tool can be used for formal verification and as a preprocessing for approximate reduction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' While exactness is thus an important feature, it can also be viewed as a limitation since some models have only a few exact reductions (if any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Therefore, one intriguing direction for future research is to produce a “relaxed” version of our algorithm to find approximate lumpings together with rigorous error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' For existing results on approximate lumping, see [37, 19] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Interestingly, the core linear algebraic problem of our algorithm, finding common invariant subspaces, has been recently studied from the perspective of approximate but rigorous computation in [20, 10] motivated by factoring linear differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We expect the ideas from these papers to be useful in our context as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We would like to thank David E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Speyer for his clear and detailed note [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' We would like to thank Mirco Tribastone for helpful discussions and Rongwei Yang for discussions about Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' GP was supported by the Paris Ile-de-France region (via project “XOR”) and partially supported by NSF grants DMS-1853482, DMS-1760448, and DMS-1853650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' AD was supported by the Max Planck Institute for Informatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content=' Antoulas, Approximation of Large-Scale Dynamical Systems, Adv.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} +page_content='1007/ 978-1-4615-3188-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFJT4oBgHgl3EQf1y2a/content/2301.11653v1.pdf'} diff --git a/QdFJT4oBgHgl3EQfJizI/content/tmp_files/2301.11461v1.pdf.txt b/QdFJT4oBgHgl3EQfJizI/content/tmp_files/2301.11461v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ffbbccc6b141989efa2679214fe5a58269baf894 --- /dev/null +++ b/QdFJT4oBgHgl3EQfJizI/content/tmp_files/2301.11461v1.pdf.txt @@ -0,0 +1,1575 @@ +LEARNING TO GENERATE ALL FEASIBLE ACTIONS +Mirco Theile1,3*, Daniele Bernardini1∗, Raphael Trumpp1 +Cristina Piazza2, Marco Caccamo1, Alberto L. Sangiovanni-Vincentelli3 +1TUM School of Engineering and Design, Technical University of Munich +2Dept. of Informatics, Technical University of Munich +3Dept. of Electrical Engineering and Computer Sciences, University of California, Berkeley +{mirco.theile,daniele.bernardini,raphael.trumpp, +cristina.piazza,mcaccamo}@tum.de,alberto@berkeley.edu +ABSTRACT +Several machine learning (ML) applications are characterized by searching for an +optimal solution to a complex task. The search space for this optimal solution is +often very large, so large in fact that this optimal solution is often not computable. +Part of the problem is that many candidate solutions found via ML are actually +infeasible and have to be discarded. Restricting the search space to only the feasible +solution candidates simplifies finding an optimal solution for the tasks. Further, the +set of feasible solutions could be re-used in multiple problems characterized by +different tasks. In particular, we observe that complex tasks can be decomposed into +subtasks and corresponding skills. We propose to learn a reusable and transferable +skill by training an actor to generate all feasible actions. The trained actor can then +propose feasible actions, among which an optimal one can be chosen according to +a specific task. The actor is trained by interpreting the feasibility of each action as +a target distribution. The training procedure minimizes a divergence of the actor’s +output distribution to this target. We derive the general optimization target for +arbitrary f-divergences using a combination of kernel density estimates, resampling, +and importance sampling. We further utilize an auxiliary critic to reduce the +interactions with the environment. A preliminary comparison to related strategies +shows that our approach learns to visit all the modes in the feasible action space, +demonstrating the framework’s potential for learning skills that can be used in +various downstream tasks. +1 +INTRODUCTION +Complex tasks can often be decomposed into multiple subtasks, with corresponding skills that solve +these subtasks. Learning reusable and transferable skills is an active area of research (Kalashnikov +et al. (2021); Chebotar et al. (2021); Deisenroth et al. (2014)). However, given a subtask, learning +or even defining the corresponding skill is not straightforward. Consider a robotic scenario where a +robot is tasked to grasp an object and handle it in downstream tasks. Different downstream tasks can +have different optimal grasps if the object has multiple feasible grasping poses. Therefore, a grasping +skill cannot be trained based on optimality definitions of individual tasks. However, a grasping +algorithm that learned all feasible grasps could support all possible downstream tasks even without +explicit knowledge thereof during training. The downstream tasks can then select their respective +optimal grasp among the proposed feasible options. Therefore, we consider a skill to be defined by +the set of all feasible actions of a subtask. +We propose a novel method to train a generative neural network to generate all feasible actions +of a subtask by interacting with an environment. The interaction loop is adopted from Contextual +Bandit (CB) (Langford et al. (2008)) and Reinforcement Learning (RL) (Sutton & Barto (2018)): the +environment presents a state for which the actor selects an action, which is tested in the environment, +yielding either a success or failure outcome. As in CB, we limit ourselves to one-step interactions as +opposed to sequential multi-step interactions common in RL. However, we do not minimize regret, +∗These authors contributed equally +1 +arXiv:2301.11461v1 [cs.LG] 26 Jan 2023 + +typically done in CB. Instead, we optimize the final policy as in RL. Unlike CB and RL, the approach +does not aim to find one optimal solution for a given problem but aims to learn all feasible ones. +By interpreting the feasibility of each action given a state as a posterior probability distribution over +the actions, a target probability density function (pdf) is defined. The actor is trained to minimize a +divergence of its output distribution to this target pdf. The training algorithm in the method proposed +can be used with any given f-divergence, including Reverse Kullback-Leibler (RKL), Forward +Kullback-Leibler (FKL), and Jensen-Shannon (JS). The possibility to use FKL and JS is instrumental +in visiting all the modes of the posterior distribution, as RKL is known to collapse into a single mode +(Jerfel et al. (2021)). The training algorithm presented in this paper uses Kernel Density Estimation +(KDE) to estimate the pdf of the actor and Monte Carlo integration with importance sampling to +estimate the normalization of the target. The divergences are estimated using samples from a proposal +distribution which is a separate KDE based on the original samples of the actor. This resampling step +is necessary for convergence, which is discussed in Section 3.3. As interactions with the environment +are typically costly, an auxiliary critic network imitating the environment is trained simultaneously. +The critic’s feasibility estimate of an action is then used to form the target distribution. +The learning algorithm has been tested on a planar robotic grasping problem. We test FKL, RKL, and +JS divergences and compare them to implementations of maximum entropy (ME) RL and Generative +Adversarial Networks (GANs). Besides accuracy, we measure how many grasping modes, i.e., +disconnected regions in the action space, are visited by each approach. Generating actions in all +grasping modes can ensure that the learned skill is reusable and transferable for various downstream +tasks. +The contributions of this paper are the following: +• Design of a new learning method for generative neural network models to explicitly learn to +generate all feasible actions. +• Introduction of a novel gradient estimator for f-divergences that takes advantage of KDEs, +resampling, and importance sampling. +• Application of the proposed learning algorithm to a 2D robotic grasping problem, comparing +the proposed gradient estimators for f-divergences with related methods. +The rest of this work is structured as follows. Section 2 discusses the related work. Section 3 +describes the optimization problem followed by the methodology in Section 4. The evaluation setup +is described in Section 5 and the results are presented in Section 6. Section 7 concludes and gives an +outlook on future work. +2 +RELATED WORK +CBs have been successfully applied to several interactive learning problems with discrete action +spaces (Langford & Zhang (2007); Agarwal et al. (2014); Foster & Rakhlin (2020); Simchi-Levi +& Xu (2021)). In several cases, the context and action spaces have been embedded in a linear +multidimensional action space. The embedding keeps the interaction linear while the action and +context embeddings can be non-linear (Chernozhukov et al. (2019); Foster et al. (2020); Zhu et al. +(2022)). Recently, there has been an increased interest in extending the approach to continuous action +spaces. However, most works are limited to 1D actions (Chernozhukov et al. (2019); Majzoubi et al. +(2020); Zhu & Mineiro (2022)). +Learning from an interactive environment is also the focus of RL (Sutton & Barto (2018)). Many RL +approaches use Actor-Critic (AC) architectures, among which the Soft Actor-Critic (SAC) algorithm +(Haarnoja et al. (2018)) is most related to our work. In SAC, the state-action value function of the +critic is transformed into an energy-based distribution (Haarnoja et al. (2017)), yielding the target +of the actor’s distribution. SAC uses RKL as the loss function for the actor, which yields maximum +entropy RL. Through a reparameterization trick, which usually uses the family of Gaussians, the RKL +is minimized through a direct gradient from the critic. +GANs propose a similar architecture to AC, training a generator and discriminator adversarially. This +adversarial training is equivalent to minimizing the JS divergence (Goodfellow et al. (2014)) and +has been extended to arbitrary f-divergences (Nowozin et al. (2016)). Conditional GANs (Mirza +2 + +& Osindero (2014)) offer an alternative solution to the posterior sampling problem, as a generator +conditioned on a given state can be trained to provide successful actions adversarially. However, the +problem analyzed in our paper is not naturally adversarial, as actions that have not yet been tested in +the interactive environment should not be implicitly rejected. The discriminator learns to discriminate +between tested successful actions from untested ones, providing the generator with inconsistent +gradients. +Expert knowledge is used in Imitation Learning (IL) to derive a policy from demonstration data. +The policy may be learned in a supervised manner in behavior cloning (Pomerleau (1988)) or as a +combination of Inverse Reinforcement Learning (IRL) (Ng et al. (2000)) to learn a reward function +and a subsequent RL procedure. Ho & Ermon (2016) introduced Generative Adversarial Imitation +Learning (GAIL), mitigating the intermediate IRL step by using a generative adversary. As discussed +in Li et al. (2017), the policy learned by GAIL tends to interpolate between modes leading to +erroneous behavior in multimodal settings. Using f-divergence minimization, the authors in Ke +et al. (2020); Ghasemipour et al. (2020) intentionally collapse modes to avoid interpolation. IL and +adversarial approaches require large amounts of expert data. However, expert data is limited in an +interactive environment. Additionally, given that we aim to find all feasible actions, even more expert +data representative of all feasible actions would be required. +Posterior sampling has been a long-standing problem in statistics. State-of-the-art methods in +Bayesian statistics rely on Markov Chain Monte Carlo (MCMC) algorithms (Hastings (1970); Gelfand +& Smith (1990)), eliminating the need to normalize the distribution which is often an intractable +problem (Kruschke (2015)). Variational Inference (VI) relies instead on fitting the posterior with a +family of parametric probability distributions that can then be sampled from (Jordan et al. (1999); +Wainwright & Jordan (2008)). Neural samplers offer another alternative by approximating the +posterior with a generative neural network (Nowozin et al. (2016); Hu et al. (2018)). Normalizing +flows also infer the pdf for each sample using invertible mappings (Rezende & Mohamed (2015); +Tabak & Turner (2013); Tabak & Vanden-Eijnden (2010)). While this approach does not require +density estimates, it limits its applicability to specific neural network designs. +For robotic grasping, Kalashnikov et al. (2018) propose using Deep Reinforcement Learning (DRL) to +find optimal grasps through interaction with multiple real-world robots. If the goal is to find grasping +poses explicitly to be used as the target of a classical controller, supervised learning techniques are +often utilized (Kleeberger et al. (2020)). To support various downstream tasks, it would be necessary +to find all feasible grasps. To this end, the action space is typically discretized and grasping success +is estimated for each discrete action through heat-maps. This can be learned supervised (Kumra +et al. (2020); Morrison et al. (2020)) or self-supervised (Zeng et al. (2020)). Zeng et al. (2020) +explicitly utilize structure given by spatial equivariances. We aim to find a solution that needs neither +discretization nor to make use of the structure as these requirements restrict applicability to planar +picking in carefully crafted environments. +3 +OPTIMIZATION PROBLEM +3.1 +PROBLEM FORMULATION +An interactive environment, simulated or physical, is defined as a function g : S × A �→ {0, 1}, +where S is the state space of the problem, and A is the corresponding action space. For all s ∈ S +we associate a feasible action space A+ +s such that g(s, a) = 1, ∀a ∈ A+ +s and an infeasible action +space A− +s such that g(s, a) = 0, ∀a ∈ A− +s , with A+ +s ∪ A− +s = A. The goal of this work is to find +a state-dependent surjective map πs : Z → A+ +s , referred to as policy, where Z is a latent space of +appropriate dimensionality. For a given state and uniform sampling from the latent space Z, the pdf +of πs is a function qs : A �→ R, which is the posterior distribution qs(a) = q(a|s). For the same +state, the distribution of the feasible actions according to the environment can be defined as +ps(a) = +g(s, a) +� +A g(s, a) da, +(1) +which is the true posterior ps(a) = p(a|s). The optimal solution satisfies Df (ps || qs) = 0, where +Df is an f-divergence, for example from Table 1. This implies ps = qs, therefore the support of qs is +equal to the support of ps, which is A+ +s by definition. Thus, the optimal policy is the solution to the +3 + +f(t) +f ′(t) +Jensen-Shannon (JS) +1 +2 +� +(t + 1) log +� +2 +t+1 +� ++ t log(t) +� +1 +2 log +� +2t +t+1 +� +Forward Kullback-Leibler (FKL) +− log(t) +− 1 +t +Reverse Kullback-Leibler (RKL) +t log(t) +log(t) + 1 +Table 1: Non-exhaustive list of f-divergences and the corresponding first derivative for gradient +estimators. The f-divergences are obtained by substituting the f functions above in equation 3 and +setting t = qθ/p. The conventions for p, q, FKL and RKL assume that p is the target distribution, q is +the model, and the FKL divergence is +� +p log(p/q). +optimization problem: +˜πs = argminπs∼Π Df (ps || qs) , +(2) +with Π being an arbitrary family of distributions. To generalize over all states s ∈ S, the policy can be +modeled as a neural sampler πθ : S × Z �→ A, with a corresponding pdf qθ(a|s), where θ indicates +the parameters of the neural network. Assuming that the environment can be used efficiently for +repeated controlled experiments, i.e., testing several actions for the same state, the above optimization +problem can be solved directly on the environment. If this is not possible, a critic network can be +used to imitate the environment, which is further discussed in Section 4. Note that the system state is +often only partially observable, and the action must be inferred from an observation. For simplicity of +notation in the following derivation of the gradients, we assume that the state is directly observable, +and we omit the state and action dependence of q and p. +3.2 +F-DIVERGENCES +The f-divergence between two pdfs p and q is a generalization of the Kullback-Leibler (KL) divergence +and has the form (Liese & Vajda (2006)) +Df(p || qθ) = +� +A +p f +�qθ +p +� +da, +(3) +where f : (0, ∞) → R is a convex function. Different choices of f lead to well known divergences +as summarized in Table 1. The gradients of the f-divergence w.r.t. θ can be estimated commuting +the derivative with the integral (L’Ecuyer (1995)) and using the score function gradient estimator +(Kleijnen & Rubinstein (1996)) as +∂ +∂θDf = ∂ +∂θ +� +A +p f +�qθ +p +� +da = +� +A +p f ′ +�qθ +p +� 1 +p +∂ +∂θqθ da = +� +A +qθ f ′ +�qθ +p +� ∂ +∂θ log qθ da, (4) +using the fact that p does not depend on θ. Since qθ is normalized to 1 and thus ∂θ +� +A q da = +� +A q ∂θ log q da = 0, a Lagrangian term λ can be added to the gradient estimator: +∂ +∂θDf = +� +A +qθ +� +f ′ +�qθ +p +� ++ λ +� ∂ +∂θ log qθ da. +(5) +If the support of qθ includes all of A the above formula can be rewritten as the expectation on qθ as +∂ +∂θDf = Eqθ +�� +f ′ +�qθ +p +� ++ λ +� ∂ +∂θ log qθ +� +. +(6) +Sampling from a proposal distribution q′, the expectation can be computed with importance sampling +(Robert & Casella (2004); Liu (2001)) as +∂ +∂θDf ≈ Eq′ +�qθ +q′ +� +f ′ +�qθ +p +� ++ λ +� ∂ +∂θ log qθ +� +. +(7) +3.3 +GRADIENT ESTIMATION +Given a sample a ∼ qθ, it is not possible to directly evaluate qθ(a) as it is not available in closed +form. Therefore, qθ needs to be estimated to compute the gradients of the f-divergence. Given N +4 + +sampled actions ai ∼ qθ, qθ can be approximated with a KDE by +qθ(a) ≈ ˆqθ,σ(a) = 1 +N +� +ai∼qθ +kσ(a − ai), +(8) +where kσ is a Gaussian kernel with a diagonal bandwidth matrix σ. The KDE makes the estimate of +the expectation possible. Using equation 6, computing the expectation value as the average over the +samples yields +∂ +∂θDf ≈ 1 +N +� +ai∼qθ +� +f ′ +� ˆqθ,σ +p +� ++ λ +� ∂ +∂θ log ˆqθ,σ. +(9) +This gradient estimator turned out not to converge in our experiments. While a systematic investigation +of the convergence issue was not completed, we suspect two primary reasons for this. First, the +support qθ usually does not cover the whole action space A, which is necessary for the expectation +formulation in equation 6. Second, evaluating qθ(ai) based on a KDE, which uses aj as supports, has +a bias for j = i. +Adding Gaussian noise to the samples gives full support in A and reduces the bias at the support +points of the KDE, which lead to convergence in the experiments. The new smoothed samples are +given by a∗ +j = ai + ϵ for mi ≤ j < m(i + 1) and ϵ ∼ N(0, σ′), where m indicates the number of +smoothed samples drawn for each original sample. This is equivalent to sampling from a KDE with +ai as supports and σ′ as bandwidth. The gradient, using importance sampling in equation 7, can be +rewritten after resampling as follows +∂ +∂θDf ≈ 1 +M +� +a∗ +j ∼ˆqθ,σ′ +ˆqθ,σ +ˆqθ,σ′ +� +f ′ +� ˆqθ,σ +p +� ++ λ +� ∂ +∂θ log ˆqθ,σ, +(10) +with M = mN. Additionally, equation 10 requires an estimate of p, which in turn requires an +estimate of the volume in equation 1 +� +A +g(a) da ≈ 1 +M +� +a∗ +j +g(a∗ +j) +ˆqθ,σ′(a∗ +j). +(11) +This estimation is similar to self-normalized importance sampling (Murphy (2012)) but uses the +proposal distribution. The bandwidth σ′ of the proposal distribution is a hyper-parameter. Setting +σ′ = c σ, experiments show that c > 1 helps convergence. Intuitively, a larger bandwidth enables the +exploration of nearby modes in the action space. Specific estimators for the different f-divergences +can be obtained substituting f ′ from Table 1 into equation 10. A summary of the gradient estimators +used in this work is given in Table 2. +4 +METHODOLOGY +The derivation in Section 3.3 assumes that the training could be performed directly on the interactive +environment. To train the actor, multiple actions have to be evaluated for the same state. Typically, +this is not possible, either because of reproducibility in real experiments or computational cost for +simulations. An auxiliary neural network ξφ : S × A → R with parameters φ, can be trained to +imitate the environment g. The policy can then be trained to match the distribution of the feasible +actions according to this auxiliary neural network. We refer to πθ and ξφ as generative actor and +critic, respectively. The neural network architectures are presented in Appendix B. +The learning algorithm presented in this paper is inspired by RL and CB. At every training step, the +environment generates a state for which the actor proposes an action. The action is evaluated in +the environment yielding success or failure. The state, action, and feasibility form an experience +stored in a replay memory. The critic is trained on random samples of experiences from the replay +memory with a cross-entropy loss on the outcome. The actor trains on a batch of states from the +memory. For each state, multiple actions are sampled from the actor, used as support for ˆqθ,σ and +ˆqθ,σ′. New samples are drawn from the proposal distribution ˆqθ,σ′. These samples are evaluated +by the critic ξφ, and the gradients are computed according to equation 10. The algorithm of the +interaction loop can be found in Appendix C. While the general interaction loop is standard in RL, +two changes have proven beneficial to the convergence: balanced replay memory and maximum +uncertainty collection. Additionally, an action optimization can take advantage of the density estimate +to improve performance after training. +5 + +Loss +Actor Gradient Estimator +λ +JS +1 +2M +� +a∗ +j +ˆqθ,σ +ˆqθ,σ′ log +� +2ˆqθ,σ +p+ˆqθ,σ +� +∂ +∂θ log ˆqθ,σ +0 +FKL +- 1 +M +� +a∗ +j +p +ˆqθ,σ′ +∂ +∂θ log ˆqθ,σ +0 +RKL +1 +M +� +a∗ +j +ˆqθ,σ +ˆqθ,σ′ log +� +ˆqθ,σ +p +� +∂ +∂θ log ˆqθ,σ +-1 +GAN +1 +N +� +ai +∂ +∂a log(1 − ξφ) ∂ +∂θai +- +ME +1 +N +� +ai +∂ +∂θ log ˆqθ,σ − ∂ +∂a log ξφ ∂ +∂θai +- +Table 2: Gradient estimators of different losses and choice of Lagrangian multiplier λ. +4.1 +BALANCED REPLAY MEMORY +Since the environment yields either a success or a failure, the critic is a binary classifier that suffers +from unbalanced data when being trained. Its memory dataset continuously grows through the +interactions between the actor and the environment. In the beginning, the actor performs poorly, +yielding primarily experiences with failure labels, with the opposite at the end of the training. This +labeling bias prevented the critic from distinguishing between success and failure outcomes, making +convergence impossible. To avoid the critic from biasing towards failure or success labels, we use +two replay memories, one for failures and one for successes. When training the critic, half of the +experiences are sampled from the positive replay memory and the other half from the negative replay +memory. With this strategy, the labeling bias can be mitigated. The potentially amplified classification +bias (e.g., complicated shapes have more failure labels) did not appear to hinder convergence. This +memory can be prefilled with imitation examples to bootstrap the critic learning. While it is possible +to minimize the use of expert knowledge, this paper focuses on the main learning method, while the +impact of imitation learning will be analyzed in future work. +4.2 +MAXIMUM-UNCERTAINTY COLLECTION +Given one state of the environment, the actor can generate several candidate actions. Depending on +the training stage and the state, these proposed actions can have a different degree of information +for the critic. Selecting actions for which the critic predicts ξ ≈ 0.5, i.e., it cannot decide between +success and failure, can provide higher information content. This strategy has proven to improve the +convergence in our tests. +4.3 +ACTION OPTIMIZATION +Optimal performance in an environment with multiple disconnected sets of feasible actions would +require a multimodal distribution with a probability density of zero between the modes. Since the +actor is a continuous neural network and the latent space is continuous, the actor cannot generate +only positive actions. However, the actor aims to minimize the probability density for actions in the +gaps between the modes. The probability density at each action can be estimated using the KDE, and +the actions with the lowest corresponding density can be rejected. The accuracy of actors with strong +multimodal performance like FKL is significantly increased from action optimization as shown in +Section 6. +5 +EXPERIMENTAL SETUP +5.1 +ROBOTIC GRASPING SIMULATION +The proposed architecture was tested in a simplified robotic grasping simulation. We assume a +vertical configuration of a parallel gripper with three degrees of freedom x, y, and α and an object +that is an extrusion of a 2D shape. The simulator generates five different shapes with varying position, +angle, color, and geometric parameters. The success of a grasp is determined by the relative position +and alignment of the gripper to the outline of the object as seen from a camera positioned above the +experiment. Details about the simulator can be found in Appendix A. +6 + +(a) Problem +x +α +y +(b) Ground Truth +x +(c) Critic +x +(d) Actor +Figure 1: Critic classification and actor distribution trained with JS compared with the ground truth. +Five example grasps are shown in the problem and their associated locations in the ground truth. The +figures show projections onto the x-y plane (top row) and the x-α plane (bottom row). +The choice of this simulation, as opposed to existing robotic simulations, was motivated by efficiency +while iterating through several combinations of models and parameters. The target distribution fidelity +is not of primary concern in this work. The focus is instead on the capability of the proposed method +to learn all feasible actions. +5.2 +COMPARISON +In the evaluation, we are comparing different f-divergences with each other and with two other +approaches. The analyzed f-divergences are the FKL, RKL, and JS divergences. The two other +approaches are an ME RL algorithm similar to SAC in Haarnoja et al. (2018), which trains the actor +to minimize +min +θ +Es∼M,z∼Z [log qθ(πθ(s, z)|s) − ξφ(s, πθ(s, z))] , +(12) +with M being the replay memory. The critic is trained as described in Section 4. Instead of using the +reparameterization trick with a known distribution to estimate the entropy, we use the KDE. The other +approach is an implementation of a conditional GAN (Mirza & Osindero (2014)) with a growing +dataset. The min-max optimization problem is given through +min +θ +max +φ +Es,a∼Mp,z∼Z [log(ξφ(s, a)) − log(1 − ξφ(s, πθ(s, z)))] , +(13) +with the positive replay memory Mp, which grows through successful interactions with the environ- +ment. An asterisk is added (e.g., JS*) when using action optimization, rejecting 10% of the proposed +actions with the lowest probability density. The actor gradient estimators for all approaches are listed +in Table 2. +In the following section, we only compare with approaches that do not explicitly utilize the structure +of the problem, as the intention of the proposed approach is to be generally applicable in a continuous +CB problem setting. A comparison with a widely used approach from the grasping literature is +conducted in Appendix D. +6 +RESULTS +For each configuration, 3 agents were trained for 1 million interaction steps with the environment, +taking approximately 48 hours on a single NVIDIA A100 GPU. At the start of the training, 80k +examples, including positives and negatives, for randomly generated shapes were added to the +balanced replay memory to bootstrap the critic and discriminator learning. +Figure 1 shows the problem, the ground truth feasible picking positions, the critic estimate, and +a heat-map of the actor’s proposed actions. All figures are projections taking the maximum over +7 + +H +8 +Box +(a) Problem +(b) Truth +(c) JS +(d) FKL +(e) RKL +(f) GAN +(g) ME +Figure 2: Qualitative comparison of the implemented algorithms, showing action heat-maps on three +different states, with the last state never been observed during training. +the dimension that is not shown. In the problem visualization in Figure 1a, five feasible picks are +shown in different colors, which correspond to the markers in Figure 1b. These markers highlight the +complex multimodality of the problem. While it appears that, e.g., red and purple are in the same +mode in the x-y projection, it is visible in the x-α projection that they are not directly connected. +Figure 1c shows that the critic has an approximate understanding of the feasible regions of the action +space, showing five modes clearly in the x-y projection. The actor distribution in Figure 1d also +shows all five modes, while the output is significantly sharper in the centers of the modes. This is due +to the use of the KDEs and the choice of bandwidth σ. +In the qualitative comparison in Figure 2 the actor distributions of the different algorithms are shown +for three different shapes. While the H and 8 shapes were trained on, the Box shape has not been seen +during training. The different subfigures show the action heat maps of all implemented algorithms, +showing only the x-y projections. The H-row shows that JS and FKL learned all five modes, with +JS having the fewest samples in the connecting area. RKL and the GAN show two modes. The ME +implementation collapses in a single mode. The 8-row and the Box-row show a similar pattern with +the most pronounced spread of the action distributions in JS and FKL and mostly collapsed action +regions in the other approaches. +To quantify the multimodal capabilities, and thus the transferability of the learned skill, each algo- +rithm’s accuracy and shares of modes on all shapes were evaluated. For each shape, 1024 random +states were generated that differ in pose, color, and geometry (example variations can be seen in the +Appendix A). For each state, 1024 actions were sampled from the different actors. The actions were +then evaluated, and the mode of each action was recorded. The modes were then ranked and averaged +over all the states of that shape by frequency. By averaging the ranks instead of the modes, the last +rank shows the average ratio of the least frequent mode for each state. +Figure 3 shows the shares of each rank for the H and Box shapes for all the algorithms, with +the asterisk indicating that action optimization was applied. This figure presents the multimodal +capabilities of the JS and FKL algorithms, which are the only ones with the last ranked mode present +for the H and with significantly more pronounced modes than the others for the Box. Therefore, only +JS and FKL are capable of learning a transferable skill according to our definition. The generalization +capability of the GAN implementation is significantly lower than the others, as seen on the Box shape. +To quantify the performance, Table 3 shows the accuracy (feasible actions generated over total actions +generated) for each shape and the last ranked mode for the H, T, and Box shapes. The table shows +that ME has solid performance on all shapes but fails to find the different modes. The GAN algorithm +performs well with some modes present, but overall it is weaker than the others. RKL has high +scores but mostly fails at finding all the modes. FKL shows good performance in mode finding, +with an overall accuracy similar to RKL. JS is on par with the ME accuracy with the addition that +it repeatably finds all the modes. Generally, action optimization improves accuracy but does not +help mode finding, slightly decreasing the least ranked mode for most approaches. The maximum +8 + +(a) H Shape +(b) Box Shape +Figure 3: Gripping rank comparison, with the ratio of picks for each ranked mode or failure in %. +JS* +JS +FKL* +FKL +RKL* +RKL +GAN* GAN +ME* +ME +Score +H +96.01.0 91.21.3 92.50.4 85.41.1 86.93.6 83.63.5 85.82.0 83.52.7 96.60.7 95.91.2 +T +96.50.5 93.31.0 92.51.0 88.00.9 97.60.9 95.71.2 81.62.5 79.71.7 93.91.9 93.61.7 +8 +86.92.3 82.91.7 83.40.7 77.70.7 82.83.2 79.22.5 70.34.5 66.95.7 85.50.5 84.21.9 +Spoon 97.40.6 96.90.5 93.61.9 90.62.3 97.41.1 97.61.1 94.72.1 95.12.3 92.11.5 92.21.6 +Box +62.22.3 61.72.7 53.87.4 52.17.1 53.52.9 53.03.4 34.28.2 33.17.2 65.76.9 65.96.8 +Avg +87.80.8 85.20.8 83.20.7 78.80.6 83.62.1 81.82.1 73.32.3 71.72.7 86.82.1 86.41.8 +Mode +H +9.70.7 +10.70.3 9.80.5 +9.40.6 +0.00.0 +0.00.0 +0.20.4 +0.20.3 +0.00.0 +0.00.0 +T +11.00.7 12.30.6 15.80.6 15.30.3 0.40.3 +0.70.4 +1.12.3 +1.42.6 +0.00.0 +0.00.0 +Box +5.50.8 +6.91.1 +6.81.2 +7.21.4 +0.70.8 +1.31.3 +0.00.0 +0.00.0 +0.00.0 +0.00.0 +Table 3: Comparison on all shapes with the mean of the grasping success ratio in % on top and the +least ranked mode in % on the bottom, with the maximum deviations over the 3 runs in superscript. +deviations in the superscript show that all approaches learn reliably with the GAN having the highest +performance deviations among runs. +7 +CONCLUSION AND FUTURE WORK +This work proposes to learn a skill by training a generator to generate all feasible actions for a subtask. +To this end, the output distribution of the generator is learned to match a uniform distribution over +the feasible action space. While learning within a 2D grasping simulation, the method shows stable +convergence for FKL, RKL, and JS. As expected, FKL is more efficient in visiting all the modes. +JS has the highest accuracy while reliably visiting all the modes. The proposed learning strategy +expands the current state-of-the-art training within multimodal interactive environments by showing +competitive accuracy while visiting all the modes. Since the proposed approach can visit all the +modes, it learns the skill of grasping independently of a specific downstream task. In future work, we +will investigate how downstream tasks can choose their optimal action among the proposed feasible +options. As it is not dependent on the structure of the problem, we will further utilize it for a 6D +grasping problem as well as for other applications. +Some limitations have emerged during experimentation. Currently, many imitation examples are +required to bootstrap the critic’s learning. A possibility to mitigate this could be the progressive tuning +of the KDEs or learning their parameters during training. This approach could favor exploration +initially and divergence estimation later in training. A complementary strategy could be using +curriculum learning techniques that start with simple problems where solutions are less sparse in the +action space. Furthermore, the proposed approach may not be very effective in high-dimensional +problems as the sampling requirements for the estimator grow exponentially. The limit on the degrees +of freedom will be explored in future work. A mitigation of this issue can come from the rich literature +9 + +100 +4.0 +3.4 +4.1 +7.5 +8.8 +13.1 +14.6 +14.2 +16.4 +16.5 +9.7 +9.8 +10.7 +3.2 +4.9 +9.4 +4.2 +80 +5.1 +16.1 +15.2 +17.9 +13.1 +15.5 +13.1 +17.5 +14.2 +60 +20.2 +19.1 +18.8 +17.6 +96.6 +95.9 +40 +23.1 +22.3 +21.3 +20.5 +67.0 +Rank 1 +65.6 +64.2 +61.3 +Rank2 +Rank3 +20 +Rank4 +27.0 +26.1 +25.0 +23.8 +Rank 5 +Fail +0 +JS* +JS +FKL* +FKL +RKL* +RKL +GAN* +（ +GAN +ME* +ME100 +34.3 +34.1 +37.8 +38.3 +80 +46.2 +46.5 +47.9 +47.0 +65.8 +66.9 +60 +5.5 +6.9 +13.3 +14.7 +11.3 +3.5 +人 +6.8 +12.0 +4.4 +7.2 +12.1 +11.1 +12.0 +40 +11.0 +18.1 +52.2 +50.9 +17.5 +3.8 +3.7 +15.2 +Rank1 +14.6 +Rank 2 +20 +37.2 +35.4 +Rank 3 +30.3 +29.3 +27.3 +25.3 +Rank 4 +20.7 +19.3 +Fail +0 +JS +FKL* +FKL +RKL* +RKL +GAN* +GAN +*sr +ME* +MEon non-parametric density estimation in higher-dimensional problems and its applicability to the +proposed method. 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We assume +a vertical configuration of a parallel gripper with three degrees of freedom x, y, and α and assume +that the object is an extrusion of the 2D shape in the observation. The action space is constrained to +the center 78 × 78 pixel region to avoid undefined behavior at the border of the RGB image. The +angle of the grasp is in [0, π) as the gripper is symmetrical, and thus a full revolution is not necessary. +The success of a grasp is only determined by the relative position and alignment of the gripper to +the outline of the object, as seen from a camera positioned above the experiment. We developed an +algorithm that, given the alignment of the gripper, i.e., x, y, and α and a simulated picture of the +object from a fixed camera, provides a success/failure outcome in a deterministic and reproducible +manner. Given the maximum aperture of the parallel gripper l and the width of the gripper claws w, +the simulation analyzes the cropped image content of dimensions l × w between the gripper claws +before the claws close on the object. The simulation checks if the object is sufficiently present and +equidistant from the claws and aligned within parameterized margins. Figure 4 shows successful +grasping poses and the respective gripper content for the objects that are trained on. +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Figure 4: Feasible gripper positions (red) for different variations of the shapes (H-shape, 8-shape, +Spoon, and T-shape) used in training, with a detailed view of the area between the gripper to the right +of each figure. +B +NEURAL NETWORK ARCHITECTURES +... +... +(128x128x5) +(43x43x32) +(22x22x64) +(11x11x128) Shared MLP +(121x(128+d)) +Latent Input (d) +FC +Figure 5: Before processing, the image is augmented with positional encoding resulting in 5 total +channels {r, g, b, x, y}. The network’s input layer (in gray) is a 5x5 embedding layer with stride 3, +followed by 7 residual blocks (in yellow) with a bottleneck. The output is processed by 3 layers of +"pixelwise" shared MLPs (in brown), with the features being concatenated with a latent input (in +purple) of length d. The latent input is a random sample from Z for the actor and the action to be +evaluated for the critic. Four (for the actor) or three (for the critic) fully connected layers (in blue) +output the action and the score, respectively. +The neural network design was guided by simplicity and inspired by GANs. Features that rely on +domain-specific knowledge are avoided to evaluate better the learning method presented in the paper. +13 + +The structure of the actor and critic neural networks are illustrated in Figure 5. The residual feature +extraction network (He et al. (2016)) is shared between the actor and critic. +As a peculiarity of the network and the loss, the actor’s inferred action has four components, +[x, y, r sin α, r cos α], with r ∈ [0, +√ +2]. The angle can be extracted trivially with the arctan of +the ratio of the third and fourth action components. As the scale factor r does not change the angle, +the critic receives the normalized action [x, y, sin α, cos α] as input. To avoid the actor from reaching +the singularity at r = 0 and the distribution q being spread along the radius, g(s, a) and ξ(s, a) are +scaled with an unnormalized Gaussian on the radius, centered at 0.5 with the standard deviation of +0.4. +C +ALGORITHM AND HYPERPARAMETERS +Algorithm 1: Jenson-Shannon training loop +1 Initialize M with imitaition and random examples and initialize θ, φ +2 for 1 to Training Steps do +// Training steps are 1, 000, 000 in experiments +3 +for 1 to Interaction Steps do // Interaction steps are 1 in experiments +4 +s ← Generate a new problem +5 +zi ← Sample uniformly in Z, +∀i ∈ [1, U] +6 +ai ← πθ(s, zi), +∀i ∈ [1, U] +7 +ˆri ← ξφ(s, ai), +∀i ∈ [1, U] +8 +j ← arg mini∈[1,U] |0.5 − ˆri| // Get action with highest uncertainty +9 +r ← g(s, aj) +10 +if r == 1 then +11 +Store (s, aj, r) in Mp +12 +else +13 +Store (s, aj, r) in Mn +14 +end +15 +end +16 +for 1 to Critic Steps do +// Critic steps are 2 in experiments +17 +(si, ai, ri)L +i=1 ← Sample half from Mp and half from Mn +18 +φ ← φ − αφ∇φ +�L +i=1 ri log(ξφ(si, ai)) + (1 − ri) log(1 − ξφ(si, ai)) +19 +end +20 +for 1 to Actor Steps do +// Actor steps are 1 in experiments +21 +for k = 1 to K do +22 +sk ← Sample from M +23 +zi ← Sample uniformly in Z, +∀i ∈ [1, N] +24 +ai ← πθ(sk, zi), +∀i ∈ [1, N] +25 +ϵj ∼ N(0, σ′), +∀j ∈ [1, M] +26 +a∗ +j ← stop_gradient(a⌈j/m⌉) + ϵj, +∀j ∈ [1, M] +// Resample from KDE +27 +ˆqj ← 1 +N +�N +i=1 kσ(a∗ +j − ai), +∀j ∈ [1, M] // Evaluate KDE on samples +28 +ˆq′ +j ← 1 +N +�N +i=1 kσ′(a∗ +j − ai), +∀j ∈ [1, M] +// Evaluate proposal pdf +29 +ˆrj ← ξφ(sk, a∗ +j), +∀j ∈ [1, M] +30 +ˆV ← +1 +M +�M +j=1 +ˆrj +ˆq′ +j +// MC integration with importance sampling +31 +ˆpj ← ˆrj +ˆV , +∀j ∈ [1, M] +32 +gk ← +1 +2M +�M +j=1 +ˆqj +ˆq′ +j log +� +2ˆqj +ˆqj+ˆpj +� +∇θ log(ˆqj) +// gradient trace +33 +end +34 +θ ← θ − αθ 1 +K +�K +k=1 gk +35 +end +36 end +14 + +Parameter +Value +Description +N +128 +Minibatch size +M +256 +Resampling size +m +2 +Samples per KDE support point (M/N) +U +64 +Maximum uncertainty proposals +K +16 +Actor batch size +L +32 +Critic batch size +σ +diag(0.025, 0.025, 0.4, 0.4) +KDE bandwidth +σ′ +diag(0.075, 0.075, 1.2, 1.2) +Sampling KDE bandwidth +|Mp| +160,000 +Positive replay memory size +|Mn| +160,000 +Negative replay memory size +|M| +320,000 +Total replay memory size +Table 4: Hyperparameters +D +OBSERVATION VARIATION EXPERIMENTS +D.1 +SETUP +Figure 6: Different distortions are applied, showing a colored chess board for illustration and an +example shape under all distortions. +To highlight the difference between the proposed approach and related work of robotic grasping, we +investigate how distortions of the observation affect the performance. The distortions investigated are +a rotation, projection, and rotation + projection as shown in Figure 6. These distortions correspond to +different camera perspectives. We train a new agent for 106 training steps for each distortion and +approach in the following comparison. +We compare with a common approach in the literature (Zeng et al. (2020)) that make use of spatial +equivariances. The approach utilizes fully convolutional networks to output a probability of success +for each action of a discretized action space. We implement two variants. In the first one, just as in +Zeng et al. (2020), the observation is fed into the neural network multiple times with different rotations. +The neural network then only needs to output a one-channel image containing the probability of +success of each discretized x, y action for the given rotation of the image. This approach thus +makes use of translation equivariance by using a convolutional neural network (CNN) and rotation +equivariance. In the experiments, we denote it as the heat-map approach (H). The second variant +estimates for each observation the success for different rotations by outputting a multi-channel +image indicating the success estimate of each discretized x, y, α action explicitly. Thus it only takes +advantage of translation equivariance. It is called stacked heat-map (SH) in the following. +The approaches are implemented using fully convolutional networks with an hourglass structure, +adopting the beginning of the Resnet in Figure 5 and adding the same structure in reverse order with +nearest-neighbor upsampling. Both approaches predict grasping success for 78x78 pixels with 16 +rotation angles. They are trained on a cross-entropy loss on the grasping outcome sampled from +the balanced replay buffer. The replay buffer is also filled with imitation learning examples, and +maximum uncertainty sampling is applied. For evaluation, the success estimate of each discretized +15 + +Rotated + +Normal +Rotated +Projected +Projectedaction is used as its probability to be sampled. To increase accuracy, an inverted temperature factor +increases the difference between higher and lower score actions. Specifically, the actions are sampled +according to +q(a|s) = +exp(β log ξ(s, a)) +� +∀a∈Ad exp(β log ξ(s, a)), +(14) +with ξ being the fully convolutional network with s as input and as output shape the discretized action +space Ad. The inverted temperature was set to β = 100. +D.2 +RESULTS +(a) H Shape +(b) Box Shape +Figure 7: Gripping rank comparison, with the ratio of picks for each ranked mode or failure in %. +Normal +Rotated +Projected +Rotated + Projected +JS* +H +SH +JS* +H +SH +JS* +H +SH +JS* +H +SH +Score +H +96.7 +92.6 +97.7 +95.1 +92.0 +96.6 +96.2 +24.0 +0.9 +97.9 21.6 +0.8 +T +97.1 +93.5 +98.2 +95.5 +94.3 +98.3 +96.2 +31.6 +0.8 +98.0 27.0 +0.7 +8 +86.4 +90.4 +97.2 +85.5 +87.4 +95.5 +85.3 +15.4 +0.9 +88.8 14.3 +0.8 +Spoon +97.7 +93.8 +98.8 +96.5 +94.5 +98.6 +97.1 +33.8 +0.7 +97.8 33.2 +0.6 +Box +62.3 +90.7 +96.2 +55.9 +88.4 +89.4 +54.7 +21.1 +1.2 +53.5 19.3 +1.1 +Avg +88.0 +92.2 +97.6 +85.7 +91.3 +95.7 +85.9 +25.2 +0.9 +87.2 23.1 +0.8 +Mode +H +9.1 +9.3 +10.0 +8.6 +7.5 +4.6 +8.0 +0.2 +0.0 +8.5 +0.0 +0.0 +T +11.5 +18.7 +18.5 +10.4 +14.8 +9.8 +11.8 +0.5 +0.1 +14.3 +0.2 +0.1 +Box +5.5 +16.7 +16.8 +4.3 +9.5 +1.4 +5.1 +0.3 +0.1 +5.3 +0.1 +0.1 +Table 5: Comparison of the proposed Jensen-Shannon approach with approaches from the literature. +The results are shown in Figure 7 and Table 5, which compare the performance of the proposed +Jensen-Shannon (JS) approach with the heat-map (H) and stacked heat-map (HS) approaches on +the four different distortions. As expected, the specifically crafted H and SH approaches perform +better on the original problem than the generic approach proposed in the paper. In that scenario, no +scene understanding is required, and only local features need to be considered to estimate grasping +success. Therefore, the approaches are expected to generalize well to unseen shapes, as seen for the +Box-Shape, since the grasping success depends only on gripper alignment. They only need to learn +to imitate the grasping success heuristic shown in Figure 4. +Interestingly, rotating the observation does not seem to impact their performance. However, under +projection and projection + rotation, both approaches fail to learn to grasp. The heat-map approach +16 + +100 +3.3 +2.3 +3.4 +3.8 +2.I +4.9 +7.4 +8.0 +Rank 1 +4.6 +8.5 +10.0 +9.1 +8.0 +8.6 +Rank 2 +9.3 +7.5 +9.8 +Rank 3 +15.6 +15.1 +80 +16.6 +16.5 +14.711.8 +Rank 4 +16.3 +16.0 +Rank 5 +Fail +17.1 +19.2 +19.6 +19.3 +76.0 +20.1 +20.5 +78.4 +60 +19.8 +24.8 +99.1 +99.2 +23.3 +23.2 +23.9 +23.7 +40 +23.1 +23.7 +22.4 +1.8 +41.4 +20 +4.3 +3.1 +32.3 +30.7 +30.3 +28.9 +27.7 +27.0 +24.8 +17.1 +17.6 +0 +JS* H SH +JS* H SH +JS* H SH +IS* H SH +Normal +Rotated +Projected +Rotated+ +Projected100 +3.8 +9.3 +Rank 1 +11.610.6 +Rank 2 +16.8 +1.4 +6.3 +9.5 +Rank 3 +16.7 +37.7 +80 +44.1 +Rank 4 +45.3 +46.5 +Fail +16.721.0 +21.8 +20.7 +5.5 +78.9 +60 +80.7 +4.3 +5.1 +11.5 +98.8 +5.3 +24.5 +98.9 +8.9 +26.7 +10.6 +10.1 +24.9 +40 +17.9 +15.3 +15.8 +60.7 +15.3 +20 +37.6 +4.5 +3.4 +30.8 +27.428.4 +27.4 +23.3 +22.8 +14.9 +15.0 +0 +JS* H SH +JS* H SH +JS* H SH +JS* H SH +Normal +Rotated +Projected +Rotated+ +Projectedstill learns to grasp the objects occasionally, while the stacked heat-map approach does not learn +anything. Our proposed approach learns well for all distortions. The generalization on the Box-Shape +decreases a bit but the general performance remains similar. Surprisingly, the results for rotation + +projection on the training shapes are above the variance in the main results in Table 3. An explanation +for the performance increase could be that the region in the input observation that the object can be in +is smaller than in the normal case, as can be seen by the area between the colored tiles in Figure 6. +The reduced region could slightly ease scene understanding, leading to improved results. +In general, the performance of our proposed approach does not depend on the distortion as it does not +explicitly use the spatial structure. Its design does not depend on the specifics of the experiment at +all. It can, therefore, learn independently of the distortion applied as long as the object is still fully +observable. +17 + diff --git a/QdFJT4oBgHgl3EQfJizI/content/tmp_files/load_file.txt b/QdFJT4oBgHgl3EQfJizI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52e5cb05415ea0ed7f7935bd5126a6559f43e363 --- /dev/null +++ b/QdFJT4oBgHgl3EQfJizI/content/tmp_files/load_file.txt @@ -0,0 +1,1176 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf,len=1175 +page_content='LEARNING TO GENERATE ALL FEASIBLE ACTIONS Mirco Theile1,3*, Daniele Bernardini1∗, Raphael Trumpp1 Cristina Piazza2, Marco Caccamo1, Alberto L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Sangiovanni-Vincentelli3 1TUM School of Engineering and Design, Technical University of Munich 2Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' of Informatics, Technical University of Munich 3Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' of Electrical Engineering and Computer Sciences, University of California, Berkeley {mirco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='theile,daniele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='bernardini,raphael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='trumpp, cristina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='piazza,mcaccamo}@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='de,alberto@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='edu ABSTRACT Several machine learning (ML) applications are characterized by searching for an optimal solution to a complex task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The search space for this optimal solution is often very large, so large in fact that this optimal solution is often not computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Part of the problem is that many candidate solutions found via ML are actually infeasible and have to be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Restricting the search space to only the feasible solution candidates simplifies finding an optimal solution for the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Further, the set of feasible solutions could be re-used in multiple problems characterized by different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In particular, we observe that complex tasks can be decomposed into subtasks and corresponding skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We propose to learn a reusable and transferable skill by training an actor to generate all feasible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The trained actor can then propose feasible actions, among which an optimal one can be chosen according to a specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The actor is trained by interpreting the feasibility of each action as a target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The training procedure minimizes a divergence of the actor’s output distribution to this target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We derive the general optimization target for arbitrary f-divergences using a combination of kernel density estimates, resampling, and importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We further utilize an auxiliary critic to reduce the interactions with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' A preliminary comparison to related strategies shows that our approach learns to visit all the modes in the feasible action space, demonstrating the framework’s potential for learning skills that can be used in various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 1 INTRODUCTION Complex tasks can often be decomposed into multiple subtasks, with corresponding skills that solve these subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Learning reusable and transferable skills is an active area of research (Kalashnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Chebotar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Deisenroth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' However, given a subtask, learning or even defining the corresponding skill is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Consider a robotic scenario where a robot is tasked to grasp an object and handle it in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Different downstream tasks can have different optimal grasps if the object has multiple feasible grasping poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Therefore, a grasping skill cannot be trained based on optimality definitions of individual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' However, a grasping algorithm that learned all feasible grasps could support all possible downstream tasks even without explicit knowledge thereof during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The downstream tasks can then select their respective optimal grasp among the proposed feasible options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Therefore, we consider a skill to be defined by the set of all feasible actions of a subtask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We propose a novel method to train a generative neural network to generate all feasible actions of a subtask by interacting with an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The interaction loop is adopted from Contextual Bandit (CB) (Langford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2008)) and Reinforcement Learning (RL) (Sutton & Barto (2018)): the environment presents a state for which the actor selects an action, which is tested in the environment, yielding either a success or failure outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' As in CB, we limit ourselves to one-step interactions as opposed to sequential multi-step interactions common in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' However, we do not minimize regret, ∗These authors contributed equally 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='11461v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='LG] 26 Jan 2023 typically done in CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Instead, we optimize the final policy as in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Unlike CB and RL, the approach does not aim to find one optimal solution for a given problem but aims to learn all feasible ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' By interpreting the feasibility of each action given a state as a posterior probability distribution over the actions, a target probability density function (pdf) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The actor is trained to minimize a divergence of its output distribution to this target pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The training algorithm in the method proposed can be used with any given f-divergence, including Reverse Kullback-Leibler (RKL), Forward Kullback-Leibler (FKL), and Jensen-Shannon (JS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The possibility to use FKL and JS is instrumental in visiting all the modes of the posterior distribution, as RKL is known to collapse into a single mode (Jerfel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The training algorithm presented in this paper uses Kernel Density Estimation (KDE) to estimate the pdf of the actor and Monte Carlo integration with importance sampling to estimate the normalization of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The divergences are estimated using samples from a proposal distribution which is a separate KDE based on the original samples of the actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This resampling step is necessary for convergence, which is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' As interactions with the environment are typically costly, an auxiliary critic network imitating the environment is trained simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The critic’s feasibility estimate of an action is then used to form the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The learning algorithm has been tested on a planar robotic grasping problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We test FKL, RKL, and JS divergences and compare them to implementations of maximum entropy (ME) RL and Generative Adversarial Networks (GANs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Besides accuracy, we measure how many grasping modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=', disconnected regions in the action space, are visited by each approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Generating actions in all grasping modes can ensure that the learned skill is reusable and transferable for various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The contributions of this paper are the following: Design of a new learning method for generative neural network models to explicitly learn to generate all feasible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Introduction of a novel gradient estimator for f-divergences that takes advantage of KDEs, resampling, and importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Application of the proposed learning algorithm to a 2D robotic grasping problem, comparing the proposed gradient estimators for f-divergences with related methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The rest of this work is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Section 2 discusses the related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Section 3 describes the optimization problem followed by the methodology in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The evaluation setup is described in Section 5 and the results are presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Section 7 concludes and gives an outlook on future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 2 RELATED WORK CBs have been successfully applied to several interactive learning problems with discrete action spaces (Langford & Zhang (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Foster & Rakhlin (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Simchi-Levi & Xu (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In several cases, the context and action spaces have been embedded in a linear multidimensional action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The embedding keeps the interaction linear while the action and context embeddings can be non-linear (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Foster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Recently, there has been an increased interest in extending the approach to continuous action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' However, most works are limited to 1D actions (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Majzoubi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Zhu & Mineiro (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Learning from an interactive environment is also the focus of RL (Sutton & Barto (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Many RL approaches use Actor-Critic (AC) architectures, among which the Soft Actor-Critic (SAC) algorithm (Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2018)) is most related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In SAC, the state-action value function of the critic is transformed into an energy-based distribution (Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2017)), yielding the target of the actor’s distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' SAC uses RKL as the loss function for the actor, which yields maximum entropy RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Through a reparameterization trick, which usually uses the family of Gaussians, the RKL is minimized through a direct gradient from the critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' GANs propose a similar architecture to AC, training a generator and discriminator adversarially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This adversarial training is equivalent to minimizing the JS divergence (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2014)) and has been extended to arbitrary f-divergences (Nowozin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Conditional GANs (Mirza 2 & Osindero (2014)) offer an alternative solution to the posterior sampling problem, as a generator conditioned on a given state can be trained to provide successful actions adversarially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' However, the problem analyzed in our paper is not naturally adversarial, as actions that have not yet been tested in the interactive environment should not be implicitly rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The discriminator learns to discriminate between tested successful actions from untested ones, providing the generator with inconsistent gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Expert knowledge is used in Imitation Learning (IL) to derive a policy from demonstration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The policy may be learned in a supervised manner in behavior cloning (Pomerleau (1988)) or as a combination of Inverse Reinforcement Learning (IRL) (Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2000)) to learn a reward function and a subsequent RL procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Ho & Ermon (2016) introduced Generative Adversarial Imitation Learning (GAIL), mitigating the intermediate IRL step by using a generative adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' As discussed in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2017), the policy learned by GAIL tends to interpolate between modes leading to erroneous behavior in multimodal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Using f-divergence minimization, the authors in Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Ghasemipour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020) intentionally collapse modes to avoid interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' IL and adversarial approaches require large amounts of expert data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' However, expert data is limited in an interactive environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Additionally, given that we aim to find all feasible actions, even more expert data representative of all feasible actions would be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Posterior sampling has been a long-standing problem in statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' State-of-the-art methods in Bayesian statistics rely on Markov Chain Monte Carlo (MCMC) algorithms (Hastings (1970);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Gelfand & Smith (1990)), eliminating the need to normalize the distribution which is often an intractable problem (Kruschke (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Variational Inference (VI) relies instead on fitting the posterior with a family of parametric probability distributions that can then be sampled from (Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Wainwright & Jordan (2008)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Neural samplers offer another alternative by approximating the posterior with a generative neural network (Nowozin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Normalizing flows also infer the pdf for each sample using invertible mappings (Rezende & Mohamed (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Tabak & Turner (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Tabak & Vanden-Eijnden (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' While this approach does not require density estimates, it limits its applicability to specific neural network designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' For robotic grasping, Kalashnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2018) propose using Deep Reinforcement Learning (DRL) to find optimal grasps through interaction with multiple real-world robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' If the goal is to find grasping poses explicitly to be used as the target of a classical controller, supervised learning techniques are often utilized (Kleeberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To support various downstream tasks, it would be necessary to find all feasible grasps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To this end, the action space is typically discretized and grasping success is estimated for each discrete action through heat-maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This can be learned supervised (Kumra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020)) or self-supervised (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020) explicitly utilize structure given by spatial equivariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We aim to find a solution that needs neither discretization nor to make use of the structure as these requirements restrict applicability to planar picking in carefully crafted environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 3 OPTIMIZATION PROBLEM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1 PROBLEM FORMULATION An interactive environment, simulated or physical, is defined as a function g : S × A �→ {0, 1}, where S is the state space of the problem, and A is the corresponding action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' For all s ∈ S we associate a feasible action space A+ s such that g(s, a) = 1, ∀a ∈ A+ s and an infeasible action space A− s such that g(s, a) = 0, ∀a ∈ A− s , with A+ s ∪ A− s = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The goal of this work is to find a state-dependent surjective map πs : Z → A+ s , referred to as policy, where Z is a latent space of appropriate dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' For a given state and uniform sampling from the latent space Z, the pdf of πs is a function qs : A �→ R, which is the posterior distribution qs(a) = q(a|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' For the same state, the distribution of the feasible actions according to the environment can be defined as ps(a) = g(s, a) � A g(s, a) da, (1) which is the true posterior ps(a) = p(a|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The optimal solution satisfies Df (ps || qs) = 0, where Df is an f-divergence, for example from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This implies ps = qs, therefore the support of qs is equal to the support of ps, which is A+ s by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Thus, the optimal policy is the solution to the 3 f(t) f ′(t) Jensen-Shannon (JS) 1 2 � (t + 1) log � 2 t+1 � + t log(t) � 1 2 log � 2t t+1 � Forward Kullback-Leibler (FKL) − log(t) − 1 t Reverse Kullback-Leibler (RKL) t log(t) log(t) + 1 Table 1: Non-exhaustive list of f-divergences and the corresponding first derivative for gradient estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The f-divergences are obtained by substituting the f functions above in equation 3 and setting t = qθ/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The conventions for p, q, FKL and RKL assume that p is the target distribution, q is the model, and the FKL divergence is � p log(p/q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' optimization problem: ˜πs = argminπs∼Π Df (ps || qs) , (2) with Π being an arbitrary family of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To generalize over all states s ∈ S, the policy can be modeled as a neural sampler πθ : S × Z �→ A, with a corresponding pdf qθ(a|s), where θ indicates the parameters of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Assuming that the environment can be used efficiently for repeated controlled experiments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=', testing several actions for the same state, the above optimization problem can be solved directly on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' If this is not possible, a critic network can be used to imitate the environment, which is further discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Note that the system state is often only partially observable, and the action must be inferred from an observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' For simplicity of notation in the following derivation of the gradients, we assume that the state is directly observable, and we omit the state and action dependence of q and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='2 F-DIVERGENCES The f-divergence between two pdfs p and q is a generalization of the Kullback-Leibler (KL) divergence and has the form (Liese & Vajda (2006)) Df(p || qθ) = � A p f �qθ p � da, (3) where f : (0, ∞) → R is a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Different choices of f lead to well known divergences as summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The gradients of the f-divergence w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' θ can be estimated commuting the derivative with the integral (L’Ecuyer (1995)) and using the score function gradient estimator (Kleijnen & Rubinstein (1996)) as ∂ ∂θDf = ∂ ∂θ � A p f �qθ p � da = � A p f ′ �qθ p � 1 p ∂ ∂θqθ da = � A qθ f ′ �qθ p � ∂ ∂θ log qθ da, (4) using the fact that p does not depend on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Since qθ is normalized to 1 and thus ∂θ � A q da = � A q ∂θ log q da = 0, a Lagrangian term λ can be added to the gradient estimator: ∂ ∂θDf = � A qθ � f ′ �qθ p � + λ � ∂ ∂θ log qθ da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (5) If the support of qθ includes all of A the above formula can be rewritten as the expectation on qθ as ∂ ∂θDf = Eqθ �� f ′ �qθ p � + λ � ∂ ∂θ log qθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (6) Sampling from a proposal distribution q′, the expectation can be computed with importance sampling (Robert & Casella (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Liu (2001)) as ∂ ∂θDf ≈ Eq′ �qθ q′ � f ′ �qθ p � + λ � ∂ ∂θ log qθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='3 GRADIENT ESTIMATION Given a sample a ∼ qθ, it is not possible to directly evaluate qθ(a) as it is not available in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Therefore, qθ needs to be estimated to compute the gradients of the f-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Given N 4 sampled actions ai ∼ qθ, qθ can be approximated with a KDE by qθ(a) ≈ ˆqθ,σ(a) = 1 N � ai∼qθ kσ(a − ai), (8) where kσ is a Gaussian kernel with a diagonal bandwidth matrix σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The KDE makes the estimate of the expectation possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Using equation 6, computing the expectation value as the average over the samples yields ∂ ∂θDf ≈ 1 N � ai∼qθ � f ′ � ˆqθ,σ p � + λ � ∂ ∂θ log ˆqθ,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (9) This gradient estimator turned out not to converge in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' While a systematic investigation of the convergence issue was not completed, we suspect two primary reasons for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' First, the support qθ usually does not cover the whole action space A, which is necessary for the expectation formulation in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Second, evaluating qθ(ai) based on a KDE, which uses aj as supports, has a bias for j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Adding Gaussian noise to the samples gives full support in A and reduces the bias at the support points of the KDE, which lead to convergence in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The new smoothed samples are given by a∗ j = ai + ϵ for mi ≤ j < m(i + 1) and ϵ ∼ N(0, σ′), where m indicates the number of smoothed samples drawn for each original sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This is equivalent to sampling from a KDE with ai as supports and σ′ as bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The gradient, using importance sampling in equation 7, can be rewritten after resampling as follows ∂ ∂θDf ≈ 1 M � a∗ j ∼ˆqθ,σ′ ˆqθ,σ ˆqθ,σ′ � f ′ � ˆqθ,σ p � + λ � ∂ ∂θ log ˆqθ,σ, (10) with M = mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Additionally, equation 10 requires an estimate of p, which in turn requires an estimate of the volume in equation 1 � A g(a) da ≈ 1 M � a∗ j g(a∗ j) ˆqθ,σ′(a∗ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (11) This estimation is similar to self-normalized importance sampling (Murphy (2012)) but uses the proposal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The bandwidth σ′ of the proposal distribution is a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Setting σ′ = c σ, experiments show that c > 1 helps convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Intuitively, a larger bandwidth enables the exploration of nearby modes in the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Specific estimators for the different f-divergences can be obtained substituting f ′ from Table 1 into equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' A summary of the gradient estimators used in this work is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 4 METHODOLOGY The derivation in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='3 assumes that the training could be performed directly on the interactive environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To train the actor, multiple actions have to be evaluated for the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Typically, this is not possible, either because of reproducibility in real experiments or computational cost for simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' An auxiliary neural network ξφ : S × A → R with parameters φ, can be trained to imitate the environment g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The policy can then be trained to match the distribution of the feasible actions according to this auxiliary neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We refer to πθ and ξφ as generative actor and critic, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The neural network architectures are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The learning algorithm presented in this paper is inspired by RL and CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' At every training step, the environment generates a state for which the actor proposes an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The action is evaluated in the environment yielding success or failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The state, action, and feasibility form an experience stored in a replay memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The critic is trained on random samples of experiences from the replay memory with a cross-entropy loss on the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The actor trains on a batch of states from the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' For each state, multiple actions are sampled from the actor, used as support for ˆqθ,σ and ˆqθ,σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' New samples are drawn from the proposal distribution ˆqθ,σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' These samples are evaluated by the critic ξφ, and the gradients are computed according to equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The algorithm of the interaction loop can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' While the general interaction loop is standard in RL, two changes have proven beneficial to the convergence: balanced replay memory and maximum uncertainty collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Additionally, an action optimization can take advantage of the density estimate to improve performance after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 5 Loss Actor Gradient Estimator λ JS 1 2M � a∗ j ˆqθ,σ ˆqθ,σ′ log � 2ˆqθ,σ p+ˆqθ,σ � ∂ ∂θ log ˆqθ,σ 0 FKL 1 M � a∗ j p ˆqθ,σ′ ∂ ∂θ log ˆqθ,σ 0 RKL 1 M � a∗ j ˆqθ,σ ˆqθ,σ′ log � ˆqθ,σ p � ∂ ∂θ log ˆqθ,σ 1 GAN 1 N � ai ∂ ∂a log(1 − ξφ) ∂ ∂θai ME 1 N � ai ∂ ∂θ log ˆqθ,σ − ∂ ∂a log ξφ ∂ ∂θai Table 2: Gradient estimators of different losses and choice of Lagrangian multiplier λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1 BALANCED REPLAY MEMORY Since the environment yields either a success or a failure, the critic is a binary classifier that suffers from unbalanced data when being trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Its memory dataset continuously grows through the interactions between the actor and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In the beginning, the actor performs poorly, yielding primarily experiences with failure labels, with the opposite at the end of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This labeling bias prevented the critic from distinguishing between success and failure outcomes, making convergence impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To avoid the critic from biasing towards failure or success labels, we use two replay memories, one for failures and one for successes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' When training the critic, half of the experiences are sampled from the positive replay memory and the other half from the negative replay memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' With this strategy, the labeling bias can be mitigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The potentially amplified classification bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=', complicated shapes have more failure labels) did not appear to hinder convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This memory can be prefilled with imitation examples to bootstrap the critic learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' While it is possible to minimize the use of expert knowledge, this paper focuses on the main learning method, while the impact of imitation learning will be analyzed in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='2 MAXIMUM-UNCERTAINTY COLLECTION Given one state of the environment, the actor can generate several candidate actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Depending on the training stage and the state, these proposed actions can have a different degree of information for the critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Selecting actions for which the critic predicts ξ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=', it cannot decide between success and failure, can provide higher information content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This strategy has proven to improve the convergence in our tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='3 ACTION OPTIMIZATION Optimal performance in an environment with multiple disconnected sets of feasible actions would require a multimodal distribution with a probability density of zero between the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Since the actor is a continuous neural network and the latent space is continuous, the actor cannot generate only positive actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' However, the actor aims to minimize the probability density for actions in the gaps between the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The probability density at each action can be estimated using the KDE, and the actions with the lowest corresponding density can be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The accuracy of actors with strong multimodal performance like FKL is significantly increased from action optimization as shown in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 5 EXPERIMENTAL SETUP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1 ROBOTIC GRASPING SIMULATION The proposed architecture was tested in a simplified robotic grasping simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We assume a vertical configuration of a parallel gripper with three degrees of freedom x, y, and α and an object that is an extrusion of a 2D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The simulator generates five different shapes with varying position, angle, color, and geometric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The success of a grasp is determined by the relative position and alignment of the gripper to the outline of the object as seen from a camera positioned above the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Details about the simulator can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 6 (a) Problem x α y (b) Ground Truth x (c) Critic x (d) Actor Figure 1: Critic classification and actor distribution trained with JS compared with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Five example grasps are shown in the problem and their associated locations in the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The figures show projections onto the x-y plane (top row) and the x-α plane (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The choice of this simulation, as opposed to existing robotic simulations, was motivated by efficiency while iterating through several combinations of models and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The target distribution fidelity is not of primary concern in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The focus is instead on the capability of the proposed method to learn all feasible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='2 COMPARISON In the evaluation, we are comparing different f-divergences with each other and with two other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The analyzed f-divergences are the FKL, RKL, and JS divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The two other approaches are an ME RL algorithm similar to SAC in Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2018), which trains the actor to minimize min θ Es∼M,z∼Z [log qθ(πθ(s, z)|s) − ξφ(s, πθ(s, z))] , (12) with M being the replay memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The critic is trained as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Instead of using the reparameterization trick with a known distribution to estimate the entropy, we use the KDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The other approach is an implementation of a conditional GAN (Mirza & Osindero (2014)) with a growing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The min-max optimization problem is given through min θ max φ Es,a∼Mp,z∼Z [log(ξφ(s, a)) − log(1 − ξφ(s, πθ(s, z)))] , (13) with the positive replay memory Mp, which grows through successful interactions with the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' An asterisk is added (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=', JS*) when using action optimization, rejecting 10% of the proposed actions with the lowest probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The actor gradient estimators for all approaches are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In the following section, we only compare with approaches that do not explicitly utilize the structure of the problem, as the intention of the proposed approach is to be generally applicable in a continuous CB problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' A comparison with a widely used approach from the grasping literature is conducted in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 6 RESULTS For each configuration, 3 agents were trained for 1 million interaction steps with the environment, taking approximately 48 hours on a single NVIDIA A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' At the start of the training, 80k examples, including positives and negatives, for randomly generated shapes were added to the balanced replay memory to bootstrap the critic and discriminator learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Figure 1 shows the problem, the ground truth feasible picking positions, the critic estimate, and a heat-map of the actor’s proposed actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' All figures are projections taking the maximum over 7 H 8 Box (a) Problem (b) Truth (c) JS (d) FKL (e) RKL (f) GAN (g) ME Figure 2: Qualitative comparison of the implemented algorithms, showing action heat-maps on three different states, with the last state never been observed during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' the dimension that is not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In the problem visualization in Figure 1a, five feasible picks are shown in different colors, which correspond to the markers in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' These markers highlight the complex multimodality of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' While it appears that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=', red and purple are in the same mode in the x-y projection, it is visible in the x-α projection that they are not directly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Figure 1c shows that the critic has an approximate understanding of the feasible regions of the action space, showing five modes clearly in the x-y projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The actor distribution in Figure 1d also shows all five modes, while the output is significantly sharper in the centers of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This is due to the use of the KDEs and the choice of bandwidth σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In the qualitative comparison in Figure 2 the actor distributions of the different algorithms are shown for three different shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' While the H and 8 shapes were trained on, the Box shape has not been seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The different subfigures show the action heat maps of all implemented algorithms, showing only the x-y projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The H-row shows that JS and FKL learned all five modes, with JS having the fewest samples in the connecting area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' RKL and the GAN show two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The ME implementation collapses in a single mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The 8-row and the Box-row show a similar pattern with the most pronounced spread of the action distributions in JS and FKL and mostly collapsed action regions in the other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To quantify the multimodal capabilities, and thus the transferability of the learned skill, each algo- rithm’s accuracy and shares of modes on all shapes were evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' For each shape, 1024 random states were generated that differ in pose, color, and geometry (example variations can be seen in the Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' For each state, 1024 actions were sampled from the different actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The actions were then evaluated, and the mode of each action was recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The modes were then ranked and averaged over all the states of that shape by frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' By averaging the ranks instead of the modes, the last rank shows the average ratio of the least frequent mode for each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Figure 3 shows the shares of each rank for the H and Box shapes for all the algorithms, with the asterisk indicating that action optimization was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This figure presents the multimodal capabilities of the JS and FKL algorithms, which are the only ones with the last ranked mode present for the H and with significantly more pronounced modes than the others for the Box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Therefore, only JS and FKL are capable of learning a transferable skill according to our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The generalization capability of the GAN implementation is significantly lower than the others, as seen on the Box shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To quantify the performance, Table 3 shows the accuracy (feasible actions generated over total actions generated) for each shape and the last ranked mode for the H, T, and Box shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The table shows that ME has solid performance on all shapes but fails to find the different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The GAN algorithm performs well with some modes present, but overall it is weaker than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' RKL has high scores but mostly fails at finding all the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' FKL shows good performance in mode finding, with an overall accuracy similar to RKL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' JS is on par with the ME accuracy with the addition that it repeatably finds all the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Generally, action optimization improves accuracy but does not help mode finding, slightly decreasing the least ranked mode for most approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The maximum 8 (a) H Shape (b) Box Shape Figure 3: Gripping rank comparison, with the ratio of picks for each ranked mode or failure in %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' JS* JS FKL* FKL RKL* RKL GAN* GAN ME* ME Score H 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='63.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='0 Table 3: Comparison on all shapes with the mean of the grasping success ratio in % on top and the least ranked mode in % on the bottom, with the maximum deviations over the 3 runs in superscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' deviations in the superscript show that all approaches learn reliably with the GAN having the highest performance deviations among runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 7 CONCLUSION AND FUTURE WORK This work proposes to learn a skill by training a generator to generate all feasible actions for a subtask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To this end, the output distribution of the generator is learned to match a uniform distribution over the feasible action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' While learning within a 2D grasping simulation, the method shows stable convergence for FKL, RKL, and JS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' As expected, FKL is more efficient in visiting all the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' JS has the highest accuracy while reliably visiting all the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The proposed learning strategy expands the current state-of-the-art training within multimodal interactive environments by showing competitive accuracy while visiting all the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Since the proposed approach can visit all the modes, it learns the skill of grasping independently of a specific downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In future work, we will investigate how downstream tasks can choose their optimal action among the proposed feasible options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' As it is not dependent on the structure of the problem, we will further utilize it for a 6D grasping problem as well as for other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Some limitations have emerged during experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Currently, many imitation examples are required to bootstrap the critic’s learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' A possibility to mitigate this could be the progressive tuning of the KDEs or learning their parameters during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This approach could favor exploration initially and divergence estimation later in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' A complementary strategy could be using curriculum learning techniques that start with simple problems where solutions are less sparse in the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Furthermore, the proposed approach may not be very effective in high-dimensional problems as the sampling requirements for the estimator grow exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The limit on the degrees of freedom will be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' A mitigation of this issue can come from the rich literature 9 100 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='0 3.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Density estimation by dual ascent of the log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Communications in Mathematical Sciences, 8:217–233, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Martin Wainwright and Michael Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Graphical models, exponential families, and variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Foundations and Trends in Machine Learning, 1:1–305, 01 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1561/ 2200000001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Andy Zeng, Shuran Song, Johnny Lee, Alberto Rodriguez, and Thomas Funkhouser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Tossingbot: Learning to throw arbitrary objects with residual physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' IEEE Transactions on Robotics, 36(4): 1307–1319, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Yinglun Zhu and Paul Mineiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Contextual bandits with smooth regret: Efficient learning in continu- ous action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 27574–27590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Yinglun Zhu, Dylan J Foster, John Langford, and Paul Mineiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Contextual bandits with large action spaces: Made practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 27428–27453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 12 A GRASPING SIMULATION The grasping simulator generates four different shapes (H, 8, Spoon, T) for training and a Box shape for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The shape position, orientation, color, and geometry parameters are randomly sampled, producing various observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The observation space is a 128 × 128 pixel RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We assume a vertical configuration of a parallel gripper with three degrees of freedom x, y, and α and assume that the object is an extrusion of the 2D shape in the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The action space is constrained to the center 78 × 78 pixel region to avoid undefined behavior at the border of the RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The angle of the grasp is in [0, π) as the gripper is symmetrical, and thus a full revolution is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The success of a grasp is only determined by the relative position and alignment of the gripper to the outline of the object, as seen from a camera positioned above the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We developed an algorithm that, given the alignment of the gripper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=', x, y, and α and a simulated picture of the object from a fixed camera, provides a success/failure outcome in a deterministic and reproducible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Given the maximum aperture of the parallel gripper l and the width of the gripper claws w, the simulation analyzes the cropped image content of dimensions l × w between the gripper claws before the claws close on the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The simulation checks if the object is sufficiently present and equidistant from the claws and aligned within parameterized margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Figure 4 shows successful grasping poses and the respective gripper content for the objects that are trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) (g) (h) Figure 4: Feasible gripper positions (red) for different variations of the shapes (H-shape, 8-shape, Spoon, and T-shape) used in training, with a detailed view of the area between the gripper to the right of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' B NEURAL NETWORK ARCHITECTURES .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (128x128x5) (43x43x32) (22x22x64) (11x11x128) Shared MLP (121x(128+d)) Latent Input (d) FC Figure 5: Before processing, the image is augmented with positional encoding resulting in 5 total channels {r, g, b, x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The network’s input layer (in gray) is a 5x5 embedding layer with stride 3, followed by 7 residual blocks (in yellow) with a bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The output is processed by 3 layers of "pixelwise" shared MLPs (in brown), with the features being concatenated with a latent input (in purple) of length d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The latent input is a random sample from Z for the actor and the action to be evaluated for the critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Four (for the actor) or three (for the critic) fully connected layers (in blue) output the action and the score, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The neural network design was guided by simplicity and inspired by GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Features that rely on domain-specific knowledge are avoided to evaluate better the learning method presented in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' 13 The structure of the actor and critic neural networks are illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The residual feature extraction network (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2016)) is shared between the actor and critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' As a peculiarity of the network and the loss, the actor’s inferred action has four components, [x, y, r sin α, r cos α], with r ∈ [0, √ 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The angle can be extracted trivially with the arctan of the ratio of the third and fourth action components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' As the scale factor r does not change the angle, the critic receives the normalized action [x, y, sin α, cos α] as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To avoid the actor from reaching the singularity at r = 0 and the distribution q being spread along the radius, g(s, a) and ξ(s, a) are scaled with an unnormalized Gaussian on the radius, centered at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='5 with the standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' C ALGORITHM AND HYPERPARAMETERS Algorithm 1: Jenson-Shannon training loop 1 Initialize M with imitaition and random examples and initialize θ, φ 2 for 1 to Training Steps do // Training steps are 1, 000, 000 in experiments 3 for 1 to Interaction Steps do // Interaction steps are 1 in experiments 4 s ← Generate a new problem 5 zi ← Sample uniformly in Z, ∀i ∈ [1, U] 6 ai ← πθ(s, zi), ∀i ∈ [1, U] 7 ˆri ← ξφ(s, ai), ∀i ∈ [1, U] 8 j ← arg mini∈[1,U] |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='5 − ˆri| // Get action with highest uncertainty 9 r ← g(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' aj) 10 if r == 1 then 11 Store (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' r) in Mp 12 else 13 Store (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' r) in Mn 14 end 15 end 16 for 1 to Critic Steps do // Critic steps are 2 in experiments 17 (si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ri)L i=1 ← Sample half from Mp and half from Mn 18 φ ← φ − αφ∇φ �L i=1 ri log(ξφ(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ai)) + (1 − ri) log(1 − ξφ(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ai)) 19 end 20 for 1 to Actor Steps do // Actor steps are 1 in experiments 21 for k = 1 to K do 22 sk ← Sample from M 23 zi ← Sample uniformly in Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ∀i ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' N] 24 ai ← πθ(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' zi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ∀i ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' N] 25 ϵj ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' σ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ∀j ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' M] 26 a∗ j ← stop_gradient(a⌈j/m⌉) + ϵj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ∀j ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' M] // Resample from KDE 27 ˆqj ← 1 N �N i=1 kσ(a∗ j − ai),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ∀j ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' M] // Evaluate KDE on samples 28 ˆq′ j ← 1 N �N i=1 kσ′(a∗ j − ai),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ∀j ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' M] // Evaluate proposal pdf 29 ˆrj ← ξφ(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' a∗ j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ∀j ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' M] 30 ˆV ← 1 M �M j=1 ˆrj ˆq′ j // MC integration with importance sampling 31 ˆpj ← ˆrj ˆV ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' ∀j ∈ [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' M] 32 gk ← 1 2M �M j=1 ˆqj ˆq′ j log � 2ˆqj ˆqj+ˆpj � ∇θ log(ˆqj) // gradient trace 33 end 34 θ ← θ − αθ 1 K �K k=1 gk 35 end 36 end 14 Parameter Value Description N 128 Minibatch size M 256 Resampling size m 2 Samples per KDE support point (M/N) U 64 Maximum uncertainty proposals K 16 Actor batch size L 32 Critic batch size σ diag(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='4) KDE bandwidth σ′ diag(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='075, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='075, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='2) Sampling KDE bandwidth |Mp| 160,000 Positive replay memory size |Mn| 160,000 Negative replay memory size |M| 320,000 Total replay memory size Table 4: Hyperparameters D OBSERVATION VARIATION EXPERIMENTS D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1 SETUP Figure 6: Different distortions are applied, showing a colored chess board for illustration and an example shape under all distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To highlight the difference between the proposed approach and related work of robotic grasping, we investigate how distortions of the observation affect the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The distortions investigated are a rotation, projection, and rotation + projection as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' These distortions correspond to different camera perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We train a new agent for 106 training steps for each distortion and approach in the following comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We compare with a common approach in the literature (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020)) that make use of spatial equivariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The approach utilizes fully convolutional networks to output a probability of success for each action of a discretized action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' We implement two variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In the first one, just as in Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' (2020), the observation is fed into the neural network multiple times with different rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The neural network then only needs to output a one-channel image containing the probability of success of each discretized x, y action for the given rotation of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' This approach thus makes use of translation equivariance by using a convolutional neural network (CNN) and rotation equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In the experiments, we denote it as the heat-map approach (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The second variant estimates for each observation the success for different rotations by outputting a multi-channel image indicating the success estimate of each discretized x, y, α action explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Thus it only takes advantage of translation equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' It is called stacked heat-map (SH) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The approaches are implemented using fully convolutional networks with an hourglass structure, adopting the beginning of the Resnet in Figure 5 and adding the same structure in reverse order with nearest-neighbor upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Both approaches predict grasping success for 78x78 pixels with 16 rotation angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' They are trained on a cross-entropy loss on the grasping outcome sampled from the balanced replay buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The replay buffer is also filled with imitation learning examples, and maximum uncertainty sampling is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' For evaluation, the success estimate of each discretized 15 Rotated + Normal Rotated Projected Projectedaction is used as its probability to be sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' To increase accuracy, an inverted temperature factor increases the difference between higher and lower score actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Specifically, the actions are sampled according to q(a|s) = exp(β log ξ(s, a)) � ∀a∈Ad exp(β log ξ(s, a)), (14) with ξ being the fully convolutional network with s as input and as output shape the discretized action space Ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The inverted temperature was set to β = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='2 RESULTS (a) H Shape (b) Box Shape Figure 7: Gripping rank comparison, with the ratio of picks for each ranked mode or failure in %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Normal Rotated Projected Rotated + Projected JS* H SH JS* H SH JS* H SH JS* H SH Score H 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='6 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='7 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='0 96.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='1 Table 5: Comparison of the proposed Jensen-Shannon approach with approaches from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The results are shown in Figure 7 and Table 5, which compare the performance of the proposed Jensen-Shannon (JS) approach with the heat-map (H) and stacked heat-map (HS) approaches on the four different distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' As expected, the specifically crafted H and SH approaches perform better on the original problem than the generic approach proposed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In that scenario, no scene understanding is required, and only local features need to be considered to estimate grasping success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Therefore, the approaches are expected to generalize well to unseen shapes, as seen for the Box-Shape, since the grasping success depends only on gripper alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' They only need to learn to imitate the grasping success heuristic shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Interestingly, rotating the observation does not seem to impact their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' However, under projection and projection + rotation, both approaches fail to learn to grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The heat-map approach 16 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content='3 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Our proposed approach learns well for all distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The generalization on the Box-Shape decreases a bit but the general performance remains similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Surprisingly, the results for rotation + projection on the training shapes are above the variance in the main results in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' An explanation for the performance increase could be that the region in the input observation that the object can be in is smaller than in the normal case, as can be seen by the area between the colored tiles in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' The reduced region could slightly ease scene understanding, leading to improved results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' In general, the performance of our proposed approach does not depend on the distortion as it does not explicitly use the spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' Its design does not depend on the specifics of the experiment at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf'} +page_content=' It can, therefore, learn independently of the distortion applied as long as the object is still fully observable.' 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https://git-lfs.github.com/spec/v1 +oid sha256:f8a318ede41d4deccd88be7e0321648a9854a3a762c6e9d5d881eb610a75910e +size 6488109 diff --git a/TtAyT4oBgHgl3EQfufmM/content/tmp_files/2301.00614v1.pdf.txt b/TtAyT4oBgHgl3EQfufmM/content/tmp_files/2301.00614v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..66cb5e35a4e30594e09d1face984ed3aa32af83d --- /dev/null +++ b/TtAyT4oBgHgl3EQfufmM/content/tmp_files/2301.00614v1.pdf.txt @@ -0,0 +1,870 @@ +A map-based model predictive control approach for train operation +Michael Hauck, Patrick Schmidt, Alexander Kobelski, Stefan Streif +Abstract— Trains are a corner stone of public transport and +play an important role in daily life. A challenging task in train +operation is to avoid skidding and sliding during fast changes of +traction conditions, which can, for example, occur due to chang- +ing weather conditions, crossings, tunnels or forest entries. The +latter depends on local track conditions and can be recorded in +a map together with other location-dependent information like +speed limits and inclination. In this paper, a model predictive +control (MPC) approach is developed. Thanks to the knowledge +of future changes of traction conditions, the approach is able to +avoid short-term skidding and sliding even under fast changes +of traction conditions. In a first step, an optimal reference +trajectory is determined by a multiple-shooting approach. In +a second step, the reference trajectory is tracked by an MPC +setup. The developed method is simulated along a track with +fast-changing traction conditions for different scenarios, like +changing weather conditions and unexpected delays. In all +cases, skidding and sliding is avoided. +I. INTRODUCTION +A key aspect in train operation is slip control, since high +slip results in flattened wheels and track damage. As a result, +it decreases the reliability of train operation [1]. +There exist a variety of approaches for anti-skid and anti- +slip control. The majority of existing methods focus on +re-adhesion control to decrease the slip after the start of +skidding or sliding. For example, a re-adhesion control for +a single-inverter multiple-induction-motors train system is +presented in [2]. In order to suppress undesired slipping, the +motor torque is controlled. In [3], a sliding-mode observer is +developed to estimate the actual traction condition, which is +used as an input for a second-order sliding-mode slip control. +In [4], a re-adhesion slip control based on excessive torque +and excessive angular momentum is presented. However, all +re-adhesion slip control approaches have the disadvantage +that they react after the emerge of undesired slipping and +thus, damage of wheels and track can not be avoided. +The emergence of undesired slipping can be tackled by +active control approaches. In contrast to the re-adhesion or +so-called ”passive” slip controls, active slip controls [1], [5], +[6] try to track the optimal adhesion value in order to avoid +undesired slipping. The adhesion value is determined under +the assumption that it depends only on slip. However, the +influence of weather and track conditions on the adhesion +The authors are with Technische Universit¨at Chemnitz, 09126 Chem- +nitz, Germany, Automatic Control and System Dynamics Lab (e-mail: +{michael.hauck, patrick.schmidt, alexander.kobelski, stefan.streif}@etit.tu- +chemnitz.de) +This research was funded by the German Ministry for Education and +Research (BMBF) in the frame of the SRCC EETCM project, grant number +03WIR1208. +©2023 the authors. This work has been submitted to ECC for publication +under a Creative Commons Licence CC-BY-NC-ND. +is neglected or at least reduced to two possible values, +which may lead to unsafe scenarios [7]. For this reason, +time-varying adhesion dynamics are introduced to tackle the +changing traction conditions between wheel and track. A +spatial iterative learning control is implemented to track the +maximum adhesion. However, the maximum adhesion con- +trols lead to severe impact and reduces the riding comfort [1]. +Another possibility to avoid undesired slipping is to use +model predictive control (MPC) approaches. MPC is a well- +known method to solve optimal control problems. Both, +input and state constraints can be incorporated easily via +this method. In addition, updated data can be included to +handle disturbances. Predictive control algorithms in train- +motion control have been previously explored for various +scenarios. For example, in [8], fuzzy predictive control is +considered and the authors show in simulations that the +performance of train safety, comfort, and other performance +indicators are improved in comparison with conventional +control technology. In [9], a predictive field-oriented control +of the induction motor is developed based on estimation of +the actual traction coefficient and measurement of actual +wheel slip. A predictive speed profile tracking algorithm +under linear constraints and under the assumption that the +traction coefficient is constant is developed in [10]. Fur- +thermore, in [11], an energy-efficient predictive train-motion +control is developed, again, under the assumption of constant +traction conditions. There, an optimal control problem is +solved backwards in time via dynamic programming, which +improves the driving mode by saving up to 40% of the +consumed energy. +To the best of the authors’ knowledge, existing predictive +slip controls assume constant traction conditions over the +prediction horizon. As a result, they fail to avoid skidding +and sliding during fast changes of the traction conditions, +which can, for example, occur during the entry in forests, +tunnels or stations. In comparison to existing approaches in +[10], [11], the current framework considers a variable traction +coefficient, which is stored in a track-condition map together +with topography data. Based on the knowledge of the future +traction, a model predictive control approach is presented in +this paper, which is able to avoid skidding and sliding even +for fast changes in the traction conditions. Recording traction +information and using it for vehicle operation has already +been successfully applied in other scenarios, see [12], [13]. +The rest of the paper is organized as follows: In Section II, +the train-motion model and acceleration constraints are in- +troduced. Additionally, the track-condition-and-topography +map is presented as well as the optimal control problem. +In Section III, a two-step solution approach is developed +arXiv:2301.00614v1 [eess.SY] 2 Jan 2023 + +to solve the given problem. Section IV provides a case- +study along with a discussion, where different situations like +change of weather and unexpected delays are considered. +Finally, the paper is concluded in Section V, presenting an +outlook on further research topics. +Notation: The set of component-wise non-negative vectors +in Rn is defined as +Rn ++ := {z = +�z1 +. . . +zn +�⊤ ∈ Rn : zi ≥ 0, ∀i = 1, . . . , n} +for n ∈ N. The set of piece-wise continuous functions +mapping [0, T] to R+ is denoted by PC0([0, T], R+), sim- +ilar for Ck([0, T], R+) as the set of k times continuously +differentiable functions, where k ≥ 0. +II. PROBLEM SETTING +In this section, a control-oriented model of the train +dynamics and restrictions of the train motion are intro- +duced. Afterwards, the concept of the track-condition-and- +topography map, its content and how it is obtained is +explained. In the end, the optimal control problem is stated. +A. Train dynamics +The train motion can be determined by a balance of force +equation [14]: +ma = −Fr(v) + Ftrain(v)u, +(1) +where m is the train mass, a is the acceleration, v is the +velocity, and u is the driving-lever position of the train. By +convention, the latter is scaled in [−1, 1]; a positive value +results in acceleration, a negative one in braking. +The resistance force in dependence of velocity v is given +as [15] +Fr(v) = 1/2ϱaircairAtrain(v−vW)2+mg sin(α)+cRmg (2) +and includes air resistance, inclination resistance, and rolling- +resistance with ϱair as air resistance coefficient, cair as air +density, Atrain as effective area for train air resistance, vW as +wind speed in driving direction, g as gravitational constant, +α as inclination, and cR as rolling resistance coefficient. +The driving power Ftrain(v) is a train-specific function, +depending on the maximum train power and an internal +limitation for small velocities [16]. In [17] and [18], the +driving power is shown for different trains and it can be +approximated by Ftrain(v) = k1e−k2v + k3, where the +coefficients ki > 0 can be determined by a least-squares +approach for a specific train. +Defining x = +�p +v�⊤, the train-motion model can be +written as the following input-affine state-space model: +˙x = +� +x2 +− Fr(x2) +m +� +� +�� +� +=:f(x;m) ++ +� +0 +k1ek2x2+k3 +m +� +� +�� +� +=:g(x;m) +u. +(3) +By fi(x; m), the i-th component of f(x; m) is denoted. Train +motion further depends on local track conditions and the +topography of the track, which are summarized in a track- +condition-and-topography map. Such a map is described in +the next section. +0 +0.3 +0.6 +0.9 +1.2 +·104 +0 +10 +20 +30 +40 +p [m] +vmax [m/s] +−2 +−1 +0 +1 +2 +α [◦] +0 +0.3 +0.6 +0.9 +1.2 +·104 +0 +0.1 +0.2 +p [m] +µ(p), good conditions +µ(p), bad conditions +Pollution +Station +Crossing +Fig. 1: Map used in case study in Section IV. Velocity limits, +inclination and time table are based on train ride from RB30 +Chemnitz to Cranzahl, Germany. Traction trajectories were +build based on measurements from [13]. +B. Track-condition-and-topography map +Usually, trains travel along the same routes with location- +dependent speed limits, inclination, weather-dependent trac- +tion parameters etc. If skidding or sliding occurs due to +location-dependent reasons at a certain point, it can be as- +sumed that this will happen again on the next trip. By record- +ing such information in a track-condition-and-topography +map, subsequent train rides can exploit this information +to improve their performance, e.g. avoiding high slip by +preemptively throttling down the engine torque. For ease of +notation, the track-condition-and-topography map is denoted +as map from now on. +In this paper, we assume a standard passenger train on +a commercially operated line with multiple stops. The train +must follow a timetable and adhere to speed limits. The track +follows the topography of the terrain, so the train ride is af- +fected by the inclination. Furthermore, the traction conditions +of the rails are subject to environmental influences. +Since trains move very fast, the resolution does not need to +be higher than one meter. For this work we assume a setup, +where the map contains: +• inclination at position p: α(p), +• maximal velocity limit at position p: vmax(p), +• two traction trajectories for maximal traction at position +p: µj(p) +j ∈ {good, bad} (based on environmental +conditions), and +• time table including arrival time tstationi at station +pstationi +Two different µ-trajectories will be used in the case study, to +show performance on good as well as bad track conditions. +Based on this information, a map is created, see Fig. 1. Bad +conditions may refer to track conditions in freezing weather +or as soon as rain begins to fall, when water mixes with +dust to a muddy film on the tracks. Note, that for future +references of µ, the maximum value for the given track +conditions (’good’ or ’bad’) is denoted by µmax. + +C. Constraints +Train motion is restricted due to physical limitations +(traction, engine, and brakes) as well as safety of passengers. +The acceleration is limited by the following restrictions: +(i) safety restrictions for the passengers during braking and +acceleration +|f2(x; m) + g2(x; m)u| ≤ amax, +(4) +with amax > 0 and (ii) the maximum traction at the current +position p +|g2(x; m)u| ≤ µmax(x1)g. +(5) +The maximum traction µmax depends on the position and +is defined in the next section. Furthermore, acceleration is +restricted by the maximum engine power and braking force. +Both are already included in the model description with +the assumption, that a driving lever position u = 1 yields +maximum engine power and u = −1 maximum breaking +force. +The total mass of the train m is the mass of its own weight +mtrain plus the load given by passengers and luggage, where +the maximal additional load is defined as mmaxload. Thus, +the following inequality holds for the mass of the train: +mmin := mtrain ≤ m ≤ mtrain + mmaxload =: mmax. (6) +The mass of the train varies due to a changing number of +passengers and luggage. It is constant between two stations, +but unknown in most cases. Therefore, the constraints (4) and +(5) have to be satisfied for all m ∈ [mmin, mmax]. Depending +on whether the train is braking or accelerating, either the +minimum mass or the maximum mass yields the highest +value in (4) and (5). Therefore, these masses are inserted +into the respective inequalities. Inequality constraints on +both, states and control are incorporated via the function +h : R2 × [−1, 1] → R7 with +h(x, u) = +� +� +� +� +� +� +� +� +� +� +f2(x; mmin) + g2(x; mmin)u − amax +−(f2(x; mmax) + g2(x; mmax)u) − amax +g2(x; mmin)u − µmax(x1)g +−g2(x; mmax)u − µmax(x1)g +x2 − vmax(x1) +−1 − u +u − 1 +� +� +� +� +� +� +� +� +� +� +. +(7) +D. Optimal control problem +Along with train-motion dynamics (1), an optimal control +problem is formulated. The solution to the optimal control +problem yields the desired lever position as a control input +for the model and is defined as: +u⋆ = arg min +u∈U +1 +2 +� tstation +t0 +u2dt +s.t. +˙x = f(x; m) + g(x; m)u, +x(t0) = x0, x(tstation) ∈ Xstation, +h(x, u) ≤ 0. +(8) +The objective function is chosen to minimize the norm of +the input within the time interval [t0, tstation], where tstation +is the desired arrival time at the station provided by the map. +As a solution to the optimal control problem, a continuous +function u⋆ ∈ U := C0([t0, tstation], [−1, 1]) is obtained. It +is assumed that a feasible solution exists for all station with +initial condition x0 ∈ {p0} × [0, vmax(p0)], where vmax(p0) +is the speed limit at p0 ∈ R+. As a terminal condition at +t = tstation, the train has to stop at the station given at +pstation. This terminal condition is replaced by a relaxed one, +which is given by Xstation increasing the amount of feasible +solutions. The tolerance for the terminal position is given as +ε1 ≥ 0. Therefore, the terminal set is given as Xstation := +[pstation − ε1, pstation + ε1] × {0}. +III. METHODS +Since the computation of u⋆ in (8) is associated with +high computing time, the optimal control problem has to +be adjusted in order to decrease the computation burden +in view of applicability. To this end, a two-step approach +is considered. In the first step, u⋆ as defined in (8) is +computed to obtain a reference trajectory uc +ref := u⋆ for +all t ∈ [t0, tstation]. Then, u⋆ is applied to system (1) to +obtain the reference trajectory xc +ref ∈ C1([t0, tstation), R2 ++). +In a second step, these trajectories are tracked online via +MPC. +A. Computation of reference trajectories +The reference trajectory for the input u is defined via (8) +and it is computed with a multiple-shooting method, which +is a powerful solution approach to tackle boundary value +problems [19]. In multiple-shooting methods, the interval +[t0, tstation] is divided into mℓ − 1 subintervals denoted as +[ti, ti+1], i = 1, 2, . . . , mℓ − 1, where mℓ is given as the +number of nodes, t1 := t0, and tmℓ := tstation. On each +of the subintervals, the given problem is solved for initial +values x(ti). In further steps, the initial values are adjusted +such that the states at the end of [ti, ti+1] coincide with +the initial value of the next subinterval, i.e. x(ti+1). At +the end, the boundary condition at the right-hand side of +[ti, ti+1] and the initial condition of [ti+1, ti+2] are the same +for i = 1, 2, . . . , mℓ − 2, which yields the desired solution +uc +ref ∈ PC0 ([t0, tstation), [−1, 1]). +Once uc +ref is computed, it is applied to system (1) to obtain +the reference trajectory xc +ref ∈ C1([t0, tstation), R2 ++), which +are tracked in the second step of the procedure, namely the +model predictive control. Since these trajectories provide a +feasible solution, the prediction horizon can be reduced in +order to decrease the computation time, since minimizing the +distance to the trajectory ensures the satisfaction of the ter- +minal condition. To decrease the computational burden of the +tracking MPC, the system dynamics (3) are first linearized +at the reference trajectories xc +ref and uc +ref. Afterwards, the +linearized model is discretized for the implementation. +B. Linearization and discretization +Using Taylor approximation, the linearized model with +states xlin := x − xc +ref and control ulin := u − uc +ref reads + +as +˙xlin = +� +0 +1 +a21(t) +a22(t) +� +� +�� +� +=Alin(t) +xlin + +� +0 +b2(t) +� +� +�� +� +=blin(t) +ulin. +(9) +The remaining entries are calculated as +a21(t) = ∂f2(x; m) +∂α +∂α +∂x1 += −g cos(αref(t)) dα +dx1 +with αref(t) as inclination at xc +ref1(t) and +∂α +∂x1 given from +the map, as well as +a22(t) = − 1 +mϱaircairAtrain(xc +ref2(t) − vW) +− 1 +mk1k2e−k2xc +ref2(t)uc +ref(t) +and +b2(t) = k1e−k2xc +ref2(t) + k3 +m +. +After linearizing the train-motion dynamics (1), the ob- +tained model (9) is discretized. The sampled states are +defined as +xd +refi = +�xc +refi(tstep) +xc +refi(2tstep) +xc +refi(3tstep) +. . .�⊤ +for i = 1, 2, where ud +ref is defined analogously. +The resulting discrete-time model is given by +xd(k + 1) = Ad(k)xd(k) + bd(k)ud(k), +(10) +where +Ad(k) = eAlin(ktstep)·tstep +and +bd(k) = +� tstep +0 +eAlin(ktstep)·(tstep−τ)blin(ktstep) dτ. +C. Tracking MPC +The linearized and discretized model is used in the optimal +control problem along with the reference trajectories from +Section III-B. An MPC setup with moving horizon N is +considered to track xd +ref1, which results in minimizing the +difference between the current position at t = ktstep and +xd +ref1(k). This difference is defined as xd +1(k) in (10). In order +to increase the amount of feasible solutions, the terminal +constraint in (8) is replaced by terminal costs. Therefore, the +cost function of the optimal control problem (8) is enlarged +by terminal costs to obtain the resulting tracking MPC as +min +U +1 +2U ⊤RU + 1 +2X⊤ +1 QX1 +s.t. +xd(k + 1) = Ad(k)xd(k) + bd(k)ud(k), xd(0) = x0, +h(xd(k), ud(k)) ≤ 0 +∀k = j, . . . , j + N − 1. +(11) +In the objective function, U, X1 +∈ +RN +are weighted +via +R, Q +∈ +RN×N, R, Q +≻ +0. +At +time +instant +j, +X1 += +� +xd +1(j) +. . . +xd +1(j + N − 1) +�⊤ +includes +xd +ref1(j), . . . , xd +ref1(j + N − 1) +as +reference +trajectory +values. +Analogously, +the +input +vector +is +defined +as +U = +� +ud(j) +. . . +ud(j + N − 1) +�⊤. +The solution of (11) is denoted as U ⋆. Once the optimal +control sequence is obtained, only the first element of U ⋆ is +applied to the system (1), which yields a new initial value +for the next iteration. Afterwards, the horizon is shifted and, +in particular, the reference values in X1 are updated. This +iterative scheme is applied until the arrival station is reached. +Note that if the remaining prediction length is smaller than +N, then N is reduced to the remaining length. +IV. IMPLEMENTATION AND CASE-STUDY +In this section, the proposed multiple-shooting method +and the tracking MPC are implemented and simulated for +the RB30, traveling from Chemnitz to Cranzahl, Germany. +Physical parameters of the train, as well as the used tuning +parameters of the multiple-shooting approach and the track- +ing MPC are listed. With these parameters, three different +scenarios (e.g. behavior for good traction conditions, weather +change, and delays) are simulated and analyzed. +A. Parameters +For the simulations presented in this section, the train- +specific model parameters and physical constants shown in +Table I are used. They result from train data sheets, whereas +ki are determined by a least-squares approach. +TABLE I: Train-specific model parameter. +Parameter +Value +ϱair +1.2041 kg/m3 +Atrain +10 m2 +cair +0.85 +cR +0.002 +k1 +1.516 × 105 kgm/s2 +k2 +0.1147 1/s +k3 +1.564 × 104 kgm/s2 +mtrain +68 200 kg +mmaxload +20 500 kg +The train-track-specific parameters like the speed limit +vmax(x1), the inclination α(x1), and the maximal traction +µmax(x1) are listed in the map (see Section II and Fig. 1). +In the simulations, the wind speed in driving direction was +assumed by vw = 0. +For the implementation of the multiple-shooting method, +the number of subintervals mℓ is determined depending on +tstation, such that the time between two subintervals is less +than three seconds, i.e. N = ⌊tstation/3⌋. In the simulations +it turned out to be a good trade-off between accuracy and +computational effort. However, the number of subintervals +can be increased, since the computation is done offline. For +the implementation of the MPC setup, the discretization time +tstep = 1 second and the horizon length N = 20 are used. +The mass of the train m = 78200kg is determined as the sum +of the empty train mass and an average passenger weight +value. The weighting matrices are chosen as R = 0.01IN +and Q = IN, where IN is the N × N identity matrix. These +values are tuned manually, such that tracking of optimal +states x and the optimal input u is fulfilled in all considered +simulations. + +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +·104 +0 +10 +20 +30 +x1 [m] +x2 [m/s] +vmax +xref2 +xMPC2 +Fig. 2: Comparison of reference velocity according to +multiple-shooting approach and the velocity determined by +tracking MPC for the first three railway sections. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +·104 +0 +1 +2 +x1 [m] +|x2 − xref2| [m/s] +Fig. 3: Resulting tracking error between the reference tra- +jectory xref2 and the calculated velocity x2 according to +tracking MPC for good traction conditions. +B. Simulation results +In a first scenario, the behavior of the train is simulated +and analyzed for good traction conditions. Afterwards, a +change in the traction conditions due to fallen leaves or +frozen railway resulting in a different µmax(x1) is analyzed. +In a third simulation, the effects of an unplanned delay are +investigated. +In Fig. 2, the reference velocity according to multiple- +shooting approach as well as the resulting velocity deter- +mined by tracking MPC is shown. The difference between +them is shown in Fig. 3. The resulting tracking error between +the reference trajectory xref1 and the train position x1 +according to tracking MPC is shown in Fig. 4. The tracking +MPC approach is able to track the reference with a maximal +error of around 5 m. +Furthermore, the satisfaction of the constraints (7) is +considered. The first two elements h1(x, u) and h2(x, u) de- +scribe the safety constraints, whereas h3(x, u) and h4(x, u) +describe maximum traction. In Fig. 5, the maximum of +the safety constraints as well as the traction constraints are +visualized. It can be seen that the safety constraints are more +restrictive than traction constraints. However, all constraints +are satisfied since all hi(x, u) ≤ 0 along the whole track. +In a next step, the influence of changing traction conditions +between two stations is analyzed. There exist two possible +cases, namely that the new traction conditions are better +or worse than the assumed ones. The first case is not +problematic, since the bad weather reference trajectory can +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +·104 +0 +2 +4 +x1 [m] +|x1 − xref1| [m] +Fig. 4: Resulting tracking error between the reference tra- +jectory xref1 and the calculated position x1 according to +tracking MPC for good traction conditions. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +·104 +−2 +−1 +0 +x1 [m] +Value of constraints +max{h1, h2} +max{h3, h4} +Fig. 5: Safety and traction constraints of tracking MPC with +good traction conditions. +be easily tracked with better traction conditions. In the other +case, a decrease of the maximal traction yields a decrease of +maximal acceleration and braking. +For this scenario, the reference trajectories for good trac- +tion conditions are tracked via MPC. However, due to an +unexpected change of the weather conditions, the traction +is changing to bad conditions immediately after leaving the +station (cf. traction trajectories in the map in Fig. 1). The +tracking error yields no significant change. More interesting +is, that the traction constraints are more restrictive at the +bigger part of the track, which can be seen in Fig. 6. +However, the constraints are satisfied along the whole track. +Therefore, the developed approach is able to avoid skidding +and sliding during changes of weather conditions. +In a third scenario, the effects of an unexpected delay +due to longer passenger changing time are analyzed. An +initial delay of tdelay = 40 seconds in the first station was +simulated under good traction conditions. With this delay, +the optimal control problem (8) is infeasible, since the train +can not reach the first station in the given time. However, the +reference trajectories are calculated offline independently of +the occurred delay, thus, they are feasible for the initial time +table. Since terminal constraints are replaced by terminal +costs in the MPC approach, the OCP (11) obtains a feasible +solution. As shown in Fig. 7, the tracking error caused by the +delay at position x1(tdelay) = 0 is around 280 meters. The +train arrives at the next station with a delay of 19 seconds, +which is smaller than the initial one. Due to waiting time in + +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +·104 +−1 +0 +x1 [m] +Value of constraints +max{h1, h2} +max{h3, h4} +Fig. 6: Safety and traction constraints of tracking MPC with +bad traction conditions. +200 +400 +600 +800 +0 +200 +400 +600 +800 +t [s] +|x1 − xref1| [m] +tdelay +Fig. 7: Tracking error with an initial delay of 40 seconds. +the station, the tracking error is zero for a short moment until +the desired departure. Until the following station, the tracking +error decreases to zero and the MPC tracks the reference +from this point on. Again, all constraints are satisfied at each +point. Therefore, especially skidding and sliding is prevented. +V. CONCLUSION AND OUTLOOK +This paper proposes a model predictive control approach +for train operation. The presented approach is able to avoid +skidding and sliding during fast changes of the traction +conditions, by using a map with traction conditions and +topography data. In order to reduce the computational effort +in online operation, the optimal control problem is solved +offline to generate reference trajectories for the position, +velocity and the lever position. Afterwards, a tracking MPC +setup follows the reference trajectories. +In a case-study, it turned out that the MPC tracks the +reference trajectory with a small tracking error while sat- +isfying all constraints. Furthermore, even with changing +weather conditions, MPC yields a feasible solution, where +traction conditions became more restrictive in this case. +Additionally, the developed approach was able to handle +unexpected delays. +In future works, a more detailed model can be considered +including, for example, motor characteristics and different +braking functions. Another future idea is an online adaption +of the map based on actual measurements. Furthermore, +information that are available shortly before they have to +be considered are important extensions. Those ones include, +e.g., blocked sections by other trains or environmental influ- +ences. +REFERENCES +[1] W.-C. Cai, D.-Y. Li, and Y.-D. Song, “A novel approach for active +adhesion control of high-speed trains under antiskid constraints,” IEEE +Transactions on Intelligent Transportation Systems, vol. 16, no. 6, pp. +3213–3222, 2015. +[2] Y. Matsumoto, N. Eguchi, and A. Kawamura, “Novel re-adhesion con- +trol for train traction system of the ”Shinkansen” with the estimation of +wheel-to-rail adhesive force,” in IECON’01. 27th Annual Conference +of the IEEE Industrial Electronics Society (Cat. No. 37243), vol. 2. +IEEE, 2001, pp. 1207–1212. +[3] M. Amodeo, A. Ferrara, R. Terzaghi, and C. Vecchio, “Wheel slip +control via second-order sliding-mode generation,” IEEE Transactions +on Intelligent Transportation Systems, vol. 11, no. 1, pp. 122–131, +2009. +[4] T. Hara and T. Koseki, “Study on re-adhesion control by monitoring +excessive angular momentum in electric railway tractions,” in 2012 +12th IEEE International Workshop on Advanced Motion Control +(AMC). +IEEE, 2012, pp. 1–6. +[5] J.-S. Hu, D. Yin, Y. Hori, and F.-R. Hu, “Electric vehicle traction con- +trol: a new mtte methodology,” IEEE Industry Applications Magazine, +vol. 18, no. 2, pp. 23–31, 2011. +[6] W. Liao, H. Chen, W. Cai, and Y. Song, “A novel active adhesion con- +trol design for high speed trains without vehicle speed measurement,” +in Proceedings of the 33rd Chinese Control Conference. IEEE, 2014, +pp. 221–226. +[7] D. Huang, W. Yang, T. Huang, N. Qin, Y. Chen, and Y. Tan, +“Iterative learning operation control of high-speed trains with adhesion +dynamics,” IEEE Transactions on Control Systems Technology, vol. 29, +no. 6, pp. 2598–2608, 2021. +[8] Y. Cao, L. Ma, and Y. Zhang, “Application of fuzzy predictive control +technology in automatic train operation,” Cluster Computing, vol. 22, +no. 6, pp. 14 135–14 144, 2019. +[9] S. Sadr, D. A. Khaburi, and J. Rodr´ıguez, “Predictive slip control for +electrical trains,” IEEE Transactions on Industrial Electronics, vol. 63, +no. 6, pp. 3446–3457, 2016. +[10] A. Molavi and F. Rashidi Fathabadi, “Robust model predictive anti- +slip controller and speed profile tracking of an electric train based on +lmi approach,” International Journal of Dynamics and Control, pp. +1–12, 2022. +[11] H. Novak, V. Leˇsi´c, and M. Vaˇsak, “Energy-efficient model predictive +train traction control with incorporated traction system efficiency,” +IEEE Transactions on Intelligent Transportation Systems, vol. 23, +no. 6, pp. 5044–5055, 2021. +[12] A. Kobelski, P. Osinenko, and S. Streif, “A method of online traction +parameter identification and mapping,” IFAC-PapersOnLine, vol. 53, +no. 2, pp. 13 933–13 938, 2020. +[13] K. Nagase, “A study of adhesion between the rails and running wheels +on main lines: results of investigations by slipping adhesion test +bogie,” Proceedings of the Institution of Mechanical Engineers, Part +F: Journal of Rail and Rapid Transit, vol. 203, no. 1, pp. 33–43, 1989. +[14] X. Yao, J. H. Park, H. Dong, L. Guo, and X. Lin, “Robust adaptive +nonsingular terminal sliding mode control for automatic train opera- +tion,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, +vol. 49, no. 12, pp. 2406–2415, 2018. +[15] D. Wende, Fahrdynamik des Schienenverkehrs. +Springer-Verlag, +2013. +[16] J. Ihme, Schienenfahrzeugtechnik. +Springer, 2016. +[17] A. Steimel, Electric traction: motive power and energy supply. +Di- +vision Deutscher Industrieverlag, 2014. +[18] J. Golling. (2020) Bahntechnik & bahnbetrieb. [Online]. Available: +https://www.bahntechnik-bahnbetrieb.de/beschleunigungsrechner/ +[19] R. Bulirsch, J. Stoer, and J. Stoer, Introduction to numerical analysis. +Springer, 2002, vol. 3. + diff --git a/TtAyT4oBgHgl3EQfufmM/content/tmp_files/load_file.txt b/TtAyT4oBgHgl3EQfufmM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08ae815d8c6673d0f28ee961d09439c2bd6dbec8 --- /dev/null +++ b/TtAyT4oBgHgl3EQfufmM/content/tmp_files/load_file.txt @@ -0,0 +1,471 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf,len=470 +page_content='A map-based model predictive control approach for train operation Michael Hauck, Patrick Schmidt, Alexander Kobelski, Stefan Streif Abstract— Trains are a corner stone of public transport and play an important role in daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' A challenging task in train operation is to avoid skidding and sliding during fast changes of traction conditions, which can, for example, occur due to chang- ing weather conditions, crossings, tunnels or forest entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The latter depends on local track conditions and can be recorded in a map together with other location-dependent information like speed limits and inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In this paper, a model predictive control (MPC) approach is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Thanks to the knowledge of future changes of traction conditions, the approach is able to avoid short-term skidding and sliding even under fast changes of traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In a first step, an optimal reference trajectory is determined by a multiple-shooting approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In a second step, the reference trajectory is tracked by an MPC setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The developed method is simulated along a track with fast-changing traction conditions for different scenarios, like changing weather conditions and unexpected delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In all cases, skidding and sliding is avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' INTRODUCTION A key aspect in train operation is slip control, since high slip results in flattened wheels and track damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' As a result, it decreases the reliability of train operation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' There exist a variety of approaches for anti-skid and anti- slip control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The majority of existing methods focus on re-adhesion control to decrease the slip after the start of skidding or sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' For example, a re-adhesion control for a single-inverter multiple-induction-motors train system is presented in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In order to suppress undesired slipping, the motor torque is controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In [3], a sliding-mode observer is developed to estimate the actual traction condition, which is used as an input for a second-order sliding-mode slip control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In [4], a re-adhesion slip control based on excessive torque and excessive angular momentum is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' However, all re-adhesion slip control approaches have the disadvantage that they react after the emerge of undesired slipping and thus, damage of wheels and track can not be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The emergence of undesired slipping can be tackled by active control approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In contrast to the re-adhesion or so-called ”passive” slip controls, active slip controls [1], [5], [6] try to track the optimal adhesion value in order to avoid undesired slipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The adhesion value is determined under the assumption that it depends only on slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' However, the influence of weather and track conditions on the adhesion The authors are with Technische Universit¨at Chemnitz, 09126 Chem- nitz, Germany, Automatic Control and System Dynamics Lab (e-mail: {michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='hauck, patrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='schmidt, alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='kobelski, stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='streif}@etit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='tu- chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='de) This research was funded by the German Ministry for Education and Research (BMBF) in the frame of the SRCC EETCM project, grant number 03WIR1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' ©2023 the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' This work has been submitted to ECC for publication under a Creative Commons Licence CC-BY-NC-ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' is neglected or at least reduced to two possible values, which may lead to unsafe scenarios [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' For this reason, time-varying adhesion dynamics are introduced to tackle the changing traction conditions between wheel and track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' A spatial iterative learning control is implemented to track the maximum adhesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' However, the maximum adhesion con- trols lead to severe impact and reduces the riding comfort [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Another possibility to avoid undesired slipping is to use model predictive control (MPC) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' MPC is a well- known method to solve optimal control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Both, input and state constraints can be incorporated easily via this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In addition, updated data can be included to handle disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Predictive control algorithms in train- motion control have been previously explored for various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' For example, in [8], fuzzy predictive control is considered and the authors show in simulations that the performance of train safety, comfort, and other performance indicators are improved in comparison with conventional control technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In [9], a predictive field-oriented control of the induction motor is developed based on estimation of the actual traction coefficient and measurement of actual wheel slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' A predictive speed profile tracking algorithm under linear constraints and under the assumption that the traction coefficient is constant is developed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Fur- thermore, in [11], an energy-efficient predictive train-motion control is developed, again, under the assumption of constant traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' There, an optimal control problem is solved backwards in time via dynamic programming, which improves the driving mode by saving up to 40% of the consumed energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' To the best of the authors’ knowledge, existing predictive slip controls assume constant traction conditions over the prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' As a result, they fail to avoid skidding and sliding during fast changes of the traction conditions, which can, for example, occur during the entry in forests, tunnels or stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In comparison to existing approaches in [10], [11], the current framework considers a variable traction coefficient, which is stored in a track-condition map together with topography data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Based on the knowledge of the future traction, a model predictive control approach is presented in this paper, which is able to avoid skidding and sliding even for fast changes in the traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Recording traction information and using it for vehicle operation has already been successfully applied in other scenarios, see [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The rest of the paper is organized as follows: In Section II, the train-motion model and acceleration constraints are in- troduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Additionally, the track-condition-and-topography map is presented as well as the optimal control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In Section III, a two-step solution approach is developed arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='00614v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='SY] 2 Jan 2023 to solve the given problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Section IV provides a case- study along with a discussion, where different situations like change of weather and unexpected delays are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Finally, the paper is concluded in Section V, presenting an outlook on further research topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Notation: The set of component-wise non-negative vectors in Rn is defined as Rn + := {z = �z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' zn �⊤ ∈ Rn : zi ≥ 0, ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' , n} for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The set of piece-wise continuous functions mapping [0, T] to R+ is denoted by PC0([0, T], R+), sim- ilar for Ck([0, T], R+) as the set of k times continuously differentiable functions, where k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' PROBLEM SETTING In this section, a control-oriented model of the train dynamics and restrictions of the train motion are intro- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Afterwards, the concept of the track-condition-and- topography map, its content and how it is obtained is explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In the end, the optimal control problem is stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Train dynamics The train motion can be determined by a balance of force equation [14]: ma = −Fr(v) + Ftrain(v)u, (1) where m is the train mass, a is the acceleration, v is the velocity, and u is the driving-lever position of the train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' By convention, the latter is scaled in [−1, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' a positive value results in acceleration, a negative one in braking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The resistance force in dependence of velocity v is given as [15] Fr(v) = 1/2ϱaircairAtrain(v−vW)2+mg sin(α)+cRmg (2) and includes air resistance, inclination resistance, and rolling- resistance with ϱair as air resistance coefficient, cair as air density, Atrain as effective area for train air resistance, vW as wind speed in driving direction, g as gravitational constant, α as inclination, and cR as rolling resistance coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The driving power Ftrain(v) is a train-specific function, depending on the maximum train power and an internal limitation for small velocities [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In [17] and [18], the driving power is shown for different trains and it can be approximated by Ftrain(v) = k1e−k2v + k3, where the coefficients ki > 0 can be determined by a least-squares approach for a specific train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Defining x = �p v�⊤, the train-motion model can be written as the following input-affine state-space model: ˙x = � x2 − Fr(x2) m � � �� � =:f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='m) + � 0 k1ek2x2+k3 m � � �� � =:g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='m) u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' (3) By fi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' m), the i-th component of f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' m) is denoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Train motion further depends on local track conditions and the topography of the track, which are summarized in a track- condition-and-topography map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Such a map is described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 104 0 10 20 30 40 p [m] vmax [m/s] −2 −1 0 1 2 α [◦] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 104 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 p [m] µ(p), good conditions µ(p), bad conditions Pollution Station Crossing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 1: Map used in case study in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Velocity limits, inclination and time table are based on train ride from RB30 Chemnitz to Cranzahl, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Traction trajectories were build based on measurements from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Track-condition-and-topography map Usually, trains travel along the same routes with location- dependent speed limits, inclination, weather-dependent trac- tion parameters etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' If skidding or sliding occurs due to location-dependent reasons at a certain point, it can be as- sumed that this will happen again on the next trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' By record- ing such information in a track-condition-and-topography map, subsequent train rides can exploit this information to improve their performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' avoiding high slip by preemptively throttling down the engine torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' For ease of notation, the track-condition-and-topography map is denoted as map from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In this paper, we assume a standard passenger train on a commercially operated line with multiple stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The train must follow a timetable and adhere to speed limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The track follows the topography of the terrain, so the train ride is af- fected by the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Furthermore, the traction conditions of the rails are subject to environmental influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Since trains move very fast, the resolution does not need to be higher than one meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' For this work we assume a setup, where the map contains: inclination at position p: α(p), maximal velocity limit at position p: vmax(p), two traction trajectories for maximal traction at position p: µj(p) j ∈ {good, bad} (based on environmental conditions), and time table including arrival time tstationi at station pstationi Two different µ-trajectories will be used in the case study, to show performance on good as well as bad track conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Based on this information, a map is created, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Bad conditions may refer to track conditions in freezing weather or as soon as rain begins to fall, when water mixes with dust to a muddy film on the tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Note, that for future references of µ, the maximum value for the given track conditions (’good’ or ’bad’) is denoted by µmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Constraints Train motion is restricted due to physical limitations (traction, engine, and brakes) as well as safety of passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The acceleration is limited by the following restrictions: (i) safety restrictions for the passengers during braking and acceleration |f2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' m) + g2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' m)u| ≤ amax, (4) with amax > 0 and (ii) the maximum traction at the current position p |g2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' m)u| ≤ µmax(x1)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' (5) The maximum traction µmax depends on the position and is defined in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Furthermore, acceleration is restricted by the maximum engine power and braking force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Both are already included in the model description with the assumption, that a driving lever position u = 1 yields maximum engine power and u = −1 maximum breaking force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The total mass of the train m is the mass of its own weight mtrain plus the load given by passengers and luggage, where the maximal additional load is defined as mmaxload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Thus, the following inequality holds for the mass of the train: mmin := mtrain ≤ m ≤ mtrain + mmaxload =: mmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' (6) The mass of the train varies due to a changing number of passengers and luggage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' It is constant between two stations, but unknown in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Therefore, the constraints (4) and (5) have to be satisfied for all m ∈ [mmin, mmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Depending on whether the train is braking or accelerating, either the minimum mass or the maximum mass yields the highest value in (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Therefore, these masses are inserted into the respective inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Inequality constraints on both, states and control are incorporated via the function h : R2 × [−1, 1] → R7 with h(x, u) = � � � � � � � � � � f2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' mmin) + g2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' mmin)u − amax −(f2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' mmax) + g2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' mmax)u) − amax g2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' mmin)u − µmax(x1)g −g2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' mmax)u − µmax(x1)g x2 − vmax(x1) −1 − u u − 1 � � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' (7) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Optimal control problem Along with train-motion dynamics (1), an optimal control problem is formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The solution to the optimal control problem yields the desired lever position as a control input for the model and is defined as: u⋆ = arg min u∈U 1 2 � tstation t0 u2dt s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' ˙x = f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' m) + g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' m)u, x(t0) = x0, x(tstation) ∈ Xstation, h(x, u) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' (8) The objective function is chosen to minimize the norm of the input within the time interval [t0, tstation], where tstation is the desired arrival time at the station provided by the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' As a solution to the optimal control problem, a continuous function u⋆ ∈ U := C0([t0, tstation], [−1, 1]) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' It is assumed that a feasible solution exists for all station with initial condition x0 ∈ {p0} × [0, vmax(p0)], where vmax(p0) is the speed limit at p0 ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' As a terminal condition at t = tstation, the train has to stop at the station given at pstation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' This terminal condition is replaced by a relaxed one, which is given by Xstation increasing the amount of feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The tolerance for the terminal position is given as ε1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Therefore, the terminal set is given as Xstation := [pstation − ε1, pstation + ε1] × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' METHODS Since the computation of u⋆ in (8) is associated with high computing time, the optimal control problem has to be adjusted in order to decrease the computation burden in view of applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' To this end, a two-step approach is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In the first step, u⋆ as defined in (8) is computed to obtain a reference trajectory uc ref := u⋆ for all t ∈ [t0, tstation].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Then, u⋆ is applied to system (1) to obtain the reference trajectory xc ref ∈ C1([t0, tstation), R2 +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In a second step, these trajectories are tracked online via MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Computation of reference trajectories The reference trajectory for the input u is defined via (8) and it is computed with a multiple-shooting method, which is a powerful solution approach to tackle boundary value problems [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In multiple-shooting methods, the interval [t0, tstation] is divided into mℓ − 1 subintervals denoted as [ti, ti+1], i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' , mℓ − 1, where mℓ is given as the number of nodes, t1 := t0, and tmℓ := tstation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' On each of the subintervals, the given problem is solved for initial values x(ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In further steps, the initial values are adjusted such that the states at the end of [ti, ti+1] coincide with the initial value of the next subinterval, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' x(ti+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' At the end, the boundary condition at the right-hand side of [ti, ti+1] and the initial condition of [ti+1, ti+2] are the same for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' , mℓ − 2, which yields the desired solution uc ref ∈ PC0 ([t0, tstation), [−1, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Once uc ref is computed, it is applied to system (1) to obtain the reference trajectory xc ref ∈ C1([t0, tstation), R2 +), which are tracked in the second step of the procedure, namely the model predictive control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Since these trajectories provide a feasible solution, the prediction horizon can be reduced in order to decrease the computation time, since minimizing the distance to the trajectory ensures the satisfaction of the ter- minal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' To decrease the computational burden of the tracking MPC, the system dynamics (3) are first linearized at the reference trajectories xc ref and uc ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Afterwards, the linearized model is discretized for the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Linearization and discretization Using Taylor approximation, the linearized model with states xlin := x − xc ref and control ulin := u − uc ref reads as ˙xlin = � 0 1 a21(t) a22(t) � � �� � =Alin(t) xlin + � 0 b2(t) � � �� � =blin(t) ulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' (9) The remaining entries are calculated as a21(t) = ∂f2(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' m) ∂α ∂α ∂x1 = −g cos(αref(t)) dα dx1 with αref(t) as inclination at xc ref1(t) and ∂α ∂x1 given from the map, as well as a22(t) = − 1 mϱaircairAtrain(xc ref2(t) − vW) − 1 mk1k2e−k2xc ref2(t)uc ref(t) and b2(t) = k1e−k2xc ref2(t) + k3 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' After linearizing the train-motion dynamics (1), the ob- tained model (9) is discretized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The sampled states are defined as xd refi = �xc refi(tstep) xc refi(2tstep) xc refi(3tstep) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='�⊤ for i = 1, 2, where ud ref is defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The resulting discrete-time model is given by xd(k + 1) = Ad(k)xd(k) + bd(k)ud(k), (10) where Ad(k) = eAlin(ktstep)·tstep and bd(k) = � tstep 0 eAlin(ktstep)·(tstep−τ)blin(ktstep) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Tracking MPC The linearized and discretized model is used in the optimal control problem along with the reference trajectories from Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' An MPC setup with moving horizon N is considered to track xd ref1, which results in minimizing the difference between the current position at t = ktstep and xd ref1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' This difference is defined as xd 1(k) in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In order to increase the amount of feasible solutions, the terminal constraint in (8) is replaced by terminal costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Therefore, the cost function of the optimal control problem (8) is enlarged by terminal costs to obtain the resulting tracking MPC as min U 1 2U ⊤RU + 1 2X⊤ 1 QX1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' xd(k + 1) = Ad(k)xd(k) + bd(k)ud(k), xd(0) = x0, h(xd(k), ud(k)) ≤ 0 ∀k = j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' , j + N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' (11) In the objective function, U, X1 ∈ RN are weighted via R, Q ∈ RN×N, R, Q ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' At time instant j, X1 = � xd 1(j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' xd 1(j + N − 1) �⊤ includes xd ref1(j), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' , xd ref1(j + N − 1) as reference trajectory values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Analogously, the input vector is defined as U = � ud(j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' ud(j + N − 1) �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The solution of (11) is denoted as U ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Once the optimal control sequence is obtained, only the first element of U ⋆ is applied to the system (1), which yields a new initial value for the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Afterwards, the horizon is shifted and, in particular, the reference values in X1 are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' This iterative scheme is applied until the arrival station is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Note that if the remaining prediction length is smaller than N, then N is reduced to the remaining length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' IMPLEMENTATION AND CASE-STUDY In this section, the proposed multiple-shooting method and the tracking MPC are implemented and simulated for the RB30, traveling from Chemnitz to Cranzahl, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Physical parameters of the train, as well as the used tuning parameters of the multiple-shooting approach and the track- ing MPC are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' With these parameters, three different scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' behavior for good traction conditions, weather change, and delays) are simulated and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Parameters For the simulations presented in this section, the train- specific model parameters and physical constants shown in Table I are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' They result from train data sheets, whereas ki are determined by a least-squares approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' TABLE I: Train-specific model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Parameter Value ϱair 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2041 kg/m3 Atrain 10 m2 cair 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='85 cR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='002 k1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='516 × 105 kgm/s2 k2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='1147 1/s k3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='564 × 104 kgm/s2 mtrain 68 200 kg mmaxload 20 500 kg The train-track-specific parameters like the speed limit vmax(x1), the inclination α(x1), and the maximal traction µmax(x1) are listed in the map (see Section II and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In the simulations, the wind speed in driving direction was assumed by vw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' For the implementation of the multiple-shooting method, the number of subintervals mℓ is determined depending on tstation, such that the time between two subintervals is less than three seconds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' N = ⌊tstation/3⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In the simulations it turned out to be a good trade-off between accuracy and computational effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' However, the number of subintervals can be increased, since the computation is done offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' For the implementation of the MPC setup, the discretization time tstep = 1 second and the horizon length N = 20 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The mass of the train m = 78200kg is determined as the sum of the empty train mass and an average passenger weight value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The weighting matrices are chosen as R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='01IN and Q = IN, where IN is the N × N identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' These values are tuned manually, such that tracking of optimal states x and the optimal input u is fulfilled in all considered simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 104 0 10 20 30 x1 [m] x2 [m/s] vmax xref2 xMPC2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 2: Comparison of reference velocity according to multiple-shooting approach and the velocity determined by tracking MPC for the first three railway sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 104 0 1 2 x1 [m] |x2 − xref2| [m/s] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 3: Resulting tracking error between the reference tra- jectory xref2 and the calculated velocity x2 according to tracking MPC for good traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Simulation results In a first scenario, the behavior of the train is simulated and analyzed for good traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Afterwards, a change in the traction conditions due to fallen leaves or frozen railway resulting in a different µmax(x1) is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In a third simulation, the effects of an unplanned delay are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 2, the reference velocity according to multiple- shooting approach as well as the resulting velocity deter- mined by tracking MPC is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The difference between them is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The resulting tracking error between the reference trajectory xref1 and the train position x1 according to tracking MPC is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The tracking MPC approach is able to track the reference with a maximal error of around 5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Furthermore, the satisfaction of the constraints (7) is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The first two elements h1(x, u) and h2(x, u) de- scribe the safety constraints, whereas h3(x, u) and h4(x, u) describe maximum traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 5, the maximum of the safety constraints as well as the traction constraints are visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' It can be seen that the safety constraints are more restrictive than traction constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' However, all constraints are satisfied since all hi(x, u) ≤ 0 along the whole track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In a next step, the influence of changing traction conditions between two stations is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' There exist two possible cases, namely that the new traction conditions are better or worse than the assumed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The first case is not problematic, since the bad weather reference trajectory can 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 104 0 2 4 x1 [m] |x1 − xref1| [m] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 4: Resulting tracking error between the reference tra- jectory xref1 and the calculated position x1 according to tracking MPC for good traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 104 −2 −1 0 x1 [m] Value of constraints max{h1, h2} max{h3, h4} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 5: Safety and traction constraints of tracking MPC with good traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' be easily tracked with better traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In the other case, a decrease of the maximal traction yields a decrease of maximal acceleration and braking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' For this scenario, the reference trajectories for good trac- tion conditions are tracked via MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' However, due to an unexpected change of the weather conditions, the traction is changing to bad conditions immediately after leaving the station (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' traction trajectories in the map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The tracking error yields no significant change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' More interesting is, that the traction constraints are more restrictive at the bigger part of the track, which can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' However, the constraints are satisfied along the whole track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Therefore, the developed approach is able to avoid skidding and sliding during changes of weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In a third scenario, the effects of an unexpected delay due to longer passenger changing time are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' An initial delay of tdelay = 40 seconds in the first station was simulated under good traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' With this delay, the optimal control problem (8) is infeasible, since the train can not reach the first station in the given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' However, the reference trajectories are calculated offline independently of the occurred delay, thus, they are feasible for the initial time table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Since terminal constraints are replaced by terminal costs in the MPC approach, the OCP (11) obtains a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 7, the tracking error caused by the delay at position x1(tdelay) = 0 is around 280 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The train arrives at the next station with a delay of 19 seconds, which is smaller than the initial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Due to waiting time in 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='4 104 −1 0 x1 [m] Value of constraints max{h1, h2} max{h3, h4} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 6: Safety and traction constraints of tracking MPC with bad traction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 200 400 600 800 0 200 400 600 800 t [s] |x1 − xref1| [m] tdelay Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' 7: Tracking error with an initial delay of 40 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' the station, the tracking error is zero for a short moment until the desired departure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Until the following station, the tracking error decreases to zero and the MPC tracks the reference from this point on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Again, all constraints are satisfied at each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Therefore, especially skidding and sliding is prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' CONCLUSION AND OUTLOOK This paper proposes a model predictive control approach for train operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' The presented approach is able to avoid skidding and sliding during fast changes of the traction conditions, by using a map with traction conditions and topography data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In order to reduce the computational effort in online operation, the optimal control problem is solved offline to generate reference trajectories for the position, velocity and the lever position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Afterwards, a tracking MPC setup follows the reference trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In a case-study, it turned out that the MPC tracks the reference trajectory with a small tracking error while sat- isfying all constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Furthermore, even with changing weather conditions, MPC yields a feasible solution, where traction conditions became more restrictive in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Additionally, the developed approach was able to handle unexpected delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' In future works, a more detailed model can be considered including, for example, motor characteristics and different braking functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Another future idea is an online adaption of the map based on actual measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Furthermore, information that are available shortly before they have to be considered are important extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' Those ones include, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=', blocked sections by other trains or environmental influ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} +page_content=' REFERENCES [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtAyT4oBgHgl3EQfufmM/content/2301.00614v1.pdf'} diff --git a/TtFAT4oBgHgl3EQf2h54/content/tmp_files/2301.08715v1.pdf.txt b/TtFAT4oBgHgl3EQf2h54/content/tmp_files/2301.08715v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4151075a277c0e96dc980b239806fa4cb88f334c --- /dev/null +++ b/TtFAT4oBgHgl3EQf2h54/content/tmp_files/2301.08715v1.pdf.txt @@ -0,0 +1,674 @@ + +1 +Degradation of crude oil and pure hydrocarbon fractions by some wild bacterial and +fungal species + +Srwa A. Mohammed*1; Taha J. Omar1; Ayad H. Hasan1,2 + + +1 Dept. of Medical Microbiology, Faculty of Science and Health, Koya University, Koya KOY45, +Kurdistan Region - F.R. Iraq + +2Department of Biomedical Sciences, College of Health Technology, Cihan University-Erbil, +Kurdistan Region, Iraq. + +* Corresponding author: Srwa A. Mohammed (srwa.ali@koyauniversity.org) + +Introduction: petroleum hydrocarbons are a major concern due to their widespread distribution +into the environment, such as soil or water, and their harmful effects on humans. (Chen et al., +2015; Wang et al., 2018). Biological processes are utilised in the bioremediation process, which +is a collection of technologies that either aid in the elimination of contaminants or make them +minimally hazardous (Silva et al., 2015). Objectives: This study was conducted to isolate and +identify bacterial and fungal species from the soil of the TaqTaq (TTOPCO) oil field in the +Kurdistan Region of Iraq, and then investigate their ability to degrade crude oil and its fractions. +Many studies agree with the present study for the biodegradation of crude oil in vitro by using +different methods and various concentrations of crude oil (Ramdass and Rampersad, 2021). + +Abstract + +The use of biodegradation as a method for cleaning up soil that has been contaminated by spilt +petroleum can be an effective strategy. So, this study investigated the existence of the wild +microorganism in soil contaminated with oil and study their ability to degrade petroleum in vitro. +Nineteen samples were collected from various locations near Taq Taq (TTOPCO) natural seeps in +the Kurdistan Region of Iraq. Morphological, cultural, biochemical tests and molecular +identification were used to identify the microbial communities, in addition, spore texture and the +colour of the fungal isolates were investigated on the fungal isolates. Out of the19 samples, 17 +indigenous bacterial strains and 5 fungal strains were successfully isolated. From the absorption +spectrophotometry, Bacillus anthracis, Bacillus cereus, Achromobacter sp. and Pseudomonas +aeruginosa for the bacterial isolates grew well on a minimal salt medium supplemented with 1% +crude oil. Results showed that these isolates mentioned above had a strong ability to degrade crude +oil by reducing the colour of 2,6-dichlorophenol indophenol (DCPIP) from deep blue to colourless. +However, for the fractions of hydrocarbon, the bacterial isolates failed and did not affect the colour +of any of the fractions. The results for fungi showed that Aspergillus lentulus and Rhizopus +arrhizus had a strong ability to degrade both crude oil and fraction F1 by reducing the colour of +DCPIP. Each fungal isolates also had a great tolerance to different concentrations of crude oil +when grown on solid MSM. This study showed these microorganisms have a strong ability to +degrade crude oil and can be used to clean up soil and the environment. + + + +2 +Keywords: Bioremediation; bacterial spp.; fungi spp.; Crude oil and fractions of crude oil; 2,6- +Dichlorophenol indophenol + +1. Introduction + +The Kurdistan Region of Iraq (KRI) is located north and northeast of the Arabian plate. The region +is one of the oil-rich areas of Iraq Shlimon, et al., (2020) Koya city is one of the oil-rich areas in +the Kurdistan Region with many reservoirs. TaqTaq (TTOPCO) reservoir is one of them. The +intensive use of petroleum, however, results in environmental disruption Xue et al. (2015). Spills +that occur during and/or as a result of petroleum extraction, storage, refining, manufacturing, +shipping, oilfield development, leakage from oil pipelines or tankers, and discharges of petroleum +hydrocarbons are also major concerns due to their widespread distribution into the environment, +such as soil or water, which affects humans’ health (Chen et al., 2015; Wang et al., 2018). Crude +oil is a complex blend of hydrophobic components such as n-alkanes, aromatics, resins, and +asphaltenes Barnes, et al. (2018). + Managing hydrocarbon contamination has become easier than before due to the development +of several new technologies in recent years. Biological processes are utilised in the bioremediation +process, which is a collection of technologies that either aid in the elimination of contaminants or +make them minimally hazardous (Silva et al., 2015). The procedure is cost-effective and can be +applied in its entirety to the area that is contaminated. Consequently, microbial remediation is a +promising method for the complete mineralisation of hydrocarbons into carbon dioxide and water +(Wang et al., 2015). + Bacteria, fungi, and yeast biodegrade hydrocarbons in the environment. Some bacterial species +can metabolise specific alkanes others break down aromatic or resin fractions of hydrocarbons in +many different manners depending on the oxygenase (Xu, et al. 2017). + Although several fungi can grow in soil, few species can survive in contaminated soils with +biodegradation efficiency ranging from 6% to 82% (Juhasz and Naidu, 2000; Das and Chandran, +2011; Acevedo et al., 2012). This study aimed to isolate and identify bacterial and fungal species +from the soil of the TaqTaq (TTOPCO) oil field and investigate their ability to degrade crude oil +and its fractions using spectrophotometry and 2,6-dichlorophenol indophenol (DCPIP) methods +and mycelial radial growth measurements. + +2. Material and Method + +2.1. Sample and sampling locations + +All the samples were collected from the Koya City Taq Taq asphalt seep (TTOPCO), which is in +the Kurdistan Region of Iraq (KRI) Five samples of contaminated soil were collected at a depth of +5 to 10 cm from each of the following four different locations: 1- Mud site (group A) the samples +were given the numbers 1–4, it was oil and water mixed samples were collected from an area close +to the drilled pool of oil at the Taq Taq oil seep. 2- Underlying and flanking region (group B) of +the Taq Taq asphalt seep flow the samples were given the numbers 5–9. 3- Ten meters away from +site number two, it had not been contaminated by any oil spills, and it was used as a control for +analysing the contaminated soil, the samples in this group were given numbers 10–14. 4-The +transportation area where the soil was contaminated with spilt oil, was 20 m away from site number +two, the collected samples were numbered 15–19. All the samples were placed into sterilised + + +3 +polyethylene bags. Approximately 2 L of oil samples were collected from operating oil wells and +stored in bottles that had been carefully sealed. The bottles and polyethylene bags were placed in +a container packed with ice until then transported to a laboratory and stored at 4°C. + +2.2. Morphological and molecular identification of isolated bacteria and fungi strains + +Pure culture technique was applied to isolate pure single colonies from each soil sample using +nutrient agar, McConkey agar, cetrimide agar and mannitol salt agar (Ozyurek and Bilkay, 2017). +Then, biochemical tests and gram stain were conducted for all the isolated single colonies (Riedel, +et al. 2019). The fungi were identified using morphological and taxonomic keys found in +mycological keys (Watanabe 2018). + The genomic DNA of the isolated bacteria and fungi was extracted by QIAwave DNA Blood +and Tissue Kit (Germany, Cat. no. 69556) according to the manufacturer’s protocol. The Nanodrop +(NanoDrop Spectro 117 432-UK) spectrophotometer was used to determine the quality and +quantity of the genomic DNA. 16S rRNA gene for bacteria that contain a highly variable region +was amplified by the PCR (Polymerase Chain Reaction) using the universal primer EubA F +(AAGGAGGTGATCCANCCRCA) and EubB R (AGAGTTTGATCMTGGCTCAG), which will +give an end product size of 1534 bp Lane, (1991). On the other hand, the internal transcribed +spacer (ITS) region for fungi was amplified using LROR (ACCCGCTGAACTTAAGC.) and LR6 +CGCCAGTTCTGCTTACC Raja, et al. (2017), which will give an end product size of 1200 bp +(Macrogen inc, South Korea company). + Bacterial amplification reactions were performed in a final volume of 25 μl of PCR reaction +mixture using Prime Q5 Hot Start High-Fidelity 2X Master Mix (Cat. no.M0494S). The PCR +reaction included 12.5μl of 2x Master Mix, 5 pmol (1 μl) of each of forward (Eub A) and reverse +(Eub B) primers, 100 ng (1 μl) template DNA, and 9.5 μl nuclease-free water. The PCR process +was performed using BIO-RAD T100TM Thermal Cycler (UK) and programmed as follows: 2 +min of initial denaturation at 98°C, followed by 25 cycles of reaction with the 50s of denaturing at +98°C, 50s of annealing at (64.3, 61, 59, 58, and 56) °C, 50s of extension at 72°C, and the final +extension was performed for 4 min at 72°C. + To amplify the ITS region of the fungal isolates the PCR reactions were performed in a final +volume of 25 μl of the reaction mixture, which included 12.5μl of 2x Master Mix, 5 pmol (1.5 μl) +of each of forward (LROR) and reverse (LR6) primer, XX ng (3 μl) template DNA, and 6.5 μl +nuclease-free water. The PCR program procedure was carried out as previously described for +bacterial isolates with an annealing temperature of 56°C and a cycling repetition of 34 cycles. + The efficiency of DNA extraction was evaluated using electrophoresis on a 1% agarose gel +(w/v) that was stained with ethidium bromide (0.5 g/ml) and had a 100bp DNA marker (DENA +100 bp plus DNA size marker II S-5091). The gel was run at 80 V for 1:30 hours in a 1X TBE +buffer. After the experiment had run its course, ultraviolet light was used to visualize the DNA +bands, and photographs were taken using UV Gel Imager SynGene 1409 Lee, et al. (2012). + All the PCR amplicons were sent out for sequencing by Macrogen inc, a South Korean +company. Sequence quality, analysis, and editing were carried out using the DNA baser assembler +tool. The 16S rDNA sequence was compared to previously identified bacterial DNA sequences +using the BLASTN in order to classify the bacterial isolates independently (http://www. +ncbi.nih.gov/BLAST). + +2.3. Crude oil and pure hydrocarbons degradation by bacteria and fungi isolates + + +4 +A minimal Salt Medium (MSM) was used for bacterial isolates (Sigma-Aldrich). 1ml/L trace +element (CaCl2.2H2O 4.77g/100ml, FeSo4.7H2O 0.37 gm/100ml, MnCl2.4H2O 0.10 gm/100ml, +Na2MoO4.2H2O 0.02 gm/100ml at pH 7) was added to the MSM, in addition to 1 ml/L vitamin +mix solution ( Pyridoxine-Hcl 10.0 mg, p-Aminobenzoic acid 5.0 mg, Lipoic acid 5.0 mg, +Nicotinic acid 5.0mg, Riboflavin acid 5.0mg, Thiamine-Hcl 5.0mg, Calcium D1-Pantothenate +5.0mg, Biotin 2.0mg, Folic acid 2.0mg, Vitamin B12 0.1mg) and 0.1gm/L yeast extract. + Minimal salt medium (MSM) and potato dextrose broth were used for the isolation and +maintenance of fungal isolates. Five grams of each collected soil sample were incubated in 250 +Erlenmeyer flasks containing 100ml freshly prepared MSM containing NaCl (0.5g/l), (NH4)2SO4 +(0.1g/L), NaNO3 (0.2 g/L), MgSO4.7H2O (0.025 g/L), K2HPO4.3H2O (1g/L) and KH2PO4 +(0.4g/L) at pH 7.0. + Crude oil was fractionated using simple distillation by pouring it into the round bottom flask +and adjusting the fractionation column and thermometer with the conical flask on one side and the +condenser on the other side. The temperature was measured with a thermometer, and four fractions +of crude oil were separated. The temperature degree of fraction number was (40-60)°C belong to +F1, (60-80) °C belong to F2, (80-100)°C belong to F3 and (100-130) °C belong to F4, respectively. +Consequently, degradation of all the fractions was conducted as mentioned above using 1% of +each fraction for bacterial and fungi isolates. + + Experiments on biodegradation were carried out in glass screw cup tubes containing 10 ml of +MSM and 1% crude oil as the sole carbon source. Prior to adding crude oil, both the media and +the crude oil were sterilized separately by autoclaving at 121°C for 15 minutes. A single colony of +the isolate was inoculated into 10 ml nutrient broth for bacteria and incubated overnight at 30°C +at 150 rpm. Following incubation, the culture was centrifuged for 10 minutes at 10000 rpm. The +bacterial suspension was ready for use after the cell pellets were washed to remove all nutrients +and re-suspended in normal saline until the OD at 600 nm was equivalent to 1. 1% of the bacterial +suspension was transferred into the MSM supplemented with crude oil. Uninoculated media was +used as a control, and all tests and controls were done in triplicate. The same procedure was +followed for pure hydrocarbon fractions (F1, F2, and F4). Two methods were used to detect crude +oil and hydrocarbon fractions degradation: 1- spectrophotometer method: the absorbency at +wavelength 600 nm was measured at 24 hours, 48 hours, one week, three weeks, and eight weeks +to determine the bacterial growth. 2- 2,6-Dichlorophenol indophenol (DCPIP) method was used +to evaluate bacterial isolates' ability to degrade crude oil and hydrocarbon fractions. 1 ml washed +bacterial cells (Optical density (OD) at 600 nm was equivalent to 1), 1% V/V crude oil and each +fraction separately, and 1% (0.6 mg/L) of redox indicator (DCPIP) were added to 10 ml MSM, +and incubated for two weeks in a shaker incubator at 30°C with 150rpm. Then, every 24 hrs, the +colour changes are monitored. Uninoculated media was used as a control (Selvakumar, 2014; +Balogun, 2015). Likewise, this method was repeated for fungal isolates. + The fungal mycelia were grown in a glass screw cup container with 10 mL of mineral salts +medium supplemented with 1% crude oil and 1% 2,6-Dichlorophenol indophenol (DCPIP) redox +as followed in bacterial degradation for 7 days at 30°C and 150 rpm in shaker incubator without +any other nutrient as additive. crude oil as a carbon source was used Barnes, (2018). + Measuring mycelial radial growth is the second method used to detect crude oil and pure +hydrocarbons degradation with fungi by the cultivation of all fungal isolates on solid MSM media +supplemented with different concentrations (5, 10, 15, 20) % of crude oil and 0% was used as a + + +5 +control. All of the plates were incubated for 7 days at 30°C. Thus, the mycelial radial growth +cm/day was measured (Anaisell, et al., 2014). + +3. Result and Discussion + +3.1. Phenotypic and molecular characterization of bacterial and fungal isolates +17 out of 19 bacterial samples could be recognized by utilizing Bergey's manual of systematic +bacteriology for phenotypic and colony identification Ozyurek (2017), while the other two samples +showed no signs of growth. In order to confirm the identity of the samples, total DNA was +extracted from all the seventeen bacterial isolates and then amplified using EubA F and EubB R +primer pair that bind specifically to the 16S rRNA gene as mentioned in section 2.2. The expected +size of the DNA fragment (1534 bp) was amplified successfully. No PCR products were detected +in the negative control as shown in (Figure 1). + + 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 + + + + + + + + + + + + + + + +Fig. 1. Partial amplification of 16S rDNA using EubA and EubB primers. Lanes 1 and 2 represent +a 100bp DNA marker (DENA 100 bp plus DNA size marker II S-5091), a negative control that +has been run without a DNA template, respectively. Lanes 3 through 19 shows ~1534bp of PCR +amplicons generated using a DNA template from number 1 through number 17 bacterial samples, +respectively. + The result showed that the following microorganisms were isolated from group A soil samples +after the classical identification and molecular confirmation. Caldibacillus thermoamylovorans +strain SSBM chromosome with accession number CP023704, Bacillus pumilus strain mv49b +KU230016, Bacillus tropicus strain NP_ 2. OP048825.1, and Pseudomonas aeruginosa strain DM +Bust3A MF599526.1, respectively. Group B soil samples included Aneurinibacillus migulanus +strain DSM 2895 with accession number NR 112214.1, Achromobacter sp. MT093185.1, +Bacillus anthracis strain FDAARGOS 695 with accession number CP054816.1, Bacillus cereus +strain T11-12 with accession number HQ333011.1, and Lysinibacillus sp. 381 with accession +number KT034471.1, respectively. Group C soil samples revealed Paenibacillus dentritiformis +strain PV3-16 with accession number MH472941.1, Aneurinibacillus migulanus strain RD with +accession number KX083693.1, Brevibacillus borstelensis strain ML13. with accession number +MN604049.1, and Bacillus paramycoides strain EFBC 17 with accession number MN793201.1, + + +6 +respectively. Group D soil samples contained Bacillus anthracis strain FDAARGOS_695 with +accession number CP054816.1, Pseudomonas stutzeri strain HA549 with accession number +KJ535356., Bacillus paramycoid 2883 with accession number MT611845.1, and Lysinibacillus +capsici strain anQ-h6 with accession number CP084108.1, respectively. Except for Bacillus +anthracis and Bacillus cereus, most Bacillus spp. are not pathogenic, and many species have been +exploited for biotechnological and industrial uses Gu, et al. (2019). + Above results are in agreement with the results reported by Chen et al. (2017) who isolated +Exiguobacterium sp. Pseudomonas aeruginosa, Alcaligenes sp., and Bacillus sp., petroleum +hydrocarbon-degrading microorganisms’ area contaminated with petroleum. And also, the study +of (Szczepaniak et al., 2015) demonstrated Petroleum hydrocarbons are compounds which +undergo decomposition in soil due to the activity of several groups of microorganisms. Several +different microbial species participate in the biodegradation of hydrocarbons, ranging from strictly +aerobic to strict anaerobic bacteria. Several Gram-positive (Rhodococcus or Bacillus) as well as +Gram-negative (Alcaligenes, Acinetobacter, Pseudomonas) species are also characterized by a +relatively broad substrate spectrum. + Five fungi isolates were isolated from the 19 soil samples. The expected size of PCR amplicons +(1200 bp) was detected successfully in the all five samples. No PCR products were detected in the +negative control as shown in (Figure 2). Therefore, to track down each fungal isolate to its exact +species, the PCR amplicons of each isolate were sent out for sequencing using the forward +(LROR). Thus, the isolate which was isolated from group A was Aspergillus lentulus 28S +ribosomal RNA with accession number XR 004500616.1, and the isolates which were isolated +from group B were Aspergillus fellis strain FM324 chromosome 3. with accession number +CP066505.1, Aspergillus luteonubrus strain MST FP2246 with accession number MT196912.1 +and Aspergillus arizonicus isolate CCF 5341 with accession number OK321187.1, respectively. +The isolate which was isolated from group C was Rhizopus arrhizus Strain SC49B03 with +accession number MW113537.1. The study of (Barnes et al., 2018) demonstrated that the ten +fungal isolates chosen were able to grow and degrade crude oil. The isolate Penicillium citrinum +NIOSN-M126 isolated from Divar mangrove sediments demonstrated the greatest ability to utilize +crude oil, followed by Aspergillus flavus NIOSN-SK56S22 isolated from Arabian Sea sediments. + +3.3. Degradation of crude oil and crude oil fractions by bacterial isolates and fungi isolates: + + +1 2 3 4 5 6 7 8 + + + + + + + + + + + + + + +7 + a b C + + + + + +Figure (3-4) Bacterial isolates that grow on MSM supplemented with 1% crude oil +Fig. 2. Partial amplification of ITS using (LROR)/(LR6) primers. Lanes 1 and 2 represent a 100bp +DNA marker (DENA 100 bp plus DNA size marker II S-5091), a negative control that has been +run without a DNA template, respectively. Lanes 3 through 7 shows ~1200 bp of PCR amplicons +generated using a DNA template from number 1 through 5 fungal samples, respectively. + +Bacteria: strains were isolated for their capacity to grow in the presence of crude oil and the +fractions of crude oil used in the investigation for detecting which bacterial isolates have a high +ability to degrade crude oil and the fractions of crude oil. Two methods were used for this purpose. + spectrophotometer method: The results showed that growth was determined through measuring +the absorbency at wavelength 600 nm by adding 1% washed bacterial cell reading OD at 600 nm +was equivalent to 1 and 1% crude oil and incubated at 30°C in a shaker incubator 150rpm. Then, +the results showed that no. (3, 4, 5, 6, 7, 8, 9, 10,11, 12, 13,14, and 16) the OD reading after 24 hrs +were (0.50, 0.43, 0.45, 0.68, 0.61, o.49, 0.60, 0.52, 0.68, 0.70, 0.46, 0.73, and 0.60) respectively. +However, the OD reading after 48 hrs was increased in some of the sample’s no. (4, 6, 7, 8, 11, +12, and 14) the OD reading at 600 nm was (0.56, 0.32, 0.69, 0.57, 0.32, 0.79, and 0.92) +respectively. Meanwhile, the OD reading was decreased in other samples. Then, the results were +shown after one week the samples number (4, 6, 7, 8, 11, 12, and 14) was recorded OD reading +(0.283, 0.299, 0.4271, 0.3544, .2947, 0.3253, and 0.5793) respectively. Furthermore, after three +weeks number (6, 7, 8, 12 and 14) OD reading was (0.21, 0.34, 0.24, 0.21, and 0.43) respectively. +Finally, after 8 weeks all the isolates decreased due to decreasing carbon sources and nutrients as +shown in Table 1. Therefore, this investigation concluded that the most potent isolates for resisting +crude oil and growing well in minimal media supplemented by crude oil were (Pseudomonas +aeruginosa, Achromobacter sp. and Bacillus anthracis, Bacillus cereus, Aneurinibacillus +migulanus, and Brevibacillus borstelensis) and (Figure 1) showed bacterial colony on solid MSM. + Degradation hydrocarbon fractions (F1, F2, F3, and F4) were managed by a simple distiller at +the laboratory and to detect hydrocarbon degradation spectrophotometer method was used for the +samples which degrade crude oil faster than others. Then, the number (4, 6,7, 8,11,12,14, 16 and +17) were undergone to this test. And the results showed that (F2 and F4) were consumed by the +isolates more than other fractions. The OD reading for isolates that consume and grow well in F2 +were (4,6, 7,12,14, and 17) the OD reading were (0.312, 0.334, 0.456, 0.594, and 0.482) +respectively, Furthermore, numbers (8,11 and 17) were consumed fraction 4 (F4) more than the +other fractions, the OD reading were (0.357, 0.3320. and 0.334) respectively. The bacterial growth +was increased until after four weeks decrease the OD reading and growth of bacteria for all the +isolates were caused by decreasing the carbon source in the media. Yu et al. (2005b) showed in +their study that Autochthonous microorganisms in sediments also possessed satisfactory +Polycyclic aromatic hydrocarbons (PAHs) degradation capability and all three PAH were +completely degraded after 4 weeks of growth. + + + + + + + + + + +8 +Fig. 3. Bacterial isolates that grown on MSM supplemented with 1% crude oil a- Bacillus cereus +b- Achromobacter sp. c- Pseudomonas sp + +A 2,6-Dichlorophenol indophenol (DCPIP) method: This method also was used to assess the +ability of bacterial isolates to degrade crude oil. The suspension was prepared as mentioned in +section 2.2. When compared to the control, the change in colour of the inoculation degradation +medium from deep blue to colourless demonstrated the ability of the bacterial isolates to degrade +crude oil. Then, the results showed that numbers (7, 8, 11, 12, and 14) have a high ability to reduce +the colour completely to colourless after 4 days, and numbers (4, 6, and 16) also can reduce the +colour but less than the numbers mentioned above as shown in (Figure 2). Many studies agree with +the present study for the biodegradation of crude oil in vitro by using different methods and +different concentrations of crude oil. Ramdass and Rampersad, (2021) showed crude oil +degradation by bacteria, fungi and yeast and they were concluded 2% of crude oil were mixed with +two media types. the result indicated that all microbes recovered were able to utilize crude oil on +both media types in vitro. Rizi et al. (2012) reported gram-positive bacteria such as Bacillus cereus +and Bacillus subtilis were predominant in degrading crude oil this paralleled with the present study +that Bacillus cereus and Bacillus spp. the most predominant in degrading crude oil. while, the +results showed the colour does not change for all the fractions (F1, F2, F3, and F4) separately for +bacterial isolates. + +Table 1. Spectrophotometer readings for bacterial isolates that cause high turbidity in MSM when +1% crude oil is added + + + + + + + + + + + + + + + + + + + + +Fungi: Five oil-degrading fungi were isolated from 19 samples of crude oil-polluted soil and +control samples. Two methods were used to detect crude oil degradation by fungi isolates. + +No. of +isolates + After +24 +hrs +After 48 +hrs +After one +week +After +3 +weeks +After 8 +weeks + +3 +0.5028 +0.231 +0.0594 +0.0887 +0.0001 +4 +0.4352 +0.564 +0.2830 +0.0568 +0.0002 +5 +0.4591 +0.152 +0.2514 +0.0343 +0.0001 +6 +0.6872 +0.327 +0.299 +0.2110 +0.0003 +7 +0.6123 +0.695 +0.4271 +0.3454 +0.0002 +8 +0.4973 +0.577 +0.3544 +0.2499 +0.0001 +9 +0.6069 +0.210 +0.0931 +0.0903 +0.0002 +10 +0.5265 +0.257 +0.1639 +0.1259 +0.0001 +11 +0.687 +0.325 +0.2947 +0.2054 +0.0001 +12 +0.7090 +0.796 +0.3253 +0.2106 +0.0002 +13 +0.4686 +0.481 +0.0553 +0.0319 +0.0002 +14 +0.7327 +0.921 +0.5793 +0.4383 +0.0002 +16 +0.6052 +0.495 +0.1906 +0.1179 +0.0001 +17 +0.3812 +0.261 +0.1232 +0.0944 +0.0001 + + +9 + 17 16 15 14 13. 12 11 10 9 8 7 6. 5 4 3 2 1 C + + +Fig. 4. Ability of the bacterial isolates which start from control © followed by number 1 through +17 to reduce 2,6-Dichlorophenol indophenol (DCPIP) by changing the colour from blue to +colourless after 4 days + + Mycelia radial growth measurement on minimal salt medium (MSM): Five isolated fungi were +screened for their ability to tolerate crude oil or petroleum hydrocarbons from 0 to 20% +concentration. All the isolated fungi showed to have a high ability to tolerate different crude oil +concentrations as well as in the control treatment (without crude oil). Rhizopus arrhizus exhibited +the highest growth rate, 6.8 cm/day at 0%, as well as (6.6, 6.6, 6.4 6.1) cm/day for crude oil +concentrations of (5,10,15, and 20%). Aspergillus lentulus presented the second highest growth +rate (5.3, 5.0, 4.4, 4.1 and 4.0) cm/ day, followed by other isolates. The experiment proved that all +five strains were capable to grow in crude oil by using crude oil as the sole carbon source. (Al- +Zaban, et al., 2021) who isolated four fungi isolates and measured mycelia radial growth on +minimal medium (MM) supplemented with crude oil at different concentrations. All isolates grew +faster in the control treatment (without crude oil) than in the other treatments supplemented with +varying concentrations of crude oil. However, he observed remarkable adaptation and were able +to survive high concentrations of crude oil of up to 20%. Reyes-César, et al. (2014) results showed +the high biodegradation potential of the Talaromyces spectabilis CCS12 strain and it can be used +in the development of novel green processes due to its strong ability to metabolize PAHs in soil. +While the present study showed Rhizopus arrhizus which isolate from a control sample has high +biodegradation potential on MSM solid media added with up to 20% of crude oil and 1% of four +fractions separately after three days of incubation and this novel and a new record for this genus +and followed by isolating Aspergillus lentulus. + The 2,6-Dichlorophenol Indophenol (DCPIP) method was used to evaluate the ability of +selected fungal strains to degrade crude oil and the hydrocarbon fractions; the results showed that +Rhizopus arrhizus. and Aspergillus lentulus. have a strong ability to degrade crude oil and reduce +the colour from deep blue to colourless and consequently, degrade the F1 fraction as well. While +other fungi isolates can degrade crude oil and change the blue colour to colourless. However, the +(DCPIP) colour does not change and remains the blue colour for the fractions when compared with +the control. Al-Hawash, et al. (2018) demonstrated that two strains of Penicillium sp. RMA1 and +RMA2 were isolated from the Rumaila oil field and can change the colour gradually from deep +blue to colourless, and this reaction suggested that Penicillium sp. RMA1 and RMA2 could +degrade crude oil. Environmental pollution caused by oil spillage is one of the century's major +issues that must be resolved. Biodegradation can emerge or limit this problem. + +4. Conclusion + + + +12F3 +12F2 +3F +10 + 0% 5%. 10% 15%. 20% + + +Fig. 5. Macroscopic images of radial growth of isolated fungal strains, in Petri dishes with +different concentrations of crude oil (0% to 20%). Line 1 is Aspergillus lentulus, line 2 Aspergillus +fellis, line 3 Aspergillus luteonubrus, line 4 Aspergillus arizonicus, and line 5 is Rhizopus arrhizus + + +This study concluded that different microorganism isolates survive and grow well in crude oil and +were affiliated with Bacillus anthracis, Bacillus cereus, Achromobacter sp., and pseudomonas +aeroginosa for bacterial isolates. Moreover, Rhizopus arrhizus and Aspergillus lentulus for fungi +isolates, this study showed these microorganisms have a strong ability to degrade crude oil and +can be used to clean up soil and the environment. However, the longitudinal investigation will +determine which enzyme is responsible for the degradation of crude petroleum and hydrocarbons, +as well as whether the transcriptome encoding this enzyme is expressed by adding crude oil or if +it is expressed simultaneously. And more studies are needed to identify other microorganisms in +the Kurdistan Region with the ability to degrade crude oil, as well as to use a mixed culture of +bacteria and fungi to observe their synergistic ability to degrade hydrocarbons. + +References + + +Al-Zaban, M.I., AlHarbi, M.A. and Mahmoud, M.A. (2021) Hydrocarbon biodegradation and +transcriptome responses of cellulase, peroxidase, and laccase encoding genes inhabiting +rhizospheric fungal isolates. Saudi Journal of Biological Sciences, 28(4); 2083-2090. + +Al-Hawash, A.B., Alkooranee, J.T., Abbood, H.A., Zhang, J., Sun, J., Zhang, X. and Ma, F. +(2018) Isolation and characterization of two crude oil-degrading fungi strains from Rumaila oil +field, Iraq. Biotechnology reports, 1 (17); 104-109. + +Anaisell, R., Angel, E.A., Francisco, J.F., Juan, M.G., Diana, V., Cortes, E. (2014). +Biodegradation of a mixture of PAHs by non-ligninolytic fungal strains isolated from crude oil- +contaminated soil. World J. Microbiol. 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Current +microbiology, 71(2); 220-228. + +Xu, X., Zhai, Z., Li, H., Wang, Q., Han, X. and Yu, H. (2017) Synergetic effect of bio- +photocatalytic hybrid system: g-C3N4 and Acinetobacter sp. JLS1 for enhanced degradation of +C16 alkane. Chemical Engineering Journal, 323; 520-529. + + Yu, S.H., Ke, L., Wong, Y.S. and Tam, N.F.Y. (2005) Degradation of polycyclic aromatic +hydrocarbons by a bacterial consortium enriched from mangrove sediments. Environment +International, 31(2); 149-154. + + diff --git a/TtFAT4oBgHgl3EQf2h54/content/tmp_files/load_file.txt b/TtFAT4oBgHgl3EQf2h54/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4fb3ccc72bf0871b06c703ec5b46a6c986f179a6 --- /dev/null +++ b/TtFAT4oBgHgl3EQf2h54/content/tmp_files/load_file.txt @@ -0,0 +1,725 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf,len=724 +page_content='1 Degradation of crude oil and pure hydrocarbon fractions by some wild bacterial and fungal species Srwa A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Mohammed*1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Taha J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Omar1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Ayad H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Hasan1,2 1 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' of Medical Microbiology, Faculty of Science and Health, Koya University, Koya KOY45, Kurdistan Region - F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Iraq 2Department of Biomedical Sciences, College of Health Technology, Cihan University-Erbil, Kurdistan Region, Iraq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Corresponding author: Srwa A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Mohammed (srwa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='ali@koyauniversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='org) Introduction: petroleum hydrocarbons are a major concern due to their widespread distribution into the environment, such as soil or water, and their harmful effects on humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Biological processes are utilised in the bioremediation process, which is a collection of technologies that either aid in the elimination of contaminants or make them minimally hazardous (Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Objectives: This study was conducted to isolate and identify bacterial and fungal species from the soil of the TaqTaq (TTOPCO) oil field in the Kurdistan Region of Iraq, and then investigate their ability to degrade crude oil and its fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Many studies agree with the present study for the biodegradation of crude oil in vitro by using different methods and various concentrations of crude oil (Ramdass and Rampersad, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Abstract The use of biodegradation as a method for cleaning up soil that has been contaminated by spilt petroleum can be an effective strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' So, this study investigated the existence of the wild microorganism in soil contaminated with oil and study their ability to degrade petroleum in vitro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Nineteen samples were collected from various locations near Taq Taq (TTOPCO) natural seeps in the Kurdistan Region of Iraq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Morphological, cultural, biochemical tests and molecular identification were used to identify the microbial communities, in addition, spore texture and the colour of the fungal isolates were investigated on the fungal isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Out of the19 samples, 17 indigenous bacterial strains and 5 fungal strains were successfully isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' From the absorption spectrophotometry, Bacillus anthracis, Bacillus cereus, Achromobacter sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' and Pseudomonas aeruginosa for the bacterial isolates grew well on a minimal salt medium supplemented with 1% crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Results showed that these isolates mentioned above had a strong ability to degrade crude oil by reducing the colour of 2,6-dichlorophenol indophenol (DCPIP) from deep blue to colourless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' However, for the fractions of hydrocarbon, the bacterial isolates failed and did not affect the colour of any of the fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The results for fungi showed that Aspergillus lentulus and Rhizopus arrhizus had a strong ability to degrade both crude oil and fraction F1 by reducing the colour of DCPIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Each fungal isolates also had a great tolerance to different concentrations of crude oil when grown on solid MSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' This study showed these microorganisms have a strong ability to degrade crude oil and can be used to clean up soil and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2 Keywords: Bioremediation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' bacterial spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' fungi spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Crude oil and fractions of crude oil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2,6- Dichlorophenol indophenol 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Introduction The Kurdistan Region of Iraq (KRI) is located north and northeast of the Arabian plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The region is one of the oil-rich areas of Iraq Shlimon, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', (2020) Koya city is one of the oil-rich areas in the Kurdistan Region with many reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' TaqTaq (TTOPCO) reservoir is one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The intensive use of petroleum, however, results in environmental disruption Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Spills that occur during and/or as a result of petroleum extraction, storage, refining, manufacturing, shipping, oilfield development, leakage from oil pipelines or tankers, and discharges of petroleum hydrocarbons are also major concerns due to their widespread distribution into the environment, such as soil or water, which affects humans’ health (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Crude oil is a complex blend of hydrophobic components such as n-alkanes, aromatics, resins, and asphaltenes Barnes, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Managing hydrocarbon contamination has become easier than before due to the development of several new technologies in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Biological processes are utilised in the bioremediation process, which is a collection of technologies that either aid in the elimination of contaminants or make them minimally hazardous (Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The procedure is cost-effective and can be applied in its entirety to the area that is contaminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Consequently, microbial remediation is a promising method for the complete mineralisation of hydrocarbons into carbon dioxide and water (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Bacteria, fungi, and yeast biodegrade hydrocarbons in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Some bacterial species can metabolise specific alkanes others break down aromatic or resin fractions of hydrocarbons in many different manners depending on the oxygenase (Xu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Although several fungi can grow in soil, few species can survive in contaminated soils with biodegradation efficiency ranging from 6% to 82% (Juhasz and Naidu, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Das and Chandran, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Acevedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' This study aimed to isolate and identify bacterial and fungal species from the soil of the TaqTaq (TTOPCO) oil field and investigate their ability to degrade crude oil and its fractions using spectrophotometry and 2,6-dichlorophenol indophenol (DCPIP) methods and mycelial radial growth measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Material and Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Sample and sampling locations All the samples were collected from the Koya City Taq Taq asphalt seep (TTOPCO), which is in the Kurdistan Region of Iraq (KRI) Five samples of contaminated soil were collected at a depth of 5 to 10 cm from each of the following four different locations: 1- Mud site (group A) the samples were given the numbers 1–4, it was oil and water mixed samples were collected from an area close to the drilled pool of oil at the Taq Taq oil seep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2- Underlying and flanking region (group B) of the Taq Taq asphalt seep flow the samples were given the numbers 5–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 3- Ten meters away from site number two, it had not been contaminated by any oil spills, and it was used as a control for analysing the contaminated soil, the samples in this group were given numbers 10–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 4-The transportation area where the soil was contaminated with spilt oil, was 20 m away from site number two, the collected samples were numbered 15–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' All the samples were placed into sterilised 3 polyethylene bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Approximately 2 L of oil samples were collected from operating oil wells and stored in bottles that had been carefully sealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The bottles and polyethylene bags were placed in a container packed with ice until then transported to a laboratory and stored at 4°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Morphological and molecular identification of isolated bacteria and fungi strains Pure culture technique was applied to isolate pure single colonies from each soil sample using nutrient agar, McConkey agar, cetrimide agar and mannitol salt agar (Ozyurek and Bilkay, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Then, biochemical tests and gram stain were conducted for all the isolated single colonies (Riedel, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The fungi were identified using morphological and taxonomic keys found in mycological keys (Watanabe 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The genomic DNA of the isolated bacteria and fungi was extracted by QIAwave DNA Blood and Tissue Kit (Germany, Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 69556) according to the manufacturer’s protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The Nanodrop (NanoDrop Spectro 117 432-UK) spectrophotometer was used to determine the quality and quantity of the genomic DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 16S rRNA gene for bacteria that contain a highly variable region was amplified by the PCR (Polymerase Chain Reaction) using the universal primer EubA F (AAGGAGGTGATCCANCCRCA) and EubB R (AGAGTTTGATCMTGGCTCAG), which will give an end product size of 1534 bp Lane, (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' On the other hand, the internal transcribed spacer (ITS) region for fungi was amplified using LROR (ACCCGCTGAACTTAAGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=') and LR6 CGCCAGTTCTGCTTACC Raja, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2017), which will give an end product size of 1200 bp (Macrogen inc, South Korea company).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Bacterial amplification reactions were performed in a final volume of 25 μl of PCR reaction mixture using Prime Q5 Hot Start High-Fidelity 2X Master Mix (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='M0494S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The PCR reaction included 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='5μl of 2x Master Mix, 5 pmol (1 μl) of each of forward (Eub A) and reverse (Eub B) primers, 100 ng (1 μl) template DNA, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='5 μl nuclease-free water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The PCR process was performed using BIO-RAD T100TM Thermal Cycler (UK) and programmed as follows: 2 min of initial denaturation at 98°C, followed by 25 cycles of reaction with the 50s of denaturing at 98°C, 50s of annealing at (64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='3, 61, 59, 58, and 56) °C, 50s of extension at 72°C, and the final extension was performed for 4 min at 72°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' To amplify the ITS region of the fungal isolates the PCR reactions were performed in a final volume of 25 μl of the reaction mixture, which included 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='5μl of 2x Master Mix, 5 pmol (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='5 μl) of each of forward (LROR) and reverse (LR6) primer, XX ng (3 μl) template DNA, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='5 μl nuclease-free water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The PCR program procedure was carried out as previously described for bacterial isolates with an annealing temperature of 56°C and a cycling repetition of 34 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The efficiency of DNA extraction was evaluated using electrophoresis on a 1% agarose gel (w/v) that was stained with ethidium bromide (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='5 g/ml) and had a 100bp DNA marker (DENA 100 bp plus DNA size marker II S-5091).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The gel was run at 80 V for 1:30 hours in a 1X TBE buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' After the experiment had run its course, ultraviolet light was used to visualize the DNA bands, and photographs were taken using UV Gel Imager SynGene 1409 Lee, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' All the PCR amplicons were sent out for sequencing by Macrogen inc, a South Korean company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Sequence quality, analysis, and editing were carried out using the DNA baser assembler tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The 16S rDNA sequence was compared to previously identified bacterial DNA sequences using the BLASTN in order to classify the bacterial isolates independently (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' ncbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='gov/BLAST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Crude oil and pure hydrocarbons degradation by bacteria and fungi isolates 4 A minimal Salt Medium (MSM) was used for bacterial isolates (Sigma-Aldrich).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 1ml/L trace element (CaCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='2H2O 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='77g/100ml, FeSo4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='7H2O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='37 gm/100ml, MnCl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='4H2O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='10 gm/100ml, Na2MoO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='2H2O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='02 gm/100ml at pH 7) was added to the MSM, in addition to 1 ml/L vitamin mix solution ( Pyridoxine-Hcl 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0 mg, p-Aminobenzoic acid 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0 mg, Lipoic acid 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0 mg, Nicotinic acid 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0mg, Riboflavin acid 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0mg, Thiamine-Hcl 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0mg, Calcium D1-Pantothenate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0mg, Biotin 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0mg, Folic acid 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0mg, Vitamin B12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1mg) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1gm/L yeast extract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Minimal salt medium (MSM) and potato dextrose broth were used for the isolation and maintenance of fungal isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Five grams of each collected soil sample were incubated in 250 Erlenmeyer flasks containing 100ml freshly prepared MSM containing NaCl (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='5g/l), (NH4)2SO4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1g/L), NaNO3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='2 g/L), MgSO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='7H2O (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='025 g/L), K2HPO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='3H2O (1g/L) and KH2PO4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='4g/L) at pH 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Crude oil was fractionated using simple distillation by pouring it into the round bottom flask and adjusting the fractionation column and thermometer with the conical flask on one side and the condenser on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The temperature was measured with a thermometer, and four fractions of crude oil were separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The temperature degree of fraction number was (40-60)°C belong to F1, (60-80) °C belong to F2, (80-100)°C belong to F3 and (100-130) °C belong to F4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Consequently, degradation of all the fractions was conducted as mentioned above using 1% of each fraction for bacterial and fungi isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Experiments on biodegradation were carried out in glass screw cup tubes containing 10 ml of MSM and 1% crude oil as the sole carbon source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Prior to adding crude oil, both the media and the crude oil were sterilized separately by autoclaving at 121°C for 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' A single colony of the isolate was inoculated into 10 ml nutrient broth for bacteria and incubated overnight at 30°C at 150 rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Following incubation, the culture was centrifuged for 10 minutes at 10000 rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The bacterial suspension was ready for use after the cell pellets were washed to remove all nutrients and re-suspended in normal saline until the OD at 600 nm was equivalent to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 1% of the bacterial suspension was transferred into the MSM supplemented with crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Uninoculated media was used as a control, and all tests and controls were done in triplicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The same procedure was followed for pure hydrocarbon fractions (F1, F2, and F4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Two methods were used to detect crude oil and hydrocarbon fractions degradation: 1- spectrophotometer method: the absorbency at wavelength 600 nm was measured at 24 hours, 48 hours, one week, three weeks, and eight weeks to determine the bacterial growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=" 2- 2,6-Dichlorophenol indophenol (DCPIP) method was used to evaluate bacterial isolates' ability to degrade crude oil and hydrocarbon fractions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 1 ml washed bacterial cells (Optical density (OD) at 600 nm was equivalent to 1), 1% V/V crude oil and each fraction separately, and 1% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='6 mg/L) of redox indicator (DCPIP) were added to 10 ml MSM, and incubated for two weeks in a shaker incubator at 30°C with 150rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Then, every 24 hrs, the colour changes are monitored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Uninoculated media was used as a control (Selvakumar, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Balogun, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Likewise, this method was repeated for fungal isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The fungal mycelia were grown in a glass screw cup container with 10 mL of mineral salts medium supplemented with 1% crude oil and 1% 2,6-Dichlorophenol indophenol (DCPIP) redox as followed in bacterial degradation for 7 days at 30°C and 150 rpm in shaker incubator without any other nutrient as additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' crude oil as a carbon source was used Barnes, (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Measuring mycelial radial growth is the second method used to detect crude oil and pure hydrocarbons degradation with fungi by the cultivation of all fungal isolates on solid MSM media supplemented with different concentrations (5, 10, 15, 20) % of crude oil and 0% was used as a 5 control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' All of the plates were incubated for 7 days at 30°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Thus, the mycelial radial growth cm/day was measured (Anaisell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Result and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=" Phenotypic and molecular characterization of bacterial and fungal isolates 17 out of 19 bacterial samples could be recognized by utilizing Bergey's manual of systematic bacteriology for phenotypic and colony identification Ozyurek (2017), while the other two samples showed no signs of growth." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' In order to confirm the identity of the samples, total DNA was extracted from all the seventeen bacterial isolates and then amplified using EubA F and EubB R primer pair that bind specifically to the 16S rRNA gene as mentioned in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The expected size of the DNA fragment (1534 bp) was amplified successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' No PCR products were detected in the negative control as shown in (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Partial amplification of 16S rDNA using EubA and EubB primers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Lanes 1 and 2 represent a 100bp DNA marker (DENA 100 bp plus DNA size marker II S-5091), a negative control that has been run without a DNA template, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Lanes 3 through 19 shows ~1534bp of PCR amplicons generated using a DNA template from number 1 through number 17 bacterial samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The result showed that the following microorganisms were isolated from group A soil samples after the classical identification and molecular confirmation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Caldibacillus thermoamylovorans strain SSBM chromosome with accession number CP023704, Bacillus pumilus strain mv49b KU230016, Bacillus tropicus strain NP_ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' OP048825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, and Pseudomonas aeruginosa strain DM Bust3A MF599526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Group B soil samples included Aneurinibacillus migulanus strain DSM 2895 with accession number NR 112214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, Achromobacter sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' MT093185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, Bacillus anthracis strain FDAARGOS 695 with accession number CP054816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, Bacillus cereus strain T11-12 with accession number HQ333011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, and Lysinibacillus sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 381 with accession number KT034471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Group C soil samples revealed Paenibacillus dentritiformis strain PV3-16 with accession number MH472941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, Aneurinibacillus migulanus strain RD with accession number KX083693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, Brevibacillus borstelensis strain ML13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' with accession number MN604049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, and Bacillus paramycoides strain EFBC 17 with accession number MN793201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, 6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Group D soil samples contained Bacillus anthracis strain FDAARGOS_695 with accession number CP054816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, Pseudomonas stutzeri strain HA549 with accession number KJ535356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', Bacillus paramycoid 2883 with accession number MT611845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, and Lysinibacillus capsici strain anQ-h6 with accession number CP084108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Except for Bacillus anthracis and Bacillus cereus, most Bacillus spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' are not pathogenic, and many species have been exploited for biotechnological and industrial uses Gu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Above results are in agreement with the results reported by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2017) who isolated Exiguobacterium sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Pseudomonas aeruginosa, Alcaligenes sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', and Bacillus sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', petroleum hydrocarbon-degrading microorganisms’ area contaminated with petroleum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' And also, the study of (Szczepaniak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2015) demonstrated Petroleum hydrocarbons are compounds which undergo decomposition in soil due to the activity of several groups of microorganisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Several different microbial species participate in the biodegradation of hydrocarbons, ranging from strictly aerobic to strict anaerobic bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Several Gram-positive (Rhodococcus or Bacillus) as well as Gram-negative (Alcaligenes, Acinetobacter, Pseudomonas) species are also characterized by a relatively broad substrate spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Five fungi isolates were isolated from the 19 soil samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The expected size of PCR amplicons (1200 bp) was detected successfully in the all five samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' No PCR products were detected in the negative control as shown in (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Therefore, to track down each fungal isolate to its exact species, the PCR amplicons of each isolate were sent out for sequencing using the forward (LROR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Thus, the isolate which was isolated from group A was Aspergillus lentulus 28S ribosomal RNA with accession number XR 004500616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, and the isolates which were isolated from group B were Aspergillus fellis strain FM324 chromosome 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' with accession number CP066505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, Aspergillus luteonubrus strain MST FP2246 with accession number MT196912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1 and Aspergillus arizonicus isolate CCF 5341 with accession number OK321187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The isolate which was isolated from group C was Rhizopus arrhizus Strain SC49B03 with accession number MW113537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The study of (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2018) demonstrated that the ten fungal isolates chosen were able to grow and degrade crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The isolate Penicillium citrinum NIOSN-M126 isolated from Divar mangrove sediments demonstrated the greatest ability to utilize crude oil, followed by Aspergillus flavus NIOSN-SK56S22 isolated from Arabian Sea sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Degradation of crude oil and crude oil fractions by bacterial isolates and fungi isolates: 1 2 3 4 5 6 7 8 7 a b C Figure (3-4) Bacterial isolates that grow on MSM supplemented with 1% crude oil Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Partial amplification of ITS using (LROR)/(LR6) primers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Lanes 1 and 2 represent a 100bp DNA marker (DENA 100 bp plus DNA size marker II S-5091), a negative control that has been run without a DNA template, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Lanes 3 through 7 shows ~1200 bp of PCR amplicons generated using a DNA template from number 1 through 5 fungal samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Bacteria: strains were isolated for their capacity to grow in the presence of crude oil and the fractions of crude oil used in the investigation for detecting which bacterial isolates have a high ability to degrade crude oil and the fractions of crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Two methods were used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' spectrophotometer method: The results showed that growth was determined through measuring the absorbency at wavelength 600 nm by adding 1% washed bacterial cell reading OD at 600 nm was equivalent to 1 and 1% crude oil and incubated at 30°C in a shaker incubator 150rpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Then, the results showed that no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (3, 4, 5, 6, 7, 8, 9, 10,11, 12, 13,14, and 16) the OD reading after 24 hrs were (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='43, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='68, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='61, o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='49, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='52, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='68, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='46, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='73, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='60) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' However, the OD reading after 48 hrs was increased in some of the sample’s no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (4, 6, 7, 8, 11, 12, and 14) the OD reading at 600 nm was (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='56, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='32, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='69, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='57, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='32, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='79, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='92) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Meanwhile, the OD reading was decreased in other samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Then, the results were shown after one week the samples number (4, 6, 7, 8, 11, 12, and 14) was recorded OD reading (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='283, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='299, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='4271, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='3544, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='2947, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='3253, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='5793) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Furthermore, after three weeks number (6, 7, 8, 12 and 14) OD reading was (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='21, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='34, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='24, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='21, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='43) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Finally, after 8 weeks all the isolates decreased due to decreasing carbon sources and nutrients as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Therefore, this investigation concluded that the most potent isolates for resisting crude oil and growing well in minimal media supplemented by crude oil were (Pseudomonas aeruginosa, Achromobacter sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' and Bacillus anthracis, Bacillus cereus, Aneurinibacillus migulanus, and Brevibacillus borstelensis) and (Figure 1) showed bacterial colony on solid MSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Degradation hydrocarbon fractions (F1, F2, F3, and F4) were managed by a simple distiller at the laboratory and to detect hydrocarbon degradation spectrophotometer method was used for the samples which degrade crude oil faster than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Then, the number (4, 6,7, 8,11,12,14, 16 and 17) were undergone to this test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' And the results showed that (F2 and F4) were consumed by the isolates more than other fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The OD reading for isolates that consume and grow well in F2 were (4,6, 7,12,14, and 17) the OD reading were (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='312, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='334, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='456, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='594, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='482) respectively, Furthermore, numbers (8,11 and 17) were consumed fraction 4 (F4) more than the other fractions, the OD reading were (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='357, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='3320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='334) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The bacterial growth was increased until after four weeks decrease the OD reading and growth of bacteria for all the isolates were caused by decreasing the carbon source in the media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2005b) showed in their study that Autochthonous microorganisms in sediments also possessed satisfactory Polycyclic aromatic hydrocarbons (PAHs) degradation capability and all three PAH were completely degraded after 4 weeks of growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Bacterial isolates that grown on MSM supplemented with 1% crude oil a- Bacillus cereus b- Achromobacter sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' c- Pseudomonas sp A 2,6-Dichlorophenol indophenol (DCPIP) method: This method also was used to assess the ability of bacterial isolates to degrade crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The suspension was prepared as mentioned in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' When compared to the control, the change in colour of the inoculation degradation medium from deep blue to colourless demonstrated the ability of the bacterial isolates to degrade crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Then, the results showed that numbers (7, 8, 11, 12, and 14) have a high ability to reduce the colour completely to colourless after 4 days, and numbers (4, 6, and 16) also can reduce the colour but less than the numbers mentioned above as shown in (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Many studies agree with the present study for the biodegradation of crude oil in vitro by using different methods and different concentrations of crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Ramdass and Rampersad, (2021) showed crude oil degradation by bacteria, fungi and yeast and they were concluded 2% of crude oil were mixed with two media types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' the result indicated that all microbes recovered were able to utilize crude oil on both media types in vitro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Rizi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2012) reported gram-positive bacteria such as Bacillus cereus and Bacillus subtilis were predominant in degrading crude oil this paralleled with the present study that Bacillus cereus and Bacillus spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' the most predominant in degrading crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' while, the results showed the colour does not change for all the fractions (F1, F2, F3, and F4) separately for bacterial isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Spectrophotometer readings for bacterial isolates that cause high turbidity in MSM when 1% crude oil is added Fungi: Five oil-degrading fungi were isolated from 19 samples of crude oil-polluted soil and control samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Two methods were used to detect crude oil degradation by fungi isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' of isolates After 24 hrs After 48 hrs After one week After 3 weeks After 8 weeks 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='5028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0001 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='4352 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0001 9 17 16 15 14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 12 11 10 9 8 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 5 4 3 2 1 C Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Ability of the bacterial isolates which start from control © followed by number 1 through 17 to reduce 2,6-Dichlorophenol indophenol (DCPIP) by changing the colour from blue to colourless after 4 days Mycelia radial growth measurement on minimal salt medium (MSM): Five isolated fungi were screened for their ability to tolerate crude oil or petroleum hydrocarbons from 0 to 20% concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' All the isolated fungi showed to have a high ability to tolerate different crude oil concentrations as well as in the control treatment (without crude oil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Rhizopus arrhizus exhibited the highest growth rate, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='8 cm/day at 0%, as well as (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='6, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='6, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1) cm/day for crude oil concentrations of (5,10,15, and 20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Aspergillus lentulus presented the second highest growth rate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='0) cm/ day, followed by other isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The experiment proved that all five strains were capable to grow in crude oil by using crude oil as the sole carbon source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (Al- Zaban, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', 2021) who isolated four fungi isolates and measured mycelia radial growth on minimal medium (MM) supplemented with crude oil at different concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' All isolates grew faster in the control treatment (without crude oil) than in the other treatments supplemented with varying concentrations of crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' However, he observed remarkable adaptation and were able to survive high concentrations of crude oil of up to 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Reyes-César, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2014) results showed the high biodegradation potential of the Talaromyces spectabilis CCS12 strain and it can be used in the development of novel green processes due to its strong ability to metabolize PAHs in soil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' While the present study showed Rhizopus arrhizus which isolate from a control sample has high biodegradation potential on MSM solid media added with up to 20% of crude oil and 1% of four fractions separately after three days of incubation and this novel and a new record for this genus and followed by isolating Aspergillus lentulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' The 2,6-Dichlorophenol Indophenol (DCPIP) method was used to evaluate the ability of selected fungal strains to degrade crude oil and the hydrocarbon fractions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' the results showed that Rhizopus arrhizus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' and Aspergillus lentulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' have a strong ability to degrade crude oil and reduce the colour from deep blue to colourless and consequently, degrade the F1 fraction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' While other fungi isolates can degrade crude oil and change the blue colour to colourless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' However, the (DCPIP) colour does not change and remains the blue colour for the fractions when compared with the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Al-Hawash, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2018) demonstrated that two strains of Penicillium sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' RMA1 and RMA2 were isolated from the Rumaila oil field and can change the colour gradually from deep blue to colourless, and this reaction suggested that Penicillium sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' RMA1 and RMA2 could degrade crude oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=" Environmental pollution caused by oil spillage is one of the century's major issues that must be resolved." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Biodegradation can emerge or limit this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Conclusion 12F3 12F2 3F 10 0% 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 10% 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 20% Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Macroscopic images of radial growth of isolated fungal strains, in Petri dishes with different concentrations of crude oil (0% to 20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Line 1 is Aspergillus lentulus, line 2 Aspergillus fellis, line 3 Aspergillus luteonubrus, line 4 Aspergillus arizonicus, and line 5 is Rhizopus arrhizus This study concluded that different microorganism isolates survive and grow well in crude oil and were affiliated with Bacillus anthracis, Bacillus cereus, Achromobacter sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', and pseudomonas aeroginosa for bacterial isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Moreover, Rhizopus arrhizus and Aspergillus lentulus for fungi isolates, this study showed these microorganisms have a strong ability to degrade crude oil and can be used to clean up soil and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' However, the longitudinal investigation will determine which enzyme is responsible for the degradation of crude petroleum and hydrocarbons, as well as whether the transcriptome encoding this enzyme is expressed by adding crude oil or if it is expressed simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' And more studies are needed to identify other microorganisms in the Kurdistan Region with the ability to degrade crude oil, as well as to use a mixed culture of bacteria and fungi to observe their synergistic ability to degrade hydrocarbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' References Al-Zaban, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', AlHarbi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' and Mahmoud, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2021) Hydrocarbon biodegradation and transcriptome responses of cellulase, peroxidase, and laccase encoding genes inhabiting rhizospheric fungal isolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Saudi Journal of Biological Sciences, 28(4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 2083-2090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Al-Hawash, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', Alkooranee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', Abbood, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=', Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' and Ma, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' (2018) Isolation and characterization of two crude oil-degrading fungi strains from Rumaila oil field, Iraq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Biotechnology reports, 1 (17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' 104-109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} +page_content=' Anaisell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtFAT4oBgHgl3EQf2h54/content/2301.08715v1.pdf'} 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tunneling out of a shallow confinement potential +Austris Akmentinsh,1 David Reifert,2 Thomas Weimann,2 +Klaus Pierz,2 Vyacheslavs Kashcheyevs,1 and Niels Ubbelohde2 +1Department of Physics, University of Latvia, 3 Jelgavas street, LV-1004 Riga, Latvia +2Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany +The ability to tune quantum tunneling is key for achieving selectivity in manipulation of individual +particles in quantum technology applications. In this work we count electron escape events out of +a time-dependent confinement potential, realized as a dynamic quantum dot in a GaAs/AlGaAs +heterostructure. A universal scaling relation of the escape probability as a function of potential +barrier rise time and depth is established and developed as a method to probe tunneling rates +over many orders of magnitude reaching limits of WKB approximation as the anharmonicity of +shallow confinement becomes relevant. Crossover to thermally activated transport is used to estimate +the single time-energy scale of the universal model. In application to metrological single electron +sources, in-situ calibrated control signals greatly extend the accessible dynamical range for probing +the quantization mechanism. Validation of the cubic potential approximation sets a foundation for +microscopic modeling of quantum tunneling devices in the shallow confinement regime. +Tunneling through a potential barrier is a hallmark +quantum phenomenon and a fundamental element for +evolving quantum technologies, enabling new and lim- +iting existing applications [1–3]. The exponential sensi- +tivity of tunneling rates to the barrier shape provides dis- +criminative power for quantum state initialization [4, 5], +read out [6], or quantum logic operations [7]. The full +range of tunneling rates that can be exploited by a par- +ticular technology is limited by time-energy uncertainty +as the minimal barrier height of a confining potential +sets the corresponding maximal tunneling rate. Identify- +ing this shallow confinement limit in microscopic tunnel- +ing of single electrons from tunable-barrier semiconduc- +tor quantum dots is the main goal of the present study. In +quantum metrology, the limits of exponential selectivity +determine precision and capability of direct realizations +of the primary current standard employing tunneling de- +vices [3, 5, 8]. +The generic abstraction in form of the +escape from a metastable state [9–11] has been key in +diverse fields, in particular macroscopic quantum tun- +neling for the development of superconducting quantum +technologies [12–14] or studies of Bose-Einstein conden- +sates [15, 16]. Here we map out rates for the last electron +escape [17] from an emerging quantum dot over many +orders of magnitude. Varying the rise time of the confin- +ing potential barrier, we employ single electron detection +to accurately count electron tunneling events. A scaling +relation for the inferred electron capture probability al- +lows to stitch together data spanning several decades of +driving speed variation yielding a method to validate the +charge capture mechanism over a greatly extended pa- +rameter range. Crossover to activated transport is used +to estimate the single energy scale which limits the max- +imal attainable tunneling rate. A minimal microscopic +model of ground state tunneling from an anharmonic +confinement potential predicts a universal scaling curve +down to the limit of shallow confinement, consistent with +experimental observations. +ω0 +−100 +0 +100 +x (nm) +−4 +−2 +0 +2 +4 +6 +8 +V (meV) +u = 0 +u = 0.5 +u = 3.0 +Vb +t +ti +t0 +-VS +−10 +0 +10 +20 +30 +D +101 +102 +103 +104 +105 +Counts +N = 0 +N = 1 +VD +VS +D +a) +b) +c) +−80 +−70 +0.00 +0.25 +0.50 +0.75 +1.00 +〈N〉 +VD (mV) +10 +−3 +10 +−2 +10 +− 1 +10 +0 +s (mV/ps) +FIG. 1. +(a) Conceptual sketch of the experiment and the +model: cubic potential V (x, t) for longitudinal confinement +induced by two gate voltages VD and VS(t) (linearly ramped, +inset), shown here for three values of dimensionless depth +u(t), with resonance energies and ground-state probability +densities of the corresponding anharmonic oscillator, com- +puted with complex dilation method. +The zero level is +fixed to the Fermi energy of the source by setting V0(t) = +µ(t) − E0(t) + Vb(t)/2. (b) Example counting histogram of +detector signal in units of average detector noise (standard +deviation). (c) Shift of the capture probability ⟨N⟩ as a func- +tion of VD due to variation of the ramp rate s. +In the experiment, the quantum dot (QD) potential is +defined in a GaAs/AlGaAs heterostructure by a shallow +etched mesa channel for confinement in the transverse +direction, and two metallic top gates inducing two tun- +neling barriers for longitudinal confinement [18]. +The +corresponding gate voltages, VS and VD, are tuned such +that a very shallow QD can emerge from the source lead +arXiv:2301.11295v1 [cond-mat.mes-hall] 26 Jan 2023 + +depending on the source gate setting VS, while being iso- +lated from the drain lead at all times. A linear voltage +ramp VS(t) = −s t, generated by a filtered digital wave- +form, is added to the source gate, while VD remains con- +stant. As shown in Fig. 1a by a sequence of snapshots, +the voltage ramp deforms the potential to raise the source +tunneling barrier, which then gradually grows and even- +tually isolates the newly created QD from the source lead +with a decoupling speed proportional to the ramp rate s. +To measure the capture probability ⟨N⟩ (only N = 0 +and N = 1 are considered for the number of captured +elementary charges N) with high accuracy at different +ramp rates, a high fidelity counting scheme is employed +[19], where the charge state of a large island serving as +the source lead is read out by a capacitively coupled de- +tector dot. Statistics for N (see Fig. 1(b)) is obtained, +by repeatedly executing a sequence of two charge state +measurements before and after a single decoupling cycle, +followed by a reset operation in which the island is briefly +connected to the ground potential. The overall repetition +rate of this sequence is 335 Hz which ensures a negligible +readout error due to the long integration time indepen- +dent of the value of s. Experiments were performed at +a base temperature of 20 mK and without a magnetic +field applied. Fig. 1(c) shows the measured ⟨N⟩ for vari- +ous values of s spanning several orders of magnitude. As +a function of VD, ⟨N⟩ transitions from 0 to 1. With in- +creasing decoupling speed, this transition appears shifted +towards more negative VD and therefore towards a shal- +lower potential and more strongly coupled QD, consistent +with the logarithmic rise-time dependence [4] observed in +a Si-based single-electron ratchet [20]. +Our baseline approximation to model ⟨N⟩ relies on +time-scale separation [21, 22] between transition rates, +which are assumed to follow instantaneously the driving +parameters, and the occupation probability P(t) of the +QD, which is eventually taken out of equilibrium with +the leads [4, 23]. The corresponding minimal rate equa- +tion [22, 24, 25] is dP/dt = Γin[1 − P] − Γout P where +Γin(t) and Γout(t) are the charging and the discharging +rates for the QD occupation number N = 0 ↔ 1 (higher +charge states N > 1 are neglected). +The detailed balance condition defines thermodynamic +activity, Γout(t)/Γin(t) = exp[µ(t)/kTL], expressed via +the electrochemical potential difference µ(t) between the +QD and the source lead at temperature TL. Large dµ/dt +makes the moment t0 for the onset of backtunneling +(lifting of Pauli blockage, µ(t0) = 0) well-defined; the +dot switches from Γin ≫ Γout to Γin ≪ Γout within +t ∈ [t0 − δt, t0 + δt] over a timescale δt = kTL/ ˙µ much +shorter than the timescale τ = −(d ln Γout/dt)−1 for the +subsequent decay of Γout(t) at t > t0 + δt [26]. In this +limit, the final occupation probability, ⟨N⟩ = P(t → ∞), +is determined only by the integral of the escape rate +Γout(t) from t0 till the QD is effectively disconnected, +Γout(∞) = 0, +⟨N⟩ = +∞ +� +−∞ +e +− +∞ +� +t +(Γin+Γout)dt′ d +dt +� +−Γin +Γin + Γout +� +dt +≈ e− +� ∞ +t0 Γout dt +(1) +as the fraction under the integral in Eq. (1) behaves as a +delta function δ(t − t0) of width δt ≪ τ. Here we assume +a fully occupied QD before the escape of the last electron +is triggered, P(t → −∞) = 1, by taking the formal limit +� ∞ +−∞[Γint′) + Γout(t′)]dt′ → ∞ in the exact solution to +the rate equation. +Since the time dependence of Γout and µ is induced +by a single parameter [25], the source voltage VS(t), the +integrands in Eq. (1) go through the same set of values +on the time axis but with different rates s′. The param- +eter s′ is determined by the nominal voltage ramp rate +s but accounts for imperfections in signal transmission. +Specifically, for a pair of ramp rates si and sj we expect +the corresponding escape rates to be related (up to an +irrelevant global shift in t) as Γ(i) +out(t) = Γ(j) +out(s′ +i t/s′ +j) if +all other external parameters are kept equal. With this, +Eq. (1) implies a testable scaling relation +[⟨Ni⟩(VD)]s′ +i = [⟨Nj⟩(VD)]s′ +j . +(2) +In particular, Eq. (2) must hold for any VD regard- +less of the functional dependence of Γout on voltages +(which we analyze and model later). In order to robustly +test Eq. (2) and infer s′(s), we first estimate the expo- +nent matrix mij = s′ +j/s′ +i for the predicted power law +⟨Ni⟩ = ⟨Nj⟩mij averaging over data points with differ- +ent VD but common (si, sj). +Diagonalizing mij yields +a single dominating eigenvalue with the corresponding +eigenvector proportional to the set of s′ +i. +The data collapse of all ⟨Ni⟩ for two example data +sets (filtered and unfiltered voltage ramps) is confirmed +in Fig. 2 by plotting −ln⟨Nj⟩, j > 1, each offset by a +factor s′ +j/s′ +1, on top of −ln⟨N1⟩. This procedure estab- +lishes the relation (insets of Fig. 2) between the nominal +ramp rates si and the inferred decoupling speeds s′ +i up +to overall normalization (which is fixed later using a mi- +croscopic model). This result confirms that Γout follows +instantaneously a single parameter which is controlled by +the external voltage ramp; the visible discretization steps +of the unfiltered waveform in the inset of Fig. 2(b) un- +derline the sensitivity of the method. Establishing this +link to external control voltages and validating the func- +tion of key drive parameters despite signal distortions is +essential for employing modulated tunneling barriers in +high-speed nanoscale devices [27, 28]. While at any given +parameter setting of s finite measurement time limits the +accurate estimation of ⟨N⟩ due to the rarity of events +as ⟨N⟩ approaches 0 or 1 with VD, the single scaling +curve empirically stitched together from measurements + +u0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +−90 +−80 +−70 +−60 +10 +−9 +10 +−8 +10 +−7 +10 +−6 +10 +−5 +10 +−4 +10 +−3 +10 +−2 +10 +−1 +M +10 +−3 +10 +−2 +10 +−1 +10 +0 +10 +−6 +10 +−5 +10 +−4 +10 +−3 +VD (mV) +s (mV/ps) +VD +c +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +−90 +−80 +−70 +−60 +10 +−9 +10 +−8 +10 +−7 +10 +−6 +10 +−5 +10 +−4 +10 +−3 +10 +−2 +10 +−1 +M +10 +−2 +10 +−1 +10 +0 +s (mV/ps) +10 +−4 +10 +−3 +u0 +VD (mV) +VD +c +a) +b) +FIG. 2. Data collapse of capture probability plotted as −( ˙u/ω0)i ln⟨Ni⟩ (symbols) as function VD for filtered (a) and unfiltered +(b) digital voltage ramps (s color mapped). Data points ⟨Nj⟩ for j > 1 appear shifted (multiplied) by s′ +j/s′ +1 = ( ˙u/ω0)j/( ˙u/ω0)1 +relative to the reference ⟨N1⟩. The dotted lines represent a fit to the universal scaling curve M(u0) given by Eq. (3) with the +inferred u0(VD) marked on the upper axes. The absolute values of the decoupling speeds ˙u/ω0 as function of the nominal ramp +rate s are shown in the inset. The intervals of u0/VD measurable for each s-parameter are indicated at the upper horizontal +axis. +at different decoupling speeds now allows to probe the +mechanism behind capture over a much larger parameter +range. In the application of such tunable barrier devices +as a quantum standard [4], verifying the robustness of +current quantization is crucial. Inferring ⟨N⟩ from the +backtunneling rate as in Eq. (2) has been widely used in +modelling the fidelity of charge capture [4, 5, 20, 23, 29– +31], yet so far relied on the linearization of ln Γout over the +relevant range of voltages. The empirical scaling curve +however shows this linearity to be violated at high speeds, +i.e. ln(− ln⟨N⟩) not linear in VD in Fig. 2. +Evidently, with increasingly negative VD we observe +the exponential growth of the integrated tunneling rate +to be capped by the disappearing source barrier to an in- +creasingly shallow dot. The confinement emerges near +the stationary inflection point of the potential energy +for which a cubic approximation is generic [11, 12, 32]. +Hence we model the microscopic potential of the QD +as V (x, t) = b x3/3 − F(t) x + V0(t), where x is mea- +sured from the uniformly moving inflection point in the +longitudinal direction and F(t) ∝ (t − ti) is the lowest +order term to capture the transition at time ti mark- +ing the formation of a barrier and a well. The starting +shape V (x, t0) at the onset of backtunneling is set by +F(t0) ∝ (t0 − ti) which is similarly assumed to be lin- +ear in the tuning gate voltage VD. Motion in the trans- +verse direction is assumed to be confined to lowest energy +mode and decoupled from x. Expansion near the inflec- +tion point implies non-trivial power laws for the barrier +height Vb = 4F 3/2b−1/2/3 ∝ (t − ti)3/2 and the linear os- +cillation frequency ω0 = (2/m)1/2(Fb)1/4 ∝ (t−ti)1/4 in- +stead of Vb ∝ t and ω0 = const often assumed [20, 30, 33] +for deeper dots (here m is the effective mass). +We +present the results in terms of a dimensionless depth, +u = Vb/(ℏω0), which counts the number of well-localized +quasibound states, and the time-independent, device- +specific Ωb = ω0 u−1/5 which sets the absolute frequency +and energy scales. The time-independent speed parame- +ter, ˙u/ω0 ∝ s′, sets an absolute scale for the decoupling +speed s′ for which the relative scale was determined ear- +lier. +In the low-temperature, quantum adiabatic modula- +tion limit we equate Γout in Eq. (1) to the decay rate Γ0 +of the resonance with the lowest real part En = E0 of the +corresponding complex energy eigenvalues En − iℏΓn/2 +of the cubic potential [34]. Γ0/ω0 is a universal function +of u (computed numerically using the complex dilation +method [35, 36]) which extrapolates non-perturbatively +the commonly used [12, 37] WKB rate ΓWKB +0 +/ω0 = +6 +� +6u/πe−36u/5 to u ∼ 1. This results in +⟨N⟩ = exp +� +−ω0 +˙u +� +u0 +[Γ0(u)/ω0(u)] du +� +, +(3) +where u0 = u(t0) is the initial depth which is sufficiently +well-defined as ˙u δt ≪ 1. +The linear relation between +F(t0) and VD gives the power-law u0 = [˜α (VD − V c +D)]5/4. +Here V c +D, ˜α and ˙u/ω0 ∝ s′ constitute fitting parame- +ters mapping the sample-specific function ⟨N⟩(VD, s′) to +a parameter-free scaling curve M(u0) ≡ −( ˙u/ω0) ln⟨N⟩ +with ⟨N⟩ as function of u0 given by Eq. (3). In Fig. 2 a +fit of Eq. (3) (dashed black lines) to the empirical scaling +shows excellent agreement with the experiment over the + +full range of probed ramp rates, including the shallow +confinement regime of u0 < 1, strong evidence for the +anharmonicity of the driven oscillator model. The shape +of M(u0) is generically derived from a ground state back- +tunneling model and contains no device-specific param- +eters. Hence experimental validation of its universality +(Fig. 2a versus 2b, see more in Fig. 4b) provides evidence +of the fundamental microscopic mechanism of electron es- +cape in contrast to the phenomenological decay cascade +model [5, 23]. The energy gap protecting this universality +is set by the device-specific scale Ωb which we estimate +in the following. +For sufficiently fast decoupling speeds, non-adiabatic +effects, such as intradot excitation [33, 38, 39] and non- +Markovian effective temperature [40–42] are predicted to +modify escape dynamics beyond tunneling out of the +ground state. +In our experiment, the quantum adia- +baticity is maintained as the increased decoupling speed +shifts the transition ⟨N⟩ = 0 ↔ 1 into the shallow limit +(u0 < 1), hence we use thermal activation in the regime +with several quasibound states (u0 > 1) to probe the +excitation spectrum and thus estimate the energy scale +ℏΩb. +Temperature-dependent broadening of the Coulomb +resonances used to read out the charge state and infer +the capture probability limits the temperature range ac- +cessible to counting to T < 2 K. Up to this temperature +however, no discernible change of the capture probability +can be observed. Higher temperatures are therefore mea- +sured using a precision current amplifier [43], detecting +the continuous current at a fixed ramp rate of 0.1 mV/ps +as the captured electrons are emitted towards the drain +by further raising the potential [4]. Fig. 3 shows ln⟨N⟩ +for temperatures up to 6 K shifted by the inferred s′ as in +Fig. 2. In comparison with the baseline of the counting +measurement (black dashed line), the good agreement +with the lowest temperatures validates the consistency +between the different measurement techniques. Further- +more, a clear crossover temperature T0[44] can be iden- +tified on the plateau, ⟨N⟩ → 1, a distinct qualitative dif- +ference to behaviour observed in the tail (VD < −75 mV +in the inset). Above this crossover the data points de- +viate from the universal scaling curve, which conversely +corroborates the ground state interpretation of the base- +temperature data and contradicts speed-dependent heat- +ing. +In order to describe the decay of the metastabil- +ity at finite temperatures, thermally activated escape +via states near the top of the barrier has to be in- +cluded in addition to tunneling out of the ground state +[11, 32, 45]. +We model the crossover at the level of +quantum transition state theory [11] without an ex- +plicit model for a heat bath: +a Boltzmann distribu- +tion with a temperature T = 1/(kBβ) controls the av- +erage over discrete resonances with decay rates Γn at +energies En ≲ Vb and a continuum above the bar- +1.0 +1.5 +2.0 +2.5 +3.0 +u0 +−80 +−75 +−70 +−65 +−60 +VD (mV) +10 +−6 +10 +−5 +10 +−4 +10 +−3 +10 +−2 +10 +−1 +M +1.5 +6.0 +−80 +−60 +VD (mV) +0 +1 +〈N〉 +1 +2 +3 +u0 +10 +−5 +10 +−3 +10 +−1 +M +1 +2 +3 +u0 +10 +−5 +10 +−3 +10 +−1 +M +0.25 +0.50 +1/T (1/K) +10 +−6 +10 +−4 +M +(i) +(ii) +a) +b) +c) +T (K) +FIG. 3. (a) Capture probability as a function of VD, inferred +from current measurements, showing thermally activated es- +cape as T is increased up to 6 K. The data points are shifted +as in Fig. 2 matching the universal scaling curve (the dotted +line). The dashed dotted line shows the model prediction with +ground state escape rate replaced by that of the first excited +state. Lower left inset: unscaled data, top right inset: Arrhe- +nius plot for u0 ≈ 2.3 compared to activated-transport models +(i) and (ii) with ℏΩb = 1.6 meV corresponding to T0 ≃ 2.9 K. +The model predictions as functions of u0 are shown in (b) and +(c) for fast (i) and slow (ii) thermalization limit, respectively. +rier, with decay rate density [45] (2πℏ)−1T (E) where +T (E) = 1/ {1 + exp [2π(Vb − E)/(ℏω0)]} is the transmis- +sion coefficient in a quadratic approximation. We use the +exact solution for the cubic potential to match the semi- +classical phase space weights between these two energy +ranges as follows [36]: +Z⟨Γ⟩ = +nb +� +n=0 +Γn ane−βEn + +� ∞ +Vb +T (E) e−βE dE +2πℏ , +(4) +Z = +nb +� +n=0 +ane−βEn + +� ∞ +Vb +ρ(E)e−βE dE . +(5) +Here the number of quantum states nb + anb += +A(u)/(2πℏ) = 18u/(5π) in the phase space area A(u) +that is classically confined (0 < E < Vb) is split into the + +integer (nb) and the fractional (0 ≤ anb < 1) parts such +that an = 1 for n < nb, and ρ(E) is the semiclassical den- +sity of states above the barrier (E > Vb) yet inside the +QD (x contributing from the potential maximum towards +the dot) [36]. +The model of the thermally activated escape from a +fixed-depth potential summarized above needs to be in- +corporated into the time-dependent problem of charge +capture. Here the decoupling timescale competes with +the thermalization time. We contrast two opposite ex- +tremes: (i) fast thermalization with respect to decou- +pling [30], where Γout(t) is replaced by ⟨Γ⟩ from Eq. +(4) with time-dependent depth u(t); (ii) slow thermaliza- +tion, where the number of confined levels nb and the dis- +crete weights ane−βEn/Z are frozen at the initial depth +u = u0 and then used in the averaging of P(∞) over +n with Γout(t) → Γn(t), neglecting the continuum con- +tribution. Fig. 3(b) and (c) shows the simulation results +for both limiting cases, which are compared to the exper- +iment at u0 ≈ 2.3 an Arrhenius plot in the inset. While +the intradot population dynamics of the thermally ex- +cited states appear to significantly affect the results, both +models reproduce the crossover on the plateau, where the +decline of the thermal excitation weight with energy no +longer outweighs the competing growth of the escape rate +(crossover from tunneling to hopping [45]). From the ex- +perimental data, we infer a value of ℏΩb = 1.6 meV and +hence fix the device-specific microscopic potential (see +also Fig. 1). With kBT0 ≃ 0.16 ℏΩb [36] this corresponds +to a crossover temperature of T0 ≃ 2.9 K. +In conclusion, matching of the counting data to the +universal scaling relation over a broad range of decou- +pling speeds (Fig. 2) and the estimation of Ωb from the +temperature dependence (inset in Fig. 3) provides an ex- +perimental technique for inferring the ground state es- +cape rate Γ0 and the barrier height Vb in physical units +down to shallow limit where Vb/ℏ ∼ Γ0 ∼ Ωb and the +confinement is eventually lost, e.g. E0 > Vb for u < 0.42, +see Fig. 4(a). Using the universal scaling curve, differ- +ent measurements can be combined in a single speed- +depth plot in Fig. 4(b), showing the inferred initial depth +u0 versus the speed ˙u/ω0 at fixed ⟨N⟩ compared to +level lines of Eq. (3). The dashed lines show two lim- +its for adiabatic electron capture: loss of confinement +(u0 < 0.42) or breakdown of adiabatic condition (which +requires small relative change in frequency over one pe- +riod, ( ˙ω0/ω0)(2π/ω0) ≪ 1 ⇒ ˙u/ω0 ≪ 5u). For ⟨N⟩ → 1 +the capture fidelity 1 − ⟨N⟩ is limited by non-adiabatic +effects, while for ⟨N⟩ → 0 the residual capture probabil- +ity is capped by the saturation of the escape rate as the +confinement is lost. +For quantum metrology applications that rely on the +escape of excess electrons and capture of the target num- +ber of electrons, this work introduces capability-defining +frequency limits for control of tunneling [3, 46]. +The +speed-depth scaling method presented here provides a +u0 +VD (mV) +Γ0 (GHz) +Vb (meV) +0 +1 +2 +3 +−90 +−80 +−70 +−60 +10 +−4 +10 +−2 +10 +0 +10 +2 +0 +1 +2 +3 +4 +5 +Γ0 +Vb +0.001 +0.010 +0.100 +0.500 +0.900 +0.990 +0.999 +10 +−7 10 +−5 10 +−3 10 +−1 +0 +1 +2 +3 +u0 +a) +b) +FIG. 4. (a) Inferred ground state tunneling rate Γ0 as a func- +tion of gate voltage compared to the barrier height Vb (width +of the line indicates the level broadening ℏΓ0). (b) Combined +speed-depth diagram: initial depth u0 is plotted as a function +of speed parameter ˙u/ω0 for levels of constant ⟨N⟩ and com- +pared to the universal scaling given by Eq. (3) (solid black +lines). Various experimental realizations (symbols) are com- +bined into a single diagram: a different QD implementation +with sinusoidal driving waveform (upwards green triangles), +linear ramp waveform as in Fig. 2a and b (blue circle, orange +square), and a set of linear ramp waveforms focused on slow +decoupling speeds (downward red triangles). 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Timoshenko, Quantum fluctua- +tions and coherence in high-precision single-electron cap- +ture, Physical Review Letters 109, 216801 (2012). +[42] I. de Vega and D. Alonso, Dynamics of non-markovian +open quantum systems, Reviews of Modern Physics 89, +015001 (2017). +[43] D. Drung, C. Krause, U. Becker, H. Scherer, and F. J. +Ahlers, Ultrastable low-noise current amplifier: A novel +device for measuring small electric currents with high +accuracy, Review of Scientific Instruments 86, 024703 +(2015). +[44] K. A. Matveev and L. I. Glazman, Coulomb blockade +of activated conduction, Physical Review B 54, 10339 +(1996). +[45] I. Affleck, Quantum-Statistical Metastability, Physical +Review Letters 46, 388 (1981). +[46] M. Kataoka, Single-electron sources, in Semiconductor +Nanodevices, edited by D. A. Ritchie (Elsevier, 2021) +Chap. 5, pp. 101–145. + diff --git a/X9FIT4oBgHgl3EQfjCvs/content/tmp_files/load_file.txt b/X9FIT4oBgHgl3EQfjCvs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..975f5154f1ac4d8118a04169943c9fe1e4658202 --- /dev/null +++ b/X9FIT4oBgHgl3EQfjCvs/content/tmp_files/load_file.txt @@ -0,0 +1,588 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf,len=587 +page_content='Universal scaling of adiabatic tunneling out of a shallow confinement potential Austris Akmentinsh,1 David Reifert,2 Thomas Weimann,2 Klaus Pierz,2 Vyacheslavs Kashcheyevs,1 and Niels Ubbelohde2 1Department of Physics, University of Latvia, 3 Jelgavas street, LV-1004 Riga, Latvia 2Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany The ability to tune quantum tunneling is key for achieving selectivity in manipulation of individual particles in quantum technology applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In this work we count electron escape events out of a time-dependent confinement potential, realized as a dynamic quantum dot in a GaAs/AlGaAs heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' A universal scaling relation of the escape probability as a function of potential barrier rise time and depth is established and developed as a method to probe tunneling rates over many orders of magnitude reaching limits of WKB approximation as the anharmonicity of shallow confinement becomes relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Crossover to thermally activated transport is used to estimate the single time-energy scale of the universal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In application to metrological single electron sources, in-situ calibrated control signals greatly extend the accessible dynamical range for probing the quantization mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Validation of the cubic potential approximation sets a foundation for microscopic modeling of quantum tunneling devices in the shallow confinement regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Tunneling through a potential barrier is a hallmark quantum phenomenon and a fundamental element for evolving quantum technologies, enabling new and lim- iting existing applications [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The exponential sensi- tivity of tunneling rates to the barrier shape provides dis- criminative power for quantum state initialization [4, 5], read out [6], or quantum logic operations [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The full range of tunneling rates that can be exploited by a par- ticular technology is limited by time-energy uncertainty as the minimal barrier height of a confining potential sets the corresponding maximal tunneling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Identify- ing this shallow confinement limit in microscopic tunnel- ing of single electrons from tunable-barrier semiconduc- tor quantum dots is the main goal of the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In quantum metrology, the limits of exponential selectivity determine precision and capability of direct realizations of the primary current standard employing tunneling de- vices [3, 5, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The generic abstraction in form of the escape from a metastable state [9–11] has been key in diverse fields, in particular macroscopic quantum tun- neling for the development of superconducting quantum technologies [12–14] or studies of Bose-Einstein conden- sates [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Here we map out rates for the last electron escape [17] from an emerging quantum dot over many orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Varying the rise time of the confin- ing potential barrier, we employ single electron detection to accurately count electron tunneling events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' A scaling relation for the inferred electron capture probability al- lows to stitch together data spanning several decades of driving speed variation yielding a method to validate the charge capture mechanism over a greatly extended pa- rameter range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Crossover to activated transport is used to estimate the single energy scale which limits the max- imal attainable tunneling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' A minimal microscopic model of ground state tunneling from an anharmonic confinement potential predicts a universal scaling curve down to the limit of shallow confinement, consistent with experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' ω0 −100 0 100 x (nm) −4 −2 0 2 4 6 8 V (meV) u = 0 u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 u = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 Vb t ti t0 VS −10 0 10 20 30 D 101 102 103 104 105 Counts N = 0 N = 1 VD VS D a) b) c) −80 −70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='00 〈N〉 VD (mV) 10 −3 10 −2 10 − 1 10 0 s (mV/ps) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (a) Conceptual sketch of the experiment and the model: cubic potential V (x, t) for longitudinal confinement induced by two gate voltages VD and VS(t) (linearly ramped, inset), shown here for three values of dimensionless depth u(t), with resonance energies and ground-state probability densities of the corresponding anharmonic oscillator, com- puted with complex dilation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The zero level is fixed to the Fermi energy of the source by setting V0(t) = µ(t) − E0(t) + Vb(t)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (b) Example counting histogram of detector signal in units of average detector noise (standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (c) Shift of the capture probability ⟨N⟩ as a func- tion of VD due to variation of the ramp rate s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In the experiment, the quantum dot (QD) potential is defined in a GaAs/AlGaAs heterostructure by a shallow etched mesa channel for confinement in the transverse direction, and two metallic top gates inducing two tun- neling barriers for longitudinal confinement [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The corresponding gate voltages, VS and VD, are tuned such that a very shallow QD can emerge from the source lead arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='11295v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='mes-hall] 26 Jan 2023 depending on the source gate setting VS, while being iso- lated from the drain lead at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' A linear voltage ramp VS(t) = −s t, generated by a filtered digital wave- form, is added to the source gate, while VD remains con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 1a by a sequence of snapshots, the voltage ramp deforms the potential to raise the source tunneling barrier, which then gradually grows and even- tually isolates the newly created QD from the source lead with a decoupling speed proportional to the ramp rate s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' To measure the capture probability ⟨N⟩ (only N = 0 and N = 1 are considered for the number of captured elementary charges N) with high accuracy at different ramp rates, a high fidelity counting scheme is employed [19], where the charge state of a large island serving as the source lead is read out by a capacitively coupled de- tector dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Statistics for N (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 1(b)) is obtained, by repeatedly executing a sequence of two charge state measurements before and after a single decoupling cycle, followed by a reset operation in which the island is briefly connected to the ground potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The overall repetition rate of this sequence is 335 Hz which ensures a negligible readout error due to the long integration time indepen- dent of the value of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Experiments were performed at a base temperature of 20 mK and without a magnetic field applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 1(c) shows the measured ⟨N⟩ for vari- ous values of s spanning several orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' As a function of VD, ⟨N⟩ transitions from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' With in- creasing decoupling speed, this transition appears shifted towards more negative VD and therefore towards a shal- lower potential and more strongly coupled QD, consistent with the logarithmic rise-time dependence [4] observed in a Si-based single-electron ratchet [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Our baseline approximation to model ⟨N⟩ relies on time-scale separation [21, 22] between transition rates, which are assumed to follow instantaneously the driving parameters, and the occupation probability P(t) of the QD, which is eventually taken out of equilibrium with the leads [4, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The corresponding minimal rate equa- tion [22, 24, 25] is dP/dt = Γin[1 − P] − Γout P where Γin(t) and Γout(t) are the charging and the discharging rates for the QD occupation number N = 0 ↔ 1 (higher charge states N > 1 are neglected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The detailed balance condition defines thermodynamic activity, Γout(t)/Γin(t) = exp[µ(t)/kTL], expressed via the electrochemical potential difference µ(t) between the QD and the source lead at temperature TL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Large dµ/dt makes the moment t0 for the onset of backtunneling (lifting of Pauli blockage, µ(t0) = 0) well-defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' the dot switches from Γin ≫ Γout to Γin ≪ Γout within t ∈ [t0 − δt, t0 + δt] over a timescale δt = kTL/ ˙µ much shorter than the timescale τ = −(d ln Γout/dt)−1 for the subsequent decay of Γout(t) at t > t0 + δt [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In this limit, the final occupation probability, ⟨N⟩ = P(t → ∞), is determined only by the integral of the escape rate Γout(t) from t0 till the QD is effectively disconnected, Γout(∞) = 0, ⟨N⟩ = ∞ � −∞ e − ∞ � t (Γin+Γout)dt′ d dt � −Γin Γin + Γout � dt ≈ e− � ∞ t0 Γout dt (1) as the fraction under the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (1) behaves as a delta function δ(t − t0) of width δt ≪ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Here we assume a fully occupied QD before the escape of the last electron is triggered, P(t → −∞) = 1, by taking the formal limit � ∞ −∞[Γint′) + Γout(t′)]dt′ → ∞ in the exact solution to the rate equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Since the time dependence of Γout and µ is induced by a single parameter [25], the source voltage VS(t), the integrands in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (1) go through the same set of values on the time axis but with different rates s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The param- eter s′ is determined by the nominal voltage ramp rate s but accounts for imperfections in signal transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Specifically, for a pair of ramp rates si and sj we expect the corresponding escape rates to be related (up to an irrelevant global shift in t) as Γ(i) out(t) = Γ(j) out(s′ i t/s′ j) if all other external parameters are kept equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' With this, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (1) implies a testable scaling relation [⟨Ni⟩(VD)]s′ i = [⟨Nj⟩(VD)]s′ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (2) In particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (2) must hold for any VD regard- less of the functional dependence of Γout on voltages (which we analyze and model later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In order to robustly test Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (2) and infer s′(s), we first estimate the expo- nent matrix mij = s′ j/s′ i for the predicted power law ⟨Ni⟩ = ⟨Nj⟩mij averaging over data points with differ- ent VD but common (si, sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Diagonalizing mij yields a single dominating eigenvalue with the corresponding eigenvector proportional to the set of s′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The data collapse of all ⟨Ni⟩ for two example data sets (filtered and unfiltered voltage ramps) is confirmed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2 by plotting −ln⟨Nj⟩, j > 1, each offset by a factor s′ j/s′ 1, on top of −ln⟨N1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' This procedure estab- lishes the relation (insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2) between the nominal ramp rates si and the inferred decoupling speeds s′ i up to overall normalization (which is fixed later using a mi- croscopic model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' This result confirms that Γout follows instantaneously a single parameter which is controlled by the external voltage ramp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' the visible discretization steps of the unfiltered waveform in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2(b) un- derline the sensitivity of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Establishing this link to external control voltages and validating the func- tion of key drive parameters despite signal distortions is essential for employing modulated tunneling barriers in high-speed nanoscale devices [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' While at any given parameter setting of s finite measurement time limits the accurate estimation of ⟨N⟩ due to the rarity of events as ⟨N⟩ approaches 0 or 1 with VD, the single scaling curve empirically stitched together from measurements u0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 −90 −80 −70 −60 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 M 10 −3 10 −2 10 −1 10 0 10 −6 10 −5 10 −4 10 −3 VD (mV) s (mV/ps) VD c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 −90 −80 −70 −60 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 M 10 −2 10 −1 10 0 s (mV/ps) 10 −4 10 −3 u0 VD (mV) VD c a) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Data collapse of capture probability plotted as −( ˙u/ω0)i ln⟨Ni⟩ (symbols) as function VD for filtered (a) and unfiltered (b) digital voltage ramps (s color mapped).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Data points ⟨Nj⟩ for j > 1 appear shifted (multiplied) by s′ j/s′ 1 = ( ˙u/ω0)j/( ˙u/ω0)1 relative to the reference ⟨N1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The dotted lines represent a fit to the universal scaling curve M(u0) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (3) with the inferred u0(VD) marked on the upper axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The absolute values of the decoupling speeds ˙u/ω0 as function of the nominal ramp rate s are shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The intervals of u0/VD measurable for each s-parameter are indicated at the upper horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' at different decoupling speeds now allows to probe the mechanism behind capture over a much larger parameter range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In the application of such tunable barrier devices as a quantum standard [4], verifying the robustness of current quantization is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Inferring ⟨N⟩ from the backtunneling rate as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (2) has been widely used in modelling the fidelity of charge capture [4, 5, 20, 23, 29– 31], yet so far relied on the linearization of ln Γout over the relevant range of voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The empirical scaling curve however shows this linearity to be violated at high speeds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' ln(− ln⟨N⟩) not linear in VD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Evidently, with increasingly negative VD we observe the exponential growth of the integrated tunneling rate to be capped by the disappearing source barrier to an in- creasingly shallow dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The confinement emerges near the stationary inflection point of the potential energy for which a cubic approximation is generic [11, 12, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Hence we model the microscopic potential of the QD as V (x, t) = b x3/3 − F(t) x + V0(t), where x is mea- sured from the uniformly moving inflection point in the longitudinal direction and F(t) ∝ (t − ti) is the lowest order term to capture the transition at time ti mark- ing the formation of a barrier and a well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The starting shape V (x, t0) at the onset of backtunneling is set by F(t0) ∝ (t0 − ti) which is similarly assumed to be lin- ear in the tuning gate voltage VD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Motion in the trans- verse direction is assumed to be confined to lowest energy mode and decoupled from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Expansion near the inflec- tion point implies non-trivial power laws for the barrier height Vb = 4F 3/2b−1/2/3 ∝ (t − ti)3/2 and the linear os- cillation frequency ω0 = (2/m)1/2(Fb)1/4 ∝ (t−ti)1/4 in- stead of Vb ∝ t and ω0 = const often assumed [20, 30, 33] for deeper dots (here m is the effective mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' We present the results in terms of a dimensionless depth, u = Vb/(ℏω0), which counts the number of well-localized quasibound states, and the time-independent, device- specific Ωb = ω0 u−1/5 which sets the absolute frequency and energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The time-independent speed parame- ter, ˙u/ω0 ∝ s′, sets an absolute scale for the decoupling speed s′ for which the relative scale was determined ear- lier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In the low-temperature, quantum adiabatic modula- tion limit we equate Γout in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (1) to the decay rate Γ0 of the resonance with the lowest real part En = E0 of the corresponding complex energy eigenvalues En − iℏΓn/2 of the cubic potential [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Γ0/ω0 is a universal function of u (computed numerically using the complex dilation method [35, 36]) which extrapolates non-perturbatively the commonly used [12, 37] WKB rate ΓWKB 0 /ω0 = 6 � 6u/πe−36u/5 to u ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' This results in ⟨N⟩ = exp � −ω0 ˙u � u0 [Γ0(u)/ω0(u)] du � , (3) where u0 = u(t0) is the initial depth which is sufficiently well-defined as ˙u δt ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The linear relation between F(t0) and VD gives the power-law u0 = [˜α (VD − V c D)]5/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Here V c D, ˜α and ˙u/ω0 ∝ s′ constitute fitting parame- ters mapping the sample-specific function ⟨N⟩(VD, s′) to a parameter-free scaling curve M(u0) ≡ −( ˙u/ω0) ln⟨N⟩ with ⟨N⟩ as function of u0 given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2 a fit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (3) (dashed black lines) to the empirical scaling shows excellent agreement with the experiment over the full range of probed ramp rates, including the shallow confinement regime of u0 < 1, strong evidence for the anharmonicity of the driven oscillator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The shape of M(u0) is generically derived from a ground state back- tunneling model and contains no device-specific param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Hence experimental validation of its universality (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2a versus 2b, see more in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 4b) provides evidence of the fundamental microscopic mechanism of electron es- cape in contrast to the phenomenological decay cascade model [5, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The energy gap protecting this universality is set by the device-specific scale Ωb which we estimate in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' For sufficiently fast decoupling speeds, non-adiabatic effects, such as intradot excitation [33, 38, 39] and non- Markovian effective temperature [40–42] are predicted to modify escape dynamics beyond tunneling out of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In our experiment, the quantum adia- baticity is maintained as the increased decoupling speed shifts the transition ⟨N⟩ = 0 ↔ 1 into the shallow limit (u0 < 1), hence we use thermal activation in the regime with several quasibound states (u0 > 1) to probe the excitation spectrum and thus estimate the energy scale ℏΩb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Temperature-dependent broadening of the Coulomb resonances used to read out the charge state and infer the capture probability limits the temperature range ac- cessible to counting to T < 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Up to this temperature however, no discernible change of the capture probability can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Higher temperatures are therefore mea- sured using a precision current amplifier [43], detecting the continuous current at a fixed ramp rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='1 mV/ps as the captured electrons are emitted towards the drain by further raising the potential [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 3 shows ln⟨N⟩ for temperatures up to 6 K shifted by the inferred s′ as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In comparison with the baseline of the counting measurement (black dashed line), the good agreement with the lowest temperatures validates the consistency between the different measurement techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Further- more, a clear crossover temperature T0[44] can be iden- tified on the plateau, ⟨N⟩ → 1, a distinct qualitative dif- ference to behaviour observed in the tail (VD < −75 mV in the inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Above this crossover the data points de- viate from the universal scaling curve, which conversely corroborates the ground state interpretation of the base- temperature data and contradicts speed-dependent heat- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In order to describe the decay of the metastabil- ity at finite temperatures, thermally activated escape via states near the top of the barrier has to be in- cluded in addition to tunneling out of the ground state [11, 32, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' We model the crossover at the level of quantum transition state theory [11] without an ex- plicit model for a heat bath: a Boltzmann distribu- tion with a temperature T = 1/(kBβ) controls the av- erage over discrete resonances with decay rates Γn at energies En ≲ Vb and a continuum above the bar- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 u0 −80 −75 −70 −65 −60 VD (mV) 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='0 −80 −60 VD (mV) 0 1 〈N〉 1 2 3 u0 10 −5 10 −3 10 −1 M 1 2 3 u0 10 −5 10 −3 10 −1 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='50 1/T (1/K) 10 −6 10 −4 M (i) (ii) a) b) c) T (K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (a) Capture probability as a function of VD, inferred from current measurements, showing thermally activated es- cape as T is increased up to 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The data points are shifted as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2 matching the universal scaling curve (the dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The dashed dotted line shows the model prediction with ground state escape rate replaced by that of the first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Lower left inset: unscaled data, top right inset: Arrhe- nius plot for u0 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='3 compared to activated-transport models (i) and (ii) with ℏΩb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='6 meV corresponding to T0 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='9 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The model predictions as functions of u0 are shown in (b) and (c) for fast (i) and slow (ii) thermalization limit, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' rier, with decay rate density [45] (2πℏ)−1T (E) where T (E) = 1/ {1 + exp [2π(Vb − E)/(ℏω0)]} is the transmis- sion coefficient in a quadratic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' We use the exact solution for the cubic potential to match the semi- classical phase space weights between these two energy ranges as follows [36]: Z⟨Γ⟩ = nb � n=0 Γn ane−βEn + � ∞ Vb T (E) e−βE dE 2πℏ , (4) Z = nb � n=0 ane−βEn + � ∞ Vb ρ(E)e−βE dE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (5) Here the number of quantum states nb + anb = A(u)/(2πℏ) = 18u/(5π) in the phase space area A(u) that is classically confined (0 < E < Vb) is split into the integer (nb) and the fractional (0 ≤ anb < 1) parts such that an = 1 for n < nb, and ρ(E) is the semiclassical den- sity of states above the barrier (E > Vb) yet inside the QD (x contributing from the potential maximum towards the dot) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The model of the thermally activated escape from a fixed-depth potential summarized above needs to be in- corporated into the time-dependent problem of charge capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Here the decoupling timescale competes with the thermalization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' We contrast two opposite ex- tremes: (i) fast thermalization with respect to decou- pling [30], where Γout(t) is replaced by ⟨Γ⟩ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (4) with time-dependent depth u(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (ii) slow thermaliza- tion, where the number of confined levels nb and the dis- crete weights ane−βEn/Z are frozen at the initial depth u = u0 and then used in the averaging of P(∞) over n with Γout(t) → Γn(t), neglecting the continuum con- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 3(b) and (c) shows the simulation results for both limiting cases, which are compared to the exper- iment at u0 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='3 an Arrhenius plot in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' While the intradot population dynamics of the thermally ex- cited states appear to significantly affect the results, both models reproduce the crossover on the plateau, where the decline of the thermal excitation weight with energy no longer outweighs the competing growth of the escape rate (crossover from tunneling to hopping [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' From the ex- perimental data, we infer a value of ℏΩb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='6 meV and hence fix the device-specific microscopic potential (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' With kBT0 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='16 ℏΩb [36] this corresponds to a crossover temperature of T0 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='9 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' In conclusion, matching of the counting data to the universal scaling relation over a broad range of decou- pling speeds (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2) and the estimation of Ωb from the temperature dependence (inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 3) provides an ex- perimental technique for inferring the ground state es- cape rate Γ0 and the barrier height Vb in physical units down to shallow limit where Vb/ℏ ∼ Γ0 ∼ Ωb and the confinement is eventually lost, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' E0 > Vb for u < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='42, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Using the universal scaling curve, differ- ent measurements can be combined in a single speed- depth plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 4(b), showing the inferred initial depth u0 versus the speed ˙u/ω0 at fixed ⟨N⟩ compared to level lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The dashed lines show two lim- its for adiabatic electron capture: loss of confinement (u0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='42) or breakdown of adiabatic condition (which requires small relative change in frequency over one pe- riod, ( ˙ω0/ω0)(2π/ω0) ≪ 1 ⇒ ˙u/ω0 ≪ 5u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' For ⟨N⟩ → 1 the capture fidelity 1 − ⟨N⟩ is limited by non-adiabatic effects, while for ⟨N⟩ → 0 the residual capture probabil- ity is capped by the saturation of the escape rate as the confinement is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' For quantum metrology applications that rely on the escape of excess electrons and capture of the target num- ber of electrons, this work introduces capability-defining frequency limits for control of tunneling [3, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The speed-depth scaling method presented here provides a u0 VD (mV) Γ0 (GHz) Vb (meV) 0 1 2 3 −90 −80 −70 −60 10 −4 10 −2 10 0 10 2 0 1 2 3 4 5 Γ0 Vb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='999 10 −7 10 −5 10 −3 10 −1 0 1 2 3 u0 a) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (a) Inferred ground state tunneling rate Γ0 as a func- tion of gate voltage compared to the barrier height Vb (width of the line indicates the level broadening ℏΓ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (b) Combined speed-depth diagram: initial depth u0 is plotted as a function of speed parameter ˙u/ω0 for levels of constant ⟨N⟩ and com- pared to the universal scaling given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' (3) (solid black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Various experimental realizations (symbols) are com- bined into a single diagram: a different QD implementation with sinusoidal driving waveform (upwards green triangles), linear ramp waveform as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' 2a and b (blue circle, orange square), and a set of linear ramp waveforms focused on slow decoupling speeds (downward red triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' The vertical line in (a) and the horizontal dashed line in (b) mark u0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='42, the dotted line in (b) indicates the adiabatic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' benchmark for characterization of shallow quantum dots and sets the stage for the exploration of quantum exci- tation in the controlled manipulation of individual parti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Discussions with Akira Fujiwara, Nathan Johnson, Pe- ter Silvestrov, and Gento Yamahata are acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation) within the framework of Germany’s Excellence Strategy-EXC-2123 QuantumFrontiers-390837967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content='K are supported by grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' lzp-2021/1-0232 from the Latvian Council of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Chatterjee, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Stevenson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9FIT4oBgHgl3EQfjCvs/content/2301.11295v1.pdf'} +page_content=' Franceschi, A.' metadata={'source': 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Time Algorithms for Several Geometric +Optimization (With Outliers) Problems In Machine Learning⋆ +Hu Ding +School of Computer Science and Engineering, University of Science and Technology of China +He Fei, China +huding@ustc.edu.cn +Abstract. In this paper, we study several important geometric optimization problems arising in +machine learning. First, we revisit the Minimum Enclosing Ball (MEB) problem in Euclidean +space Rd. The problem has been extensively studied before, but real-world machine learning tasks +often need to handle large-scale datasets so that we cannot even afford linear time algorithms. +Motivated by the recent studies on beyond worst-case analysis, we introduce the notion of stability +for MEB, which is natural and easy to understand. Roughly speaking, an instance of MEB is stable, +if the radius of the resulting ball cannot be significantly reduced by removing a small fraction +of the input points. Under the stability assumption, we present two sampling algorithms for +computing radius-approximate MEB with sample complexities independent of the number of input +points n. In particular, the second algorithm has the sample complexity even independent of the +dimensionality d. We also consider the general case without the stability assumption. We present +a hybrid algorithm that can output either a radius-approximate MEB or a covering-approximate +MEB. Our algorithm improves the running time and the number of passes for the previous +sublinear MEB algorithms. Our method relies on two novel techniques, the Uniform-Adaptive +Sampling method and Sandwich Lemma. Furthermore, we observe that these two techniques can +be generalized to design sublinear time algorithms for a broader range of geometric optimization +problems with outliers in high dimensions, including MEB with outliers, one-class and two-class +linear SVMs with outliers, k-center clustering with outliers, and flat fitting with outliers. Our +proposed algorithms also work fine for kernels. +1 +Introduction +Many real-world machine learning tasks can be formulated as geometric optimization problems +in Euclidean space. We start with a fundamental geometric optimization problem, Minimum +Enclosing Ball (MEB), which has attracted a lot of attentions in past years. Given a set P +of n points in Euclidean space Rd, where d could be quite high, the problem of MEB is to +find a ball with minimum radius to cover all the points in P [20,45,69]. MEB finds several +important applications in machine learning [76]. For example, the popular classification model +Support Vector Machine (SVM) can be formulated as an MEB problem in high dimensional +space, if all the mapped points have the same norm by using kernel method, e.g., the popular +radial basis function kernel; this SVM is called “Core Vector Machine (CVM)” which is +currently one of the most important SVM training methods for large-scale data sets, since it +was proposed in 2005 [90]. Hence fast MEB algorithms can be used to speed up its training +procedure [29,30,90]. Recently, MEB has also been studied for preserving privacy [44,77] and +quantum cryptography [53]. +Usually, we consider the approximate solutions of MEB. If a ball covers all the n points but +has a radius larger than the optimal one, we call it a “radius-approximate solution”; if a +ball has the radius no larger than the optimal one but covers less than n points, we call it a +“covering-approximate solution” instead (the formal definitions are shown in Section 2). +In the era of big data, the dataset could be so large that we cannot even afford linear time +algorithms. This motivates us to ask the following questions: +⋆ Part of this work has appeared in [37,38]. +arXiv:2301.02870v1 [cs.DS] 7 Jan 2023 + +Is it possible to develop approximation algorithms for MEB that run in sublinear time in +the input size? And how about other high-dimensional geometric optimization problems arising +in machine learning? +It is common to assume that the input data is represented by a n × d matrix, and any +algorithm having complexity o(nd) can be considered as a sublinear time algorithm. In practice, +data items are usually represented as sparse vectors in Rd; so it can be fast to perform the +operations, like distance computing, even though the dimensionality d is high (e.g., if each +vector has s ≪ d non-zero entries, the time for computing the distance is O(s) rather than +O(d); see the concluding remarks of [30]). Moreover, the number of input points n is often +much larger than the dimensionality d in many real-world scenarios. Therefore, we are +interested in designing the algorithms that have complexities sublinear in n (or +linear in n but with small factor before it). +1.1 +Our Main Ideas and Results +Our idea for designing sublinear time MEB algorithms is inspired by the recent studies +on optimization with respect to stable instances, under the umbrella of beyond worst-case +analysis [82]. For example, several recent works introduced the notion of stability for problems +like clustering and max-cut [8,13,18]. In this paper, we give the notion of “stability” for MEB. +Roughly speaking, an instance of MEB is stable, if the radius of the resulting ball cannot be +significantly reduced by removing a small fraction of the input points (e.g., the radius cannot +be reduced by 10% if only 1% of the points are removed). The rationale behind this notion is +quite natural: if the given instance is not stable, the small fraction of points causing significant +reduction in the radius should be viewed as outliers (or we may need multiple balls to cover +the input points as the k-center clustering problem [51,61]). To the best of our knowledge, +this is the first study on MEB from the perspective of stability. +We prove an important implication of the stability assumption: informally speaking, if +an instance of MEB is stable, its center should reveal a certain extent of robustness in the +space (Section 3). Using this implication, we propose two sampling algorithms for computing +(1 + ϵ)-radius approximate MEB with sublinear time complexities (Section 4); in particular, +our second algorithm has the sample size (i.e., the number of sampled points) independent of +the number of input points n and dimensionality d (to the best of our knowledge, this is the +first algorithm achieving (1 + ϵ)-radius approximation with such a sublinear complexity). +Moreover, we have an interesting observation: the ideas developed under the stability +assumption can even help us to solve the general instance without the stability assumption, if +we relax the requirement slightly. We introduce a hybrid approach that can output either a +radius-approximate MEB or a covering-approximate MEB, depending upon whether the input +instance is sufficiently stable1 (Section 5). Also, a byproduct is that we can infer the stability +degree of the given instance from the output. It is worth noting that the simple uniform +sampling idea based on VC-dimension [58,92] can only yield a “bi-criteria” approximation, +which has errors on both the radius and the number of covered points (see the discussion on our +first sampling algorithm in Section 4.1). Comparing with the sublinear time MEB algorithm +proposed by Clarkson et al. [30], we reduce the total running time from ˜O(ϵ−2n + ϵ−1d + M) +to O(n + h(ϵ, δ) · d + M), where M is the number of non-zero entries in the input n × d matrix +and h(ϵ, δ) is a factor depending on the pre-specified radius error bound ϵ and covering error +bound δ. Thus, our improvement is significant if n ≫ d. The only tradeoff is that we allow +a covering approximation for unstable instance (given the lower bound proved by [30], it is +quite unlikely to reduce the term ϵ−2n if we keep restricting the output to be (1 + ϵ)-radius +approximation). Moreover, our algorithm only needs uniform sampling and a single pass +1 We do not need to explicitly know whether the instance is stable or not, when running our algorithm. +2 + +over the data; on the other hand, the algorithm of [30] needs ˜O(ϵ−1) passes (the details +are shown in Table 1). In addition to the stability idea, our method also relies on two key +techniques, the novel “Uniform-Adaptive Sampling” method and “Sandwich Lemma”. +Roughly speaking, the Uniform-Adaptive Sampling method can help us to bound the error +induced in each “randomized greedy selection” step; the Sandwich Lemma enables us to +estimate the objective value of each candidate and select the best one in sublinear time. +Results +Quality +Time +Number of passes Extendibility for +MEB with +outliers +Clarkson et al. [30] +(1 + ϵ)-rad. +˜O(ϵ−2n + ϵ−1d + M) +˜O(ϵ−1) +N/A +Core-sets methods +[20,29,69,79] +(1 + ϵ)-rad. +roughly O(ϵ−1nd) +or O(ϵ−1(n + d + M)) +if M = o(nd) +O(ϵ−1) +bi-criteria +approx. [22] +Numerical method [84] +(1 + ϵ)-rad. +˜O(ϵ−1/2nd) or +˜O(ϵ−1/2(n + d + M)) +if M = o(nd) +O(ϵ−1/2) +N/A +Numerical method [6] +(1 + ϵ)-rad. +˜O(nd + n +√ +d/√ϵ) +˜O(d + +� +d/ϵ) +N/A +Streaming algorithm [4,25] +1.22-rad. +O(nd/ϵ5) +one pass +N/A +This +paper +stable +instance +(1 + ϵ)-rad. +O(C1 · d) (Sec. 4.2) +uniform sampling +N/A +general +instance +(1 + ϵ)-rad. +or (1 − δ)-cov. +O +� +(n + C2)d +� +or +O(n + C2 · d + M) +if M = o(nd) (Sec. 5.3) +uniform sampling +plus a single pass +(1 + ϵ)-rad. +or (1 − δ)-cov. +(Sec. 6) +Table 1: The existing and our results for computing MEB in high dimensions. In the table, +“rad.” and “cov.” stand for “radius approximation” and “covering approximation”, respectively. +“M” is the number of non-zero entries in the input n × d matrix. The factor C1 depends on ϵ +and the stability degree of the given instance; the factor C2 depends on ϵ and δ. +Finally, we present several extensions of our result. In practice, we may assume the presence +of outliers in given datasets. In particular, as the rapid development of machine learning, the +field of adversarial machine learning has attracted a great amount of attentions [17,52]. A small +set of outliers could be added by some adversarial attacker to make the model severely deviate +and cause unexpected error (the seminal paper [16] on poisoning attacks against SVM has just +received the ICML2022 Test of Time award). To defend such poisoning attacks, we often design +robust algorithms that are resilient against outliers [65]. However, the presence of outliers +makes the problem not only non-convex but also highly combinatorial in high dimensions; +for example, if m of the input n data items are outliers (m < n), we have to consider an +exponentially large number +�n +m +� +of different possible cases when optimizing the objective +function. So we consider to design sublinear time algorithms for the following problems. +MEB with outliers. MEB with outliers is a natural generalization of the MEB problem, +where the goal is to find the minimum ball covering at least a certain fraction of input points. +We can apply MEB with outliers to solve many practical problems (e.g., outlier recognition) +in data mining and data analysis [89]. We define the stability for MEB with outliers, and +propose the sublinear time approximation algorithm. Our algorithm is the first sublinear +time algorithm for the MEB with outliers problem (comparing with the previous linear time +algorithms like [22]), to the best of our knowledge. +3 + +Other enclosing with outliers problems. Besides MEB with outliers, we observe that +our proposed techniques can be used to solve a broader range of enclosing with outliers problems. +We define a general optimization problem called minimum enclosing “x” (MEX) with +Outliers, where the “x” could be a specified kind of shape (e.g., the shape is a ball for MEB +with outliers). We prove that it is possible to generalize the Uniform-Adaptive Sampling method +and Sandwich Lemma to adapt the shape “x”, as long as it satisfies several properties. In +particular we focus on the MEX with outlier problems including flat fitting, k-center clustering, +and SVM with outliers; a common characteristic of these problems is that each of them has an +iterative algorithm based on greedy selection for its vanilla version (without outliers) that is +similar to the MEB algorithm of [20]. Though these problems have been widely studied before, +the research in terms of their sublinear time algorithms is till quite limited. +Remark 1. Because the geometric optimization problems studied in this paper are motivated +from machine learning applications, we also take into account the kernels [85]. Our proposed +algorithms only need to conduct the basic operations, like computing the distance and inner +product, on the data items. Therefore, they also work fine for kernels. +The rest of the paper is organized as follows. In Section 1.2, we summarize the +previous results that are related to our work. In Section 2, we present the important definitions +and briefly introduce the coreset construction method for MEB from [20] (which will be +used in our following algorithms and analysis). In Section 3, we prove the implication of +MEB stability. Further, in Section 4 we propose two sublinear time MEB algorithms for +stable instance. In Section 5, we propose two key techniques, Uniform-Adaptive sampling and +Sandwich lemma, and then present our sublinear time algorithm for general MEB without the +stability assumption. In Section 6, we extend the idea of Section 5 to the MEB with outliers +problem. Finally, we present the generalized Uniform-Adaptive sampling and Sandwich lemma, +together with the applications in several enclosing with outliers problems (including flat fitting, +k-center clustering, and SVM with outliers) in Section 7. +1.2 +Previous Work +The works most related to ours are [7,30]. Clarkson et al. [30] developed an elegant perceptron +framework for solving several optimization problems arising in machine learning, such as +MEB. Given a set of n points in Rd represented as an n × d matrix with M non-zero entries, +their framework can compute the MEB in ˜O( n +ϵ2 + d +ϵ ) time 2. Note that the parameter “ϵ” is +an additive error (i.e., the resulting radius is r + ϵ if r is the radius of the optimal MEB) +which can be converted into a relative error (i.e., (1 + ϵ)r) in O(M) preprocessing time. Thus, +if M = o(nd), the running time is still sublinear in the input size nd (please see Table 1). +The framework of [30] also inspires the sublinear time algorithms for training SVMs [60] and +approximating Semidefinite Programs [47]. Hayashi and Yoshida [59] presented a sampling- +based method for minimizing quadratic functions of which the MEB objective is a special case, +but it yields a large additive error O(ϵn2). +Alon et al. [7] studied the following property testing problem: given a set of n points in +some metric space, determine whether the instance is (k, b)-clusterable, where an instance is +called (k, b)-clusterable if it can be covered by k balls with radius (or diameter) b > 0. They +proposed several sampling algorithms to answer the question “approximately”. Specifically, +they distinguish between the case that the instance is (k, b)-clusterable and the case that it is +ϵ-far away from (k, b′)-clusterable, where ϵ ∈ (0, 1) and b′ ≥ b. “ϵ-far” means that more than ϵn +points should be removed so that it becomes (k, b′)-clusterable. Note that their method cannot +yield a single criterion radius-approximation or covering-approximation algorithm for the MEB +2 The asymptotic notation ˜O(f) = O +� +f · polylog( nd +ϵ ) +� +. +4 + +problem, since it will introduce unavoidable errors on the radius and the number of covered +points due to the relaxation of “ϵ-far”. But it is possible to convert it into a “bi-criteria” +approximation, where it allows approximations on both the radius and the number of uncovered +outliers (e.g., discard more than the pre-specified number of outliers). +MEB and core-set. A core-set is a small set of points that approximates the struc- +ture/shape of a much larger point set [1, 43, 80]. The core-set idea has also been used to +compute approximate MEB in high dimensional space [22,67,69,79]. B˘adoiu and Clarkson [20] +showed that it is possible to find a core-set of size ⌈2/ϵ⌉ that yields a (1+ϵ)-radius approximate +MEB. Several other methods can yield even lower core-set sizes, such as [21,67]. In fact, the +algorithm for computing the core-set of MEB is a Frank-Wolfe algorithm [46], which has been +systematically studied by Clarkson [29]. Other MEB algorithms that do not rely on core-sets +include [6,45,84]. Agarwal and Sharathkumar [4] presented a streaming (1+ +√ +3 +2 ++ ϵ)-radius ap- +proximation algorithm for computing MEB; later, Chan and Pathak [25] proved that the same +algorithm actually yields an approximation ratio less than 1.22. Very recently, Cohen-Addad +et al. [31] proposed the sublinear time algorithm for computing high dimensional power means +(e.g., geometric median and mean points) by using core-sets. +MEB with outliers and k-center clustering with outliers. The MEB with outliers +problem can be viewed as the case k = 1 of the k-center clustering with outliers problem [27]. +B˘adoiu et al. [22] extended their core-set idea to the problems of MEB and k-center clustering +with outliers, and achieved linear time bi-criteria approximation algorithms (if k is assumed to +be a constant). Huang et al. [62] and Ding et al. [41] respectively showed that simple uniform +sampling approach can yield bi-criteria approximation of k-center clustering with outliers. +Several algorithms for the low dimensional MEB with outliers have also been developed [5, +42, 54, 71]. There also exist a number of works on streaming MEB and k-center clustering +with outliers [24, 28, 72, 94]. Other related topics include robust optimization [14], robust +fitting [3,57], and optimization with uncertainty [23]. +SVM with outliers. Given two point sets P1 and P2 in Rd, the problem of Support +Vector Machine (SVM) is to find the largest margin to separate P1 and P2 (if they are +separable) [26]. SVM can be formulated as a quadratic programming problem, and a number +of efficient techniques have been developed in the past, such as the soft margin SVM [32,81], +ν-SVM [33,86], and Core-SVM [91]. There also exist a number of works on designing robust +algorithms for SVM with outliers [40,88,93]. +Flat fitting with outliers. Given an integer j ≥ 0 and a set of points in Rd, the flat +fitting problem is to find a j-dimensional flat having the smallest maximum distance to the +input points [55]; obviously, the MEB problem is a special case with j = 0. In high dimensions, +Har-Peled and Varadarajan [56] provided a linear time algorithm if j is assumed to be fixed; +their running time was further reduced by Panigrahy [79] based on a core-set approach. There +also exist several methods considering flat fitting with outliers but only for low-dimensional +case [3,57]. +Optimizations under stability. Bilu and Linial [18] showed that the Max-Cut problem +becomes easier if the given instance is stable with respect to perturbation on edge weights. +Ostrovsky et al. [78] proposed a separation condition for k-means clustering which refers to the +scenario where the clustering cost of k-means is significantly lower than that of (k − 1)-means +for a given instance, and demonstrated the effectiveness of the Lloyd heuristic [70] under the +separation condition. Balcan et al. [13] introduced the concept of approximation-stability for +finding the ground-truth of k-median and k-means clustering. Awasthi et al. [8] introduced +another notion of clustering stability and gave a PTAS for k-median and k-means clustering. +More clustering algorithms under stability assumption were studied in [9–12,68]. +5 + +Sublinear time algorithms. Besides the aforementioned sublinear MEB algorithm [30], +a number of sublinear time algorithms have been studied for the problems like clustering [35,63, +64,73,74] and property testing [15,50]. More detailed discussion on sublinear time algorithms +can be found in the survey papers [34,83]. +2 +Definitions and Preliminaries +We describe and analyze our algorithms in the unit-cost RAM model [75]. Suppose the input +is represented by an n × d matrix (i.e., n points in Rd). As mentioned in [30], it is common to +assume that each entry of the matrix can be recovered in constant time. +We let |A| denote the number of points of a given point set A in Rd, and ||x − y|| denote +the Euclidean distance between two points x and y in Rd. We use B(c, r) to denote the ball +centered at a point c with radius r > 0. Below, we give the definitions for MEB and the +notion of stability. To keep the structure of our paper more compact, we place other necessary +definitions for our extensions to Section 5, Section 6, and Section 7, respectively. +Definition 1 (Minimum Enclosing Ball (MEB)). Given a set P of n points in Rd, the +MEB problem is to find a ball with minimum radius to cover all the points in P. The resulting +ball and its radius are denoted by MEB(P) and Rad(P), respectively. +Definition 2 (Radius Approximation and Covering Approximation). Let 0 < ϵ, δ < +1. A ball B(c, r) is called a (1 + ϵ)-radius approximation of MEB(P), if the ball covers all +points in P and has radius r ≤ (1 + ϵ)Rad(P). On the other hand, the ball is called a (1 − δ)- +covering approximation of MEB(P), if it covers at least (1 − δ)n points in P and has radius +r ≤ Rad(P). +Both radius approximation and covering approximation are single-criterion approximations. +When ϵ (resp., δ) approaches to 0, the (1 + ϵ)-radius approximation (resp., (1 − δ)-covering +approximation) will approach to MEB(P). The “covering approximation” seems to be similar +to “MEB with outliers”, but actually they are quite different (see Definition 4 in Section 5). +Definition 3 ((α, β)-stable). Given a set P of n points in Rd with two parameters α and +β in (0, 1), P is an (α, β)-stable instance if (1) Rad(P \ Q) > (1 − α)Rad(P) for any +Q ⊂ P with |Q| < βn, and (2) there exists a Q′ ⊂ P with |Q′| = ⌈βn⌉ having Rad(P \ Q′) ≤ +(1 − α)Rad(P). +The intuition of Definition 3. Actually, β can be viewed as a function of α, and vice +versa. For example, given an α > 0, there always exists a β ≥ 1 +n such that P is an (α, β)-stable +instance (β ≥ 1 +n because we must remove at least one point). The property of stability indicates +that Rad(P) cannot be significantly reduced unless removing a large enough fraction of points +from P. For a fixed α, the larger β is, the more stable P should be. Similarly, for a fixed β, +the smaller α is, the more stable P should be. +Actually, our stability assumption is quite reasonable in practice. For example, if the radius +can be reduced considerably (say by α = 10%) after removing only a very small fraction (say +β = 1%) of points, it is natural to view the small fraction of points as outliers. To better +understand the notion of stability in high dimensions, we consider the following two examples. +Example (i). Suppose that the distribution of P is uniform and dense inside MEB(P). +Let α ∈ (0, 1) be a fixed number, and we study the corresponding β of P. If we want the +radius of the remaining (1 − β)n points to be as small as possible, intuitively we should remove +the outermost βn points (since P is uniform and dense). Let Q′ denote the set of outermost +βn points that has Rad(P \ Q′) ≤ (1 − α)Rad(P). Then we have |P\Q′| +|P| +≈ +V ol +� +MEB(P\Q′) +� +V ol +� +MEB(P) +� += +6 + +(Rad(P\Q′))d +(Rad(P))d +≤ (1 − α)d, where V ol(·) is the volume function. That is, 1 − β ≤ (1 − α)d and it +implies limd→∞ β = 1 when α is fixed; that means P tends to be very stable as d increases. +Example (ii). Consider a regular d-dimensional simplex P containing d + 1 points +where each pair of points have the pairwise distance equal to 1. It is not hard to obtain +Rad(P) = +� +d +2(1+d), and we denote it by rd. If we remove β(d + 1) points from P, namely +it becomes a regular d′-dimensional simplex with d′ = (1 − β)(d + 1) − 1, the new radius +rd′ = +� +d′ +2(1+d′). To achieve rd′ +rd ≤ 1 − α with a fixed α, it is easy to see that 1 − β should be +no larger than +1 +1+(2α−α2)d; this implies limd→∞ β = 1. Similar to example (i), the instance P +tends to be very stable as d increases. +Remark 2. In practice, it is difficult to know the exact value of β for a fixed α. However, the +value of β only affects the sample sizes in our proposed algorithms in Section 4, and thus only +assuming a reasonable lower bound β0 < β is already sufficient. In Section 5, we also consider +the general case without the stability assumption, where the proposed algorithm does not even +need to input β0. +2.1 +A More Careful Analysis for Core-set Construction in [20] +We first briefly introduce the core-set construction for MEB, since it will be used in our +proposed algorithms. Let 0 < ϵ < 1. The algorithm in [20] yields an MEB core-set of size +2/ϵ (for convenience, we always assume that 2/ϵ is an integer). But there is a small issue +in their paper. The analysis assumes that the exact MEB of the core-set is computed in +each iteration, but in fact one may only compute an approximate MEB. Thus, an immediate +question is whether the quality is still guaranteed with such a change. Kumar et al. [69] fixed +this issue, and showed that computing a (1 + O(ϵ2))-approximate MEB for the core-set in +each iteration still guarantees a core-set with size O(1/ϵ), where the hidden constant is larger +than 80. Clarkson [29] showed that the greedy core-set construction algorithm of MEB, as a +special case of the Frank-Wolfe algorithm, yields a core-set with size slightly larger than 4/ϵ. +Note that there exist several other methods yielding even lower core-set size [21,67], but their +construction algorithms are more complicated and thus not applicable to our problems. Below +we show that it is possible to guarantee a core-set of [20] with the size being arbitrarily close +to 2/ϵ, even if we only compute an approximate MEB in each iteration. This improves the +core-set sizes of [29,69], and the new analysis is also interesting in its own right. +For the sake of completeness, we first briefly introduce the idea of the core-set construction +algorithm in [20]. Given a point set P ⊂ Rd, the algorithm is a simple iterative procedure. +Initially, it selects an arbitrary point from P and places it into an initially empty set T. In +each of the following 2/ϵ iterations, the algorithm updates the center of MEB(T) and adds +to T the farthest point from the current center of MEB(T). Finally, the center of MEB(T) +induces a (1 + ϵ)-approximation for MEB(P). The selected set of 2/ϵ points (i.e., T) is called +the core-set of MEB. To ensure the expected improvement in each iteration, they [20] showed +that the following two inequalities hold if the algorithm always selects the farthest point to +the current center of MEB(T): +ri+1 ≥ (1 + ϵ)Rad(P) − Li; +ri+1 ≥ +� +r2 +i + L2 +i , +(1) +where ri and ri+1 are the radii of MEB(T) in the i-th and (i + 1)-th iterations, respectively, +and Li is the shifting distance of the center of MEB(T) from the i-th to (i + 1)-th iteration. +As mentioned earlier, we often compute only an approximate MEB(T) in each iteration. +In the i-th iteration, we let ci and oi denote the centers of the exact and the approximate +MEB(T), respectively. Suppose that ||ci − oi|| ≤ ξri, where ξ ∈ (0, +ϵ +1+ϵ) (we will see why this +7 + +q +oioi +cici +ci+1 +ci+1 +≤ ri+1 +≤ ri+1 +> (1 + ϵ)Rad(P) +> (1 + ϵ)Rad(P) += Li += Li +≤ ξri +≤ ξri +Fig. 1: An illustration of (2). +bound is needed later). Using another algorithm proposed in [20], one can obtain the point oi +in O( 1 +ξ2 |T|d) time. Note that we only compute oi rather than ci in each iteration. Hence we +can only select the farthest point (say q) to oi. If ||q − oi|| ≤ (1 + ϵ)Rad(P), we are done and +a (1 + ϵ)-approximation of MEB is already obtained. Otherwise, we have +(1 + ϵ)Rad(P) < ||q − oi|| ≤ ||q − ci+1|| + ||ci+1 − ci|| + ||ci − oi|| ≤ ri+1 + Li + ξri +(2) +by the triangle inequality (see Figure 1). In other words, we should replace the first inequality +of (1) by “ri+1 > (1 + ϵ)Rad(P) − Li − ξri”. Also, the second inequality of (1) still holds since +it depends only on the property of the exact MEB (see [20, Lemma 2.1]). Thus, we have +ri+1 ≥ max +�� +r2 +i + L2 +i , (1 + ϵ)Rad(P) − Li − ξri +� +. +(3) +This leads to the following theorem whose proof can be found in Section A. +Theorem 1. In the core-set construction algorithm of [20], if one computes an approximate +MEB for T in each iteration and the resulting center oi has the distance to ci less than +ξri = s +ϵ +1+ϵri for some s ∈ (0, 1), the final core-set size is bounded by z = +2 +(1−s)ϵ. Also, the +bound could be arbitrarily close to 2/ϵ when s is small enough. +We can simply set s to be any constant in (0, 1); for instance, if s = 1/3, the core-set +size will be bounded by z = 3/ϵ. Since |T| ≤ z in each iteration, the total running time is +O +� +z +� +|P|d + 1 +ξ2 zd +�� += O +� +1 +ϵ +� +|P| + 1 +ϵ3 +� +d +� +. +Remark 3. We also want to emphasize a simple observation on the above core-set construction +procedure, which will be used in our algorithms and analyses later on. The algorithm always +selects the farthest point to oi in each iteration. However, this is actually not necessary. As long +as the selected point has distance at least (1 + ϵ)Rad(P), the result presented in Theorem 1 +is still true. If no such a point exists (i.e., P \ B +� +oi, (1 + ϵ)Rad(P) +� += ∅), a (1 + ϵ)-radius +approximate MEB (i.e., the ball B +� +oi, (1 + ϵ)Rad(P) +� +) has been already obtained. +Remark 4 (kernels). If each point p ∈ P is mapped to ψ(p) in RD by some kernel function +(e.g., as the CVM [90]), where D could be +∞, we can still run the core-set algorithm of [20], +since the algorithm only needs to compute the distances and the center oi is always a convex +combination of T in each iteration; instead of returning an explicit center, the algorithm will +output the coefficients of the convex combination for the center. And similarly, our Algorithm 2 +presented in Section 4.2 also works fine for kernels. +3 +Implication of the Stability Property +In this section, we show an important implication of the stability property of Definition 3. +8 + +Theorem 2. Assume ϵ, ϵ′, β0 ∈ (0, 1). Let P be an (ϵ2, β)-stable instance of the MEB problem +with β > β0, and o be the center of its MEB. Let ˜o be a given point in Rd. Assume the number +r ≤ (1 + ϵ′2)Rad(P). If the ball B +� +˜o, r +� +covers at least (1 − β0)n points from P, the following +holds +||˜o − o|| < (2 +√ +2ϵ + +√ +3ϵ′)Rad(P). +(4) +Theorem 2 indicates that if a ball covers a large enough subset of P and its radius is +bounded, its center should be close to the center of MEB(P). Let P ′ = B +� +˜o, r +� +∩ P, and +assume o′ is the center of MEB(P ′). To bound the distance between ˜o and o, we bridge them +by the point o′ (since ||˜o − o|| ≤ ||˜o − o′|| + ||o′ − o||). The following are two key lemmas for +proving Theorem 2. +Lemma 1. The distance ||o′ − o|| ≤ +√ +2ϵRad(P). +Proof. We consider two cases: MEB(P ′) is totally covered by MEB(P) and otherwise. For +the first case (see Figure 2(a)), it is easy to see that +||o′ − o|| ≤ Rad(P) − (1 − ϵ2)Rad(P) = ϵ2Rad(P) < +√ +2ϵRad(P), +(5) +where the first inequality comes from the fact that MEB(P ′) has radius at least (1−ϵ2)Rad(P) +(Definition 3). Thus, we can focus on the second case below. +Let a be any point located on the intersection of the two spheres of MEB(P ′) and MEB(P). +Then we have the following claim. +! +!" +! +!" +# +$ +$" +% +oH +oH +! +!" +# +!" +˜o˜o +# +$ +(a) +(b) +(c) +(d) +Fig. 2: (a) The case MEB(P ′) ⊂ MEB(P); (b) an illustration under the assumption ∠ao′o < +π/2 in the proof of Claim 1; (c) the angle ∠ao′o ≥ π/2; (d) an illustration of Lemma 2. +Claim 1. The angle ∠ao′o ≥ π/2. +Proof. Suppose that ∠ao′o < π/2. Note that ∠aoo′ is always smaller than π/2 since ||o − a|| = +Rad(P) ≥ Rad(P ′) = ||o′ − a||. Therefore, o and o′ are separated by the hyperplane H that is +orthogonal to the segment o′o and passes through the point a. See Figure 2(b). Now we show +that P ′ can be covered by a ball smaller than MEB(P ′). Let oH be the point H ∩ o′o, and +t (resp., t′) be the point collinear with o and o′ on the right side of the sphere of MEB(P ′) +(resp., left side of the sphere of MEB(P); see Figure 2(b)). Then, we have +||t − oH|| + ||oH − o′|| = ||t − o′|| = ||a − o′|| < ||o′ − oH|| + ||oH − a|| +=⇒ ||t − oH|| < ||oH − a||. +(6) +Similarly, we have ||t′ − oH|| < ||oH − a||. Consequently, MEB(P) ∩ MEB(P ′) is covered by +the ball B(oH, ||oH −a||) (the “red dotted” ball in Figure 2(b)). Further, because P ′ is covered +by MEB(P) ∩ MEB(P ′) and ||oH − a|| < ||o′ − a|| = Rad(P ′), P ′ is covered by the ball +B(oH, ||oH − a||) that is smaller than MEB(P ′). This contradicts to the fact that MEB(P ′) +is the minimum enclosing ball of P ′. Thus, the claim ∠ao′o ≥ π/2 is true. +⊓⊔ +9 + +Given Claim 1, we know that ||o′ − o|| ≤ +�� +Rad(P) +�2 − +� +Rad(P ′) +�2. See Figure 2(c). +Moreover, Definition 3 implies that Rad(P ′) ≥ (1 − ϵ2)Rad(P). Therefore, we have +||o′ − o|| ≤ +�� +Rad(P) +�2 − +� +(1 − ϵ2)Rad(P) +�2 ≤ +√ +2ϵRad(P). +(7) +⊓⊔ +Lemma 2. The distance ||˜o − o′|| < ( +√ +2ϵ + +√ +3ϵ′)Rad(P). +Proof. Let L be the hyperplane orthogonal to the segment ˜oo′ and passing through the center +o′. Suppose ˜o is located on the left side of L. Then, there always exists a point b ∈ P ′ located +on the right closed semi-sphere of MEB(P ′) divided by L (this result is from [22, Lemma +2.2]; for completeness, we state the lemma in Section B). See Figure 2(d). That is, the angle +∠bo′˜o ≥ π/2. As a consequence, we have +||˜o − o′|| ≤ +� +||˜o − b||2 − ||b − o′||2. +(8) +Moreover, since ||˜o − b|| ≤ r ≤ (1 + ϵ′2)Rad(P) and ||b − o′|| = Rad(P ′) ≥ (1 − ϵ2)Rad(P), +(8) implies that ||˜o−o′|| ≤ +� +(1 + ϵ′2)2 − (1 − ϵ2)2Rad(P), where this upper bound is equal to +� +2ϵ′2 + ϵ′4 + 2ϵ2 − ϵ4Rad(P) < +� +3ϵ′2 + 2ϵ2Rad(P) < ( +√ +2ϵ + +√ +3ϵ′)Rad(P). +(9) +⊓⊔ +By triangle inequality, Lemmas 1 and 2, we immediately have +||˜o − o|| ≤ ||˜o − o′|| + ||o′ − o|| < (2 +√ +2ϵ + +√ +3ϵ′)Rad(P). +(10) +This completes the proof of Theorem 2. +4 +Sublinear Time Algorithms for MEB under Stability Assumption +Suppose ϵ ∈ (0, 1). We assume that the given instance P is an (ϵ2, β)-stable instance where β +is larger than a given lower bound β0 (i.e., β > β0). Using Theorem 2, we present two different +sublinear time sampling algorithms for computing MEB. Following most of the articles on +sublinear time algorithms (e.g., [35,73,74]), in each sampling step of our algorithms, we always +take the sample independently and uniformly at random. +4.1 +The First Algorithm +r +c +Fig. 3: An illustration for the first sampling algorithm. The red points are the samples; we +expand B(c, r) slightly and the larger ball is a radius-approximate MEB of the whole input +point set. +10 + +The first algorithm is based on the theory of VC dimension and ϵ-nets [58,92]. Roughly +speaking, we compute an approximate MEB of a small random sample (say, B(c, r)), and +expand the ball slightly; then we prove that this expanded ball is an approximate MEB +of the whole data set (see Figure 3). Our key idea is to show that B(c, r) covers at least +(1 − β0)n points and therefore c is close to the optimal center by Theorem 2. As emphasized +in Section 1.1, our result is a single-criterion approximation. If simply applying the uniform +sample idea without the stability assumption (as the ideas in [41,62]), it will yield a bi-criteria +approximation where the ball has to cover less than n points for achieving the desired bounded +radius. +Algorithm 1 MEB Algorithm I +Input: Two parameters 0 < ϵ, η < 1; an (ϵ2, β)-stable instance P of MEB problem in Rd, where β is larger +than a given lower bound β0 > 0. +1: Sample a set S of Θ( 1 +β0 · max{log 1 +η , d log +d +β0 }) points from P uniformly at random. +2: Apply any approximate MEB algorithm (such as the core-set based algorithm [20]) to compute a (1 + ϵ2)- +radius approximate MEB of S, and let the obtained ball be B(c, r). +3: Output the ball B +� +c, 1+(2 +√ +2+ +√ +3)ϵ +1−ϵ2 +r +� +. +Theorem 3. With probability 1 − η, Algorithm 1 returns a λ-radius approximate MEB of P, +where +λ = +� +1 + (2 +√ +2 + +√ +3)ϵ +� +(1 + ϵ2) +1 − ϵ2 += 1 + O(ϵ). +(11) +Before proving Theorem 3, we prove the following lemma first. +Lemma 3. Let S be a set of Θ( 1 +β0 · max{log 1 +η, d log d +β0 }) points sampled randomly and inde- +pendently from a given point set P ⊂ Rd, and B be any ball covering S. Then, with probability +1 − η, |B ∩ P| ≥ (1 − β0)|P|. +Proof. Consider the range space Σ = (P, Φ) where each range φ ∈ Φ is the complement of a +ball in the space. In a range space, a subset Y ⊂ P is a β0-net if +for any φ ∈ Φ, |P ∩ φ| +|P| +≥ β0 =⇒ Y ∩ φ ̸= ∅. +(12) +The size |S| = Θ( 1 +β0 ·max{log 1 +η, d log d +β0 }), and from [58,92] we know that S is a β0-net of P with +probability 1− η. Thus, if |B ∩ P| < (1− β0)|P|, i.e., |P \B| > β0|P|, we have S ∩ +� +P \B +� +̸= ∅. +This contradicts to the fact that S is covered by B. Consequently, |B ∩ P| ≥ (1 − β0)|P|. +⊓⊔ +Proof. (of Theorem 3) Denote by o the center of MEB(P). Since S ⊂ P and B(c, r) is a +(1 + ϵ2)-radius approximate MEB of S, we know that r ≤ (1 + ϵ2)Rad(P). Moreover, Lemma 3 +implies that |B(c, r) ∩ P| ≥ (1 − β0)|P| with probability 1 − η. Suppose it is true and let +P ′ = B(c, r) ∩ P. Then, we have the distance +||c − o|| ≤ (2 +√ +2 + +√ +3)ϵRad(P) +(13) +via Theorem 2 (we set ϵ′ = ϵ). For simplicity, we use x to denote (2 +√ +2 + +√ +3)ϵ. The inequality +(13) implies that the point set P is covered by the ball B(c, (1 + x)Rad(P)). Note that we +cannot directly return B(c, (1 + x)Rad(P)) as the final result, since we do not know the value +of Rad(P). Thus, we have to estimate the radius (1 + x)Rad(P). +11 + +Since P ′ is covered by B(c, r) and |P ′| ≥ (1 − β0)|P|, r should be at least (1 − ϵ2)Rad(P) +due to Definition 3. Hence, we have +1 + x +1 − ϵ2 r ≥ (1 + x)Rad(P). +(14) +That is, P is covered by the ball B(c, 1+x +1−ϵ2 r). Moreover, the radius +1 + x +1 − ϵ2 r ≤ 1 + x +1 − ϵ2 (1 + ϵ2)Rad(P). +(15) +This means the ball B(c, 1+x +1−ϵ2 r) is a λ-radius approximate MEB of P, where +λ = (1 + ϵ2) 1 + x +1 − ϵ2 = +� +1 + (2 +√ +2 + +√ +3)ϵ +� +(1 + ϵ2) +1 − ϵ2 +(16) +and λ = 1 + O(ϵ) if ϵ is a fixed small number in (0, 1). +⊓⊔ +Running time of Algorithm 1. For simplicity, we assume log 1 +η < d log d +β0 . If we use +the core-set based algorithm [20] to compute B(c, r) (see Remark 3), the running time of +Algorithm 1 is O +� 1 +ϵ2 (|S|d + 1 +ϵ6 d) +� += O +� d2 +ϵ2β0 log d +β0 + d +ϵ8 +� += ˜O(d2) where the hidden factor +depends on ϵ and β0. +Remark 5. If the dimensionality d is too high, the random projection technique Johnson- +Lindenstrauss (JL) transform [36] can be used to approximately preserve the radius of enclosing +ball [2,66,87]. However, it is not useful for reducing the time complexity of Algorithm 1. If we +apply the JL-transform on the sampled Θ( d +β0 log d +β0 ) points in Step 1, the JL-transform step +itself already takes Ω( d2 +β0 log d +β0 ) time. +4.2 +The Second Algorithm +Our first algorithm in Section 4.1 is simple, but has a sample size (i.e., the number of sampled +points) depending on the dimensionality d, while the second algorithm has a sample size +independent of both n and d (it is particularly important when a kernel function is applied, +because the new dimension could be very large or even +∞). We briefly overview our idea +first. +High level idea of the second algorithm: Recall our Remark 3 (ii). If we know the +value of (1 + ϵ)Rad(P), we can perform almost the same core-set construction procedure +described in Theorem 1 to achieve an approximate center of MEB(P), where the only difference +is that we add a point with distance at least (1 + ϵ)Rad(P) to oi in each iteration. In this way, +we avoid selecting the farthest point to oi, since this operation will inevitably have a linear +time complexity. To implement our strategy in sublinear time, we need to determine the value +of (1+ϵ)Rad(P) first. We propose Lemma 4 below to estimate the range of Rad(P), and then +perform a binary search on the range to determine the value of (1 + ϵ)Rad(P) approximately. +Based on the stability property, we observe that the core-set construction procedure can serve +as an “oracle” to help us to guess the value of (1 + ϵ)Rad(P) (see Algorithm 3). Let h > 0 be +a candidate. We add a point with distance at least h to oi in each iteration. We prove that the +procedure cannot continue for more than z iterations if h ≥ (1 + ϵ)Rad(P), and will continue +more than z iterations with constant probability if h < (1 − ϵ)Rad(P), where z is the size of +core-set described in Theorem 1. Also, during the core-set construction, we add the points +to the core-set via random sampling, rather than a deterministic way. A minor issue here is +that we need to replace ϵ by ϵ2 in Theorem 1, so as to achieve the overall (1 + O(ϵ))-radius +approximation in the following analysis. +12 + +Lemma 4. Given a parameter η ∈ (0, 1), one selects an arbitrary point p1 ∈ P and takes a +sample Q ⊂ P with |Q| = +1 +β0 log 1 +η uniformly at random. Let p2 = arg maxp∈Q ||p − p1||. Then, +with probability 1 − η, +Rad(P) ∈ [1 +2||p1 − p2||, +1 +1 − ϵ2 ||p1 − p2||]. +(17) +Proof. First, the lower bound of Rad(P) is obvious since ||p1 − p2|| is always no larger than +2Rad(P). Then, we consider the upper bound. Let B(p1, l) be the ball covering exactly (1−β0)n +points of P, and thus l ≥ (1 − ϵ2)Rad(P) according to Definition 3. To complete our proof, +we also need the following folklore lemma presented in [39]. +Lemma 5. +[39] Let N be a set of elements, and N′ be a subset of N with size |N′| = τ |N| +for some τ ∈ (0, 1). Given η ∈ (0, 1), if one randomly samples +ln 1/η +ln 1/(1−τ) ≤ 1 +τ ln 1 +η elements from +N, then with probability at least 1 − η, the sample contains at least one element of N′. +l +p1 +p1 +p2 +p2 +Fig. 4: An illustration of Lemma 4; the red points are the sampled set Q. +In Lemma 5, let N and N′ be the point set P and the subset P \ B(p1, l), respectively. +We know that Q contains at least one point from N′ according to Lemma 5 (by setting +τ = β0). Namely, Q contains at least one point outside B(p1, l). Moreover, because p2 = +arg maxp∈Q ||p − p1||, we have ||p1 − p2|| ≥ l ≥ (1 − ϵ2)Rad(P), i.e., Rad(P) ≤ +1 +1−ϵ2 ||p1 − p2|| +(see Figure 4 for an illustration). +⊓⊔ +Algorithm 3 serves as a subroutine in Algorithm 2. In Algorithm 3, we simply set z = 3 +ϵ2 +with s = 1/3 as described in Theorem 1 (as mentioned before, we replace ϵ by ϵ2); we compute +oi having distance less than s +ϵ2 +1+ϵ2 Rad(T) to the center of MEB(T) in Step 2(1). +Algorithm 2 MEB Algorithm II +Input: Two parameters 0 < ϵ, η0 < 1; an (ϵ2, β)-stable instance P of MEB problem in Rd, where β is larger +than a given lower bound β0 > 0. Set the interval [a, b] for Rad(P) that is obtained by Lemma 4. +1: Among the set {(1−ϵ2)a, (1+ϵ2)(1−ϵ2)a, · · · , (1+ϵ2)w(1−ϵ2)a = (1+ϵ2)b} where w = ⌈log1+ϵ2 +2 +(1−ϵ2)2 ⌉+1 = +O( 1 +ϵ2 ), perform binary search for the value h by using Algorithm 3 with z = +3 +ϵ2 and η = +η0 +2 log w . +2: Suppose that Algorithm 3 returns “no” when h = (1 + ϵ2)i0(1 − ϵ2)a and returns “yes” when h = +(1 + ϵ2)i0+1(1 − ϵ2)a. +3: Run Algorithm 3 again with h = (1 + ϵ2)i0+2a, z = +3 +ϵ2 , and η = η0/2; let ˜o be the obtained ball center of T +when the loop stops. +4: Return the ball B(˜o, r), where r = +1+(2 +√ +2+ +2 +√ +6 +√ +1−ϵ2 )ϵ +1+ϵ2 +h. +13 + +Algorithm 3 Oracle for testing h +Input: An instance P, a parameter η ∈ (0, 1), h > 0, and a positive integer z. +1: Initially, arbitrarily select a point p ∈ P and let T = {p}. +2: i = 1; repeat the following steps: +(1) Compute an approximate MEB of T and let the ball center be oi as described in Theorem 1 (replace ϵ +by ϵ2 and set s = 1/3). +(2) Sample a set Q ⊂ P with |Q| = +1 +β0 log z +η uniformly at random. +(3) Select the point q ∈ Q that is farthest to oi, and add it to T. +(4) If ||q − oi|| < h, stop the loop and output “yes”. +(5) i = i + 1; if i > z, stop the loop and output “no”. +Theorem 4. With probability 1 − η0, Algorithm 2 returns a λ-radius approximate MEB of P, +where +λ = (1 + x1)(1 + x2) +1 + ϵ2 += 1 + O(ϵ) +with +x1 = O +� +ϵ2 +1 − ϵ2 +� +, x2 = O +� +ϵ +√ +1 − ϵ2 +� +. +(18) +The running time is ˜O +� +( +1 +ϵ2β0 + 1 +ϵ8 )d +� +, where ˜O(f) = O(f · polylog( 1 +ϵ, 1 +η0 )). +Before proving Theorem 4, we provide Lemma 6 first. +Lemma 6. If h ≥ (1 + ϵ2)Rad(P), Algorithm 3 returns “yes”; else if h < (1 − ϵ2)Rad(P), +Algorithm 3 returns “no” with probability at least 1 − η. +Proof. First, we assume that h ≥ (1 + ϵ2)Rad(P). Recall the remark following Theorem 1. If +we always add a point q with distance at least h ≥ (1 + ϵ2)Rad(P) to oi, the loop 2(1)-(5) +cannot continue more than z iterations, i.e., Algorithm 3 will return “yes”. +Now, we consider the case h < (1 − ϵ2)Rad(P). Similar to the proof of Lemma 4, we +consider the ball B(oi, l) covering exactly (1 − β0)n points of P. According to Definition 3, we +know that l ≥ (1 − ϵ2)Rad(P) > h. Also, with probability 1 − η/z, the sample Q contains at +least one point outside B(oi, l) due to Lemma 5. By taking the union bound, with probability +(1 − η/z)z ≥ 1 − η, ||q − oi|| is always larger than h and eventually Algorithm 3 will return +“no”. +⊓⊔ +Proof. (of Theorem 4) Since Algorithm 3 returns “no” when h = (1 + ϵ2)i0(1 − ϵ2)a and +returns “yes” when h = (1 + ϵ2)i0+1(1 − ϵ2)a, from Lemma 6 we know that +(1 + ϵ2)i0(1 − ϵ2)a < (1 + ϵ2)Rad(P); +(19) +(1 + ϵ2)i0+1(1 − ϵ2)a ≥ (1 − ϵ2)Rad(P). +(20) +The above inequalities together imply that +(1 + ϵ2)3 +1 − ϵ2 Rad(P) > (1 + ϵ2)i0+2a ≥ (1 + ϵ2)Rad(P). +(21) +Thus, when running Algorithm 3 with h = (1 + ϵ2)i0+2a in Step 3, the algorithm returns “yes” +(by the right hand-side of (21)). Then, consider the ball B(˜o, h). We claim that |P\B(˜o, h)| < β0n. +Otherwise, the sample Q contains at least one point outside B(˜o, h) with probability 1 − η/z +in Step 2(2) of Algorithm 3, i.e., the loop will continue. Thus, it contradicts to the fact that +the algorithm returns “yes”. Let P ′ = P ∩ B(˜o, h), and then |P ′| ≥ (1 − β0)n. Moreover, the +left hand-side of (21) indicates that +h = (1 + ϵ2)i0+2a < (1 + +8ϵ2 +1 − ϵ2 )Rad(P). +(22) +14 + +Now, we can apply Theorem 2, where we set “ϵ′” to be “ +� +8ϵ2 +1−ϵ2 ” in the theorem. Let o be the +center of MEB(P). Consequently, we have +||˜o − o|| < (2 +√ +2 + 2 +√ +6/ +� +1 − ϵ2)ϵ · Rad(P). +(23) +For simplicity, we let x1 = +8ϵ2 +1−ϵ2 and x2 = (2 +√ +2+2 +√ +6/ +√ +1 − ϵ2)ϵ. Hence, h ≤ (1+x1)Rad(P) +and ||˜o−o|| ≤ x2Rad(P) in (22) and (23). From (23), we know that P ⊂ B(˜o, (1+x2)Rad(P)). +From the right hand-side of (21), we know that (1 + x2)Rad(P) ≤ 1+x2 +1+ϵ2 h. Thus, we have +P ⊂ B +� +˜o, 1+x2 +1+ϵ2 h +� +where 1+x2 +1+ϵ2 h = +1+(2 +√ +2+ +2 +√ +6 +√ +1−ϵ2 )ϵ +1+ϵ2 +h. Also, the radius +1 + x2 +1 + ϵ2 h +≤ +���� +by (22) +(1 + x2)(1 + x1) +1 + ϵ2 +Rad(P) = λ · Rad(P). +(24) +Thus B +� +˜o, 1+x2 +1+ϵ2 h +� +is a λ-radius approximate MEB of P, and λ = 1 + O(ϵ) if ϵ is a fixed small +number in (0, 1). +Success probability. The success probability of Algorithm 3 is 1 − η. In Algorithm 2, we +set η = +η0 +2 log w in Step 1 and η = η0/2 in Step 3, respectively. We take the union bound and +the success probability of Algorithm 2 is (1 − +η0 +2 log w)log w(1 − η0/2) > 1 − η0. +Running time. As the subroutine, Algorithm 3 runs in O(z( 1 +β0 (log z +η)d + 1 +ϵ6 d)) time; +Algorithm 2 calls the subroutine O +� +log( 1 +ϵ2 ) +� +times. Note that z = O( 1 +ϵ2 ). Thus, the total +running time is ˜O +� +( +1 +ϵ2β0 + 1 +ϵ8 )d +� +. +⊓⊔ +5 +Sublinear Time Algorithm for General MEB +In Section 4, we propose the sublinear time algorithms under the stability assumption. Specif- +ically, we assume that the given instance is (ϵ2, β)-stable and β is larger than a reasonable +known lower bound β0. However, when β0’s value is unknown, we cannot not determine the +sample size for the algorithm; or we may only know a trivial lower bound, e.g., 1 +n, and then +the sample size could be too large. So in this section we consider the general case without the +stability assumption. +High-level idea. An interesting observation is that the ideas developed for stable instance +can even help us to develop a hybrid approach for MEB when the stability assumption does not +hold. First, we “suppose” the input instance is (α, β)-stable where “α” and “β” are designed +based on the pre-specified radius error bound ϵ and covering error bound δ, and compute +a “potential” (1 + ϵ)-radius approximation (say a ball B1); then we compute a “potential” +(1−δ)-covering approximation (say a ball B2), where the definition of “covering approximation” +is given in Definition 2; finally, we determine the final output based on the ratio of their radii. +Specifically, we set a threshold τ that is determined by the given radius error bound ϵ. If the +ratio is no larger than τ, we can infer that B1 is a “true” (1 + ϵ)-radius approximation and +return it; otherwise, we return B2 that is a “true” (1 − δ)-covering approximation. Moreover, +for the latter case (i.e., returning a (1 − δ)-covering approximation), we will show that our +proposed algorithm yields a radius not only being strictly smaller than Rad(P), but also having +a gap of Θ(ϵ2) · Rad(P) to Rad(P) (i.e., the returned radius is at most +� +1 − Θ(ϵ2) +� +· Rad(P)). +Our algorithm only needs uniform sampling and a single pass over the input data, where the +space complexity in memory is O(d) (the hidden factor depends on ϵ and δ); if the input data +matrix is sparse (i.e., M = o(nd)), the time complexity is sublinear. +Before presenting our algorithms, we need to show the formal definitions for the problem of +MEB with outliers first, since it will be used for computing the (1 − δ)-covering approximation. +15 + +Definition 4 (MEB with Outliers). Given a set P of n points in Rd and a small parameter +γ ∈ [0, 1), the MEB with outliers problem is to find the smallest ball that covers (1−γ)n points. +Namely, the task is to find a subset of P with size (1 − γ)n such that the resulting MEB is the +smallest among all possible choices of the subset. The obtained ball is denoted by MEB(P, γ). +For convenience, we use Popt to denote the optimal subset of P with respect to MEB(P, γ). +Namely, Popt = argQ min +� +Rad(Q) | Q ⊂ P, |Q| = (1 − γ)n +� +. From Definition 4, we can see +that the main challenge is to determine the subset of P. Similar to Definition 2, we also define +the radius approximation and covering approximation for MEB with outliers. +Definition 5 (Radius Approximation and Covering Approximation). Let 0 < ϵ, δ < +1. A ball B(c, r) is called a (1 + ϵ)-radius approximation of MEB(P, γ), if the ball covers +(1 − γ)n points of P and has radius r ≤ (1 + ϵ)Rad(Popt). On the other hand, the ball is called +a (1 − δ)-covering approximation of MEB(P, γ), if it covers at least (1 − δ − γ)n points in P +and has radius r ≤ Rad(Popt). +A bi-criteria (1 + ϵ, 1 − δ)-approximation is a ball that covers at least +� +1 − δ − γ +� +n points +and has radius at most (1 + ϵ)Rad(Popt). +Roadmap. First, we introduce two random sampling techniques in Section 5.1, which are +the keys for designing the sublinear bi-criteria approximation algorithm for MEB with outliers +in Section 5.2. Based on the bi-criteria approximation of Section 5.2, we can solve the general +MEB problem in Section 5.3. +5.1 +Two Key Lemmas for Handling Outliers +To shed some light on our ideas, consider using the core-set construction method in Section 2.1 +to compute a bi-criteria (1+ϵ, 1−δ)-approximation for an instance (P, γ) of MEB with outliers. +Let oi be the obtained ball center in the current iteration, and Q be the set of +(δ +γ)n farthest points to oi from P. A key step for updating oi is finding a point in the set +Popt ∩ Q (the formal analysis is given in Section 5.2). Actually, this can be done by performing +a random sampling from Q. However, it requires to compute the set Q in advance, which takes +an Ω(nd) time complexity. To keep the running time to be sublinear, we need to find a point +from Popt ∩ Q by a more sophisticated way. Since Popt is mixed with outliers in the set Q, +simple uniform sampling cannot realize our goal. To solve this issue, we propose a “two level” +sampling procedure which is called “Uniform-Adaptive Sampling”. Roughly speaking, we +take a random sample A of size n′ first (i.e., the uniform sampling step), and then randomly +select a point from Q′, the set of the farthest 3 +2(δ + γ)n′ points from A to oi (i.e., the adaptive +sampling step). According to Lemma 7, with probability at least (1 − η1) +δ +3(δ+γ), the selected +point belongs to Popt ∩ Q; more importantly, the sample size n′ is independent of n and d. +The key to prove Lemma 7 is to show that the size of the intersection Q′ ∩ +� +Popt ∩ Q +� +is large +enough. By setting an appropriate value for n′, we can prove a lower bound of |Q′ ∩ +� +Popt ∩Q +� +|. +Lemma 7 (Uniform-Adaptive Sampling). Let η1 ∈ (0, 1). If we sample n′ = O( 1 +δ log 1 +η1 ) +points independently and uniformly at random from P and let Q′ be the set of farthest 3 +2(δ+γ)n′ +points to oi from the sample, then, with probability at least 1 − η1, the following holds +���Q′ ∩ +� +Popt ∩ Q +���� +|Q′| +≥ +δ +3(δ + γ). +(25) +Proof. Let A denote the set of sampled n′ points from P. First, we know |Q| = (δ + γ)n +and |Popt ∩ Q| ≥ δn (since there are at most γn outliers in Q). For ease of presentation, let +16 + +λ = |Popt∩Q| +n +≥ δ. Let {xi | 1 ≤ i ≤ n′} be n′ independent random variables with xi = 1 +if the i-th sampled point of A belongs to Popt ∩ Q, and xi = 0 otherwise. Thus, E[xi] = λ +for each i. Let σ be a small parameter in (0, 1). By using the Chernoff bound, we have +Pr +� �n′ +i=1 xi /∈ (1 ± σ)λn′� +≤ e−O(σ2λn′). That is, +Pr +� +|A ∩ +� +Popt ∩ Q +� +| ∈ (1 ± σ)λn′� +≥ 1 − e−O(σ2λn′). +(26) +Similarly, we have +Pr +� +|A ∩ Q| ∈ (1 ± σ)(δ + γ)n′� +≥ 1 − e−O(σ2(δ+γ)n′). +(27) +Note that n′ = O( 1 +δ log 1 +η1 ). By setting σ < 1/2 for (26) and (27), we have +���A ∩ +� +Popt ∩ Q +���� > 1 +2δn′ +and +���A ∩ Q +��� < 3 +2(δ + γ)n′ +(28) +with probability 1 − η1. Note that Q contains all the farthest (δ + γ)n points to oi. Denote by +li the +� +(δ + γ)n + 1 +� +-th largest distance from P to oi. Then we have +A ∩ Q = {p ∈ A | ||p − oi|| > li}. +(29) +Also, since Q′ is the set of the farthest 3 +2(δ + γ)n′ points to oi from A, there exists some l′ +i > 0 +such that +Q′ = {p ∈ A | ||p − oi|| > l′ +i}. +(30) +(29) and (30) together imply that either (A∩Q) ⊆ Q′ or Q′ ⊆ (A∩Q). Since +��A∩Q +�� < 3 +2(δ+γ)n′ +and |Q′| = 3 +2(δ + γ)n′, we know +� +A ∩ Q +� +⊆ Q′. Therefore, +� +A ∩ +� +Popt ∩ Q +�� += +� +Popt ∩ +� +A ∩ Q +�� +⊆ Q′. +(31) +Also, it is obvious that +� +A ∩ +� +Popt ∩ Q +�� +⊆ +� +Popt ∩ Q +� +. +(32) +The above (31) and (32) together imply +� +A ∩ +� +Popt ∩ Q +�� +⊆ +� +Q′ ∩ +� +Popt ∩ Q +�� +. +(33) +Moreover, since Q′ ⊆ A, we have +� +Q′ ∩ +� +Popt ∩ Q +�� +⊆ +� +A ∩ +� +Popt ∩ Q +�� +. +(34) +Consequently, (33) and (34) together imply Q′ ∩ +� +Popt ∩ Q +� += A ∩ +� +Popt ∩ Q +� +and hence +���Q′ ∩ +� +Popt ∩ Q +���� +|Q′| += +���A ∩ +� +Popt ∩ Q +���� +|Q′| +≥ +δ +3(δ + γ), +(35) +where the inequality comes from the first inequality of (28) and the fact |Q′| = 3 +2(δ + γ)n′. +⊓⊔ +17 + +The random sampling method is not always guaranteed to succeed. To boost the overall +success probability, we have to repeatedly run the algorithm multiple times and each time +the algorithm will generate a candidate solution (i.e., the ball center). Consequently we have +to select the best one as our final solution. With a slight abuse of notation, we still use oi +to denote a candidate ball center; since our goal is to achieve a (1 + ϵ, 1 − δ)-approximation, +we need to compute the +� +(δ + γ)n + 1 +� +-th largest distance from P to oi, which is denoted as +li. A straightforward way is to compute the value “li” in linear time for each candidate and +return the one having the smallest li. In this section, we propose the “Sandwich Lemma” to +estimate li in sublinear time. Let B be the set of n′′ sampled points from P in Lemma 8, and ˜li +be the +� +(1+δ/γ)2γn′′ +1 +� +-th largest distance from B to oi. If we can prove the inequalities (37) +and (38) of Lemma 8, then they can imply that ˜li is a qualified estimation of li: if B(oi, li) is a +(1 + ϵ, 1 − δ)-approximation, the ball B(oi, ˜li) should be a (1 + ϵ, 1 − O(δ))-approximation. The +key idea is to prove that the ball B(oi, ˜li) is “sandwiched” by two balls B(oi, ˜l′ +i) and B(oi, li), +where ˜l′ +i is a carefully designed value satisfying +(i) ˜l′ +i ≤ ˜li ≤ li and (ii) +���P \ B(oi, ˜l′ +i) +��� ≤ (γ + O(δ))n. +(36) +See Figure 5 for an illustration. These two conditions of ˜l′ +i can imply the inequalities (37) and +(38) of Lemma 8. Similar to Lemma 7, the sample size n′′ is also independent of n and d. +˜l′ +i˜l′ +i +lili +˜li˜li +Fig. 5: The red points are the sampled n′′ points in Lemma 8, and the +� +(1 + δ/γ)2γn′′ + 1 +� +-th +farthest point is in the ring bounded by the spheres B(oi, ˜l′ +i) and B(oi, li). +Lemma 8 (Sandwich Lemma). Let η2 ∈ (0, 1) and assume δ < γ/3. If we sample n′′ = +O +� γ +δ2 log 1 +η2 +� +points independently and uniformly at random from P and let ˜li be the +� +(1 + +δ/γ)2γn′′ + 1 +� +-th largest distance from the sample to oi, then, with probability 1 − η2, the +following holds +˜li ≤ li; +(37) +���P \ B(oi, ˜li) +��� ≤ (γ + 5δ)n. +(38) +Proof. Let B denote the set of sampled n′′ points from P. For simplicity, let t = (δ + γ)n. +Assume ˜l′ +i > 0 is the value such that +���P \ B(oi, ˜l′ +i) +��� = (γ+δ)2 +γ−δ n. Recall that li is the +� +t + 1 +� +-th +largest distance from P to oi. Since (δ + γ)n < (γ+δ)2 +γ−δ n, it is easy to know ˜l′ +i ≤ li. Below, we +aim to prove that the +� +(1 + δ/γ)2γn′′ + 1 +� +-th farthest point from B is in the ring bounded by +the spheres B(oi, ˜l′ +i) and B(oi, li) (see Figure 5). +18 + +Note the size |B| = n′′ = O +� γ +δ2 log 1 +η2 +� +. Again, using the Chernoff bound (let σ = δ/2) and +the same idea for proving (28), we have +���B \ B(oi, ˜l′ +i) +��� ≥ (1 − δ +2γ )(γ + δ)2 +γ − δ n′′ > (1 − δ +γ )(γ + δ)2 +γ − δ n′′ = (1 + δ/γ)2γn′′; +(39) +���B ∩ Q +�� ≤ (1 + δ +2γ ) t +nn′′ < (1 + δ/γ) t +nn′′ = (1 + δ/γ)2γn′′, +(40) +with probability 1 − η2. Suppose that (39) and (40) both hold. Recall that ˜li is the +� +(1 + +δ/γ)2γn′′ + 1 +� +-th largest distance from the sampled points B to oi, so +���B \ B(oi, ˜li) +��� = +(1 + δ/γ)2γn′′. Together with (39), we have +���B \ B(oi, ˜li) +��� ≤ +���B \ B(oi, ˜l′ +i) +���, i.e., +˜li ≥ ˜l′ +i. +(41) +The inequality (40) implies that the +� +(1 + δ/γ)2γn′′ + 1 +� +-th farthest point (say qx) from B +to oi is not in Q. Then, we claim that B(oi, ˜li) ∩ Q = ∅. Otherwise, let qy ∈ B(oi, ˜li) ∩ Q. Then +we have +||qy − oi|| ≤ ˜li = ||qx − oi||. +(42) +Note that Q is the set of farthest t points to oi of P. So qx /∈ Q implies +||qx − oi|| < min +q∈Q ||q − oi|| ≤ ||qy − oi|| +(43) +which is in contradiction to (42). Therefore, B(oi, ˜li) ∩ Q = ∅. Further, since B(oi, li) excludes +exactly the farthest t points (i.e., Q), “B(oi, ˜li) ∩ Q = ∅” implies +˜li ≤ li. +(44) +Overall, we have ˜li ∈ [˜l′ +i, li] from (41) and (44), i.e., the +� +(1 + δ/γ)2γn′′ + 1 +� +-th farthest +point from B locates in the ring bounded by the spheres B(oi, ˜l′ +i) and B(oi, li) as shown in +Figure 5. Also, ˜li ≥ ˜l′ +i implies +���P \ B(oi, ˜li) +��� ≤ +���P \ B(oi, ˜l′ +i) +��� = (γ + δ)2 +γ − δ n < (γ + 5δ)n, +(45) +where the last equality comes from the assumption δ < γ/3. So (37) and (38) are true in +Lemma 8. +⊓⊔ +Remark 6. Actually our proposed Uniform-Adaptive Sampling method and Sandwich lemma +are quite generic, and we will show that they can be generalized to solve a broader range of +enclosing with outliers problems in Section 7. +5.2 +Sublinear Time Algorithm for Bi-criteria Approximation +In this section, we propose a sublinear time algorithm for computing a bi-criteria (1 + ϵ, 1 − δ)- +approximation for the input instance (P, γ); that is, the returned ball covers at least +� +1−δ−γ +� +n +points and has radius at most (1 + ϵ)Rad(Popt). +Recall the remark following Theorem 1. As long as the selected point has a distance to the +center of MEB(T) larger than (1 + ϵ) times the optimal radius, the expected improvement +will always be guaranteed. Following this observation, we investigate the following approach. +Suppose we run the core-set construction procedure decribed in Theorem 1 (we should +19 + +replace P by Popt in our following analysis). In the i-th step, we add an arbitrary point from +Popt \ B(oi, (1 + ϵ)Rad(Popt)) to T. We know that a (1 + ϵ)-approximation is obtained after at +most +2 +(1−s)ϵ steps, that is, Popt ⊂ B +� +oi, (1 + ϵ)Rad(Popt) +� +for some i ≤ +2 +(1−s)ϵ. +However, we need to solve two key issues for realizing the above approach: (i) how to +determine the value of Rad(Popt) and (ii) how to correctly select a point from Popt \ B(oi, (1 + +ϵ)Rad(Popt)). Actually, we can implicitly avoid the first issue via replacing (1+ϵ)Rad(Popt) by +the t-th largest distance from the points of P to oi, where we set t = (δ + γ)n for guaranteeing +a (1 + ϵ, 1 − δ)-approximation. For the second issue, we randomly select one point from the +farthest t points of P to oi, and show that it belongs to Popt \ B(oi, (1 + ϵ)Rad(Popt)) with a +certain probability. +Based on the above idea, we present a sublinear time (1+ϵ, 1−δ)-approximation algorithm +in this section. To better understand the algorithm, we show a linear time algorithm first +(Algorithm 4 in Sections 5.2.1). Note that B˘adoiu et al. [22] also presented a (1 + ϵ, 1 − δ)- +approximation algorithm but with a higher complexity, and please see our detailed analysis on +the running time at the end of Sections 5.2.1. More importantly, we can improve the running +time of Algorithm 4 to be sublinear. For this purpose, we need to avoid computing the farthest +t points to oi, since this operation will take linear time. Also, Algorithm 4 generates a set +of candidates for the solution and we need to select the best one. This process also costs +linear time. By using the techniques proposed in Section 5.1, we can solve these issues and +develop a sublinear time algorithm that has the sample complexity independent of n and d, in +Section 5.2.2. +5.2.1 +A Linear Time Algorithm In this section, we present our linear time (1 + ϵ, 1 − δ)- +approximation algorithm for MEB with outliers. +Algorithm 4 (1 + ϵ, 1 − δ)-approximation Algorithm for MEB with Outliers +Input: A point set P with n points in Rd, the fraction of outliers γ ∈ (0, 1), and the parameters 0 < ϵ, δ < 1, +z ∈ Z+. +1: Let t = (δ + γ)n. +2: Initially, randomly select a point p ∈ P and let T = {p}. +3: i = 1; repeat the following steps until i > z: +(1) Denote by ci the exact center of MEB(T). Compute the approximate center oi with a distance to ci of +less than ξRad(T) = s +ϵ +1+ϵRad(T) as described in Theorem 1, where s is set to be +ϵ +2+ϵ. +(2) Let Q be the set of farthest t points from P to oi; denote by li the (t + 1)-th largest distance from P to +oi. +(3) Randomly select a point q ∈ Q, and add it to T. +(4) i = i + 1. +4: Output the ball B(oˆi, lˆi) where ˆi = argi min{li | 1 ≤ i ≤ z}. +Theorem 5. If the input parameter z = +2 +(1−s)ϵ (we assume it is an integer for convenience), +then with probability (1 − γ)( +δ +γ+δ)z, Algorithm 4 outputs a (1 + ϵ, 1 − δ)-approximation for the +MEB with outliers problem. +Before proving Theorem 5, we present the following two lemmas first. +Lemma 9. With probability (1 − γ)( +δ +γ+δ)z, after running z rounds in Step 3 of Algorithm 4, +the obtained set T ⊂ Popt. +Proof. Initially, because |Popt|/|P| = 1 − γ, the first selected point in Step 2 belongs to Popt +with probability 1 − γ. In each of the z rounds in Step 3, the selected point belongs to Popt +20 + +with probability +δ +γ+δ, since +|Popt ∩ Q| +|Q| += 1 − |Q \ Popt| +|Q| +≥ 1 − |P \ Popt| +|Q| += 1 − +γn +(δ + γ)n = +δ +δ + γ . +(46) +Therefore, with probability (1 − γ)( +δ +γ+δ)z the whole set T ⊂ Popt. +⊓⊔ +Lemma 10. In the i-th round of Step 3 for 1 ≤ i ≤ z, at least one of the following two events +happens: (1) oi is the ball center of a (1+ϵ, 1−δ)-approximation; (2) ri+1 > (1+ϵ)Rad(Popt)− +||ci − ci+1|| − ξri, where ri is the exact radius of MEB(T) is the i-th round. +Proof. If li ≤ (1 + ϵ)Rad(Popt), then we are done. That is, the ball B(oi, li) covers (1 − δ − γ)n +points with radius li ≤ (1 + ϵ)Rad(Popt) (the first event happens). Otherwise, li > (1 + +ϵ)Rad(Popt) and we consider the second event. Let q be the point added to T in the i-th +round. Using the triangle inequality, we have +||oi − q|| ≤ ||oi − ci|| + ||ci − ci+1|| + |ci+1 − q|| ≤ ξri + ||ci − ci+1|| + ri+1. +(47) +Since li > (1+ϵ)Rad(Popt) and q lies outside of B(oi, li), i.e, ||oi −q|| ≥ li > (1+ϵ)Rad(Popt), +(47) implies that the second event happens and the proof is completed. +⊓⊔ +Proof. (of Theorem 5) Suppose that the first event of Lemma 10 never happens. As a +consequence, we obtain a series of inequalities for each pair of radii ri+1 and ri, i.e., ri+1 > +(1 + ϵ)Rad(Popt) − ||ci − ci+1|| − ξri. Assume that T ⊂ Popt in Lemma 9, i.e., each time the +algorithm correctly adds a point from Popt to T. Using the almost identical idea for proving +Theorem 1 in Section 2.1, we know that a (1 + ϵ)-approximate MEB of Popt is obtained after +at most z rounds. The success probability directly comes from Lemma 9. Overall, we obtain +Theorem 5. +⊓⊔ +Theorem 5 directly implies the following corollary. +Corollary 1. If one repeatedly runs Algorithm 4 O( +1 +1−γ (1 + γ +δ )z) times, with constant prob- +ability, the algorithm outputs a (1 + ϵ, 1 − δ)-approximation for the problem of MEB with +outliers. +Running time. In Theorem 5, we set z = +2 +(1−s)ϵ and s ∈ (0, 1). To keep z small, according +to Theorem 1, we set s = +ϵ +2+ϵ so that z = 2 +ϵ + 1 (only larger than the lower bound 2 +ϵ by 1). +For each round of Step 3, we need to compute an approximate center oi that has a distance to +the exact one less than ξri = s +ϵ +1+ϵri = O(ϵ2)ri. Using the algorithm proposed in [20], this can +be done in O( 1 +ξ2 |T|d) = O( 1 +ϵ5 d) time. Also, the set Q can be obtained in linear time by the +algorithm in [19]. In total, the time complexity for obtaining a (1 + ϵ, 1 − δ)-approximation in +Corollary 1 is +O +�C +ϵ (n + 1 +ϵ5 )d +� +, +(48) +where C = O( +1 +1−γ (1 + γ +δ ) +2 +ϵ +1). As mentioned before, B˘adoiu et al. [22] also proposed a +linear time bi-criteria approximation. However, the hidden constant of their running time is +exponential in Θ( 1 +ϵδ) that is much larger than 2 +ϵ + 1. +21 + +5.2.2 +Improvement on Running Time In this section, we show that the running time +of Algorithm 4 can be further improved to be independent of the number of points n. First, +we observe that it is not necessary to compute the set Q of the farthest t points in Step 3(2) +of the algorithm. Actually, as long as the selected point q is part of Popt ∩ Q in Step 3(3), a +(1 + ϵ, 1 − δ)-approximation is still guaranteed. The Uniform-Adaptive Sampling procedure +proposed in Section 5.1 can help us to obtain a point q ∈ Popt ∩ Q without computing the set +Q. Moreover, in Lemma 8, we show that the radius of each candidate solution can be estimated +via random sampling. Overall, we achieve a sublinear time algorithm (Algorithm 5). Following +the analysis in Section 5.2.1, we set s = +ϵ +2+ϵ so that z = +2 +(1−s)ϵ = 2 +ϵ + 1. We present the results +in Theorem 6 and Corollary 2. Comparing with Theorem 5, we have an extra (1−η1)(1−η2) in +the success probability in Theorem 6, due to the probabilities from Lemmas 7 and 8. Another +minor issue is that the covering approximation error is increased from δ to 5δ when applying +Lemma 8. Actually this issue can be easily solved by replacing δ by δ/5 in the parameters n′, +t′, n′′, and t′′, and the asymptotic complexity does not change. +Algorithm 5 Sublinear Time (1 + ϵ, 1 − δ)-approximation Algorithm for MEB with Outliers +Input: A point set P with n points in Rd, the fraction of outliers γ ∈ (0, 1), and the parameters ϵ, η1, η2 ∈ (0, 1), +δ ∈ (0, 1/3γ), and z ∈ Z+. +1: Let n′ = O( 1 +δ log +1 +η1 ), n′′ = O +� γ +δ2 log +1 +η2 +� +, t′ = 3 +2(δ/5 + γ)n′, and t′′ = (1 + +δ +5γ )2γn′′. +2: Initially, randomly select a point p ∈ P and let T = {p}. +3: i = 1; repeat the following steps until j = z: +(1) Compute the approximate MEB center oi of T. +(2) Sample n′ points uniformly at random from P, and let Q′ be the set of farthest t′ points to oi from the +sample. +(3) Randomly select a point q ∈ Q′, and add it to T. +(4) Sample n′′ points uniformly at random from P, and let ˜li be the (t′′ + 1)-th largest distance from the +sampled points to oi. +(5) i = i + 1. +4: Output the ball B(oˆi, ˜lˆi) where ˆi = argi min{˜li | 1 ≤ i ≤ z}. +Theorem 6. If the input parameter z = 2 +ϵ + 1, then with probability (1 − γ) +� +(1 − η1)(1 − +η2) +δ/5 +3(γ+δ/5) +�z, Algorithm 5 outputs a (1 + ϵ, 1 − δ)-approximation for the problem of MEB with +outliers. +To boost the success probability in Theorem 6, we need to repeatedly run Algorithm 5 +and output the best candidate. However, we need to be careful on setting the parameters. +The success probability in Theorem 6 consists of two parts, P1 = (1 − γ) +� +(1 − η1) +δ/5 +3(γ+δ/5) +�z +and P2 = (1 − η2)z, where P1 indicates the probability that {o1, · · · , oz} contains a qualified +candidate, and P2 indicates the success probability of Lemma 8 over all the z rounds. Therefore, +if we run Algorithm 5 N = O( 1 +P1 ) times, with constant probability (by taking the union +bound), the set of all the generated candidates contains at least one that yields a (1 + ϵ, 1 − δ)- +approximation; moreover, to guarantee that we can correctly estimate the resulting radii of all +the candidates via the Sandwich Lemma with constant probability, we need to set η2 = O( 1 +zN ) +(because there are O(zN) candidates). +Corollary 2. If one repeatedly runs Algorithm 5 N = O +� +1 +1−γ +� +1 +1−η1 (3 + 3γ +δ/5) +�z� +times with +setting η2 = O( 1 +zN ), with constant probability, the algorithm outputs a (1+ϵ, 1−δ)-approximation +for the problem of MEB with outliers. +22 + +The calculation of running time is similar to (48) in Section 5.2.1. We just replace n by +max{n′, n′′} = O +� γ +δ2 log 1 +η2 +� += O +� γ +δ2 log(zN) +� += ˜O +� γ +δ2ϵ +� 3, and change the value of C to be +O +� +1 +1−γ +� +1 +1−η1 (3 + 3γ +δ/5) +� 2 +ϵ +1� +. So the total running time is independent of n. +5.3 +General MEB Problem +Now we consider solving the general MEB problem without the stability assumption in this +Section. Let 0 < ϵ, δ < 1 be two given parameters. First, we view the input P as an instance +(P, δ/2) of MEB with outliers (i.e., γ = δ/2). Then, we apply the algorithm of Section 5.2 to +obtain a bi-criteria (1+ϵ2/2, 1−δ/2)-approximation solution B(c, rc) (we replace the “ϵ” by ϵ2/2 +and replace the “δ” by δ/2). The obtained ball B(c, rc) covers at least (1−δ/2−δ/2)n = (1−δ)n +points of P, and the radius +rc ≤ (1 + 1 +2ϵ2) · r−δ/2, +(49) +where r−δ/2 stands for the radius of the smallest ball that covers at least (1 − δ/2)n points of +P. +Second, we assume that the input P is an (α, β)-stable instance with α = ϵ2 and β = δ/2; +then run Algorithm 2 to obtain a candidate ball center ˜o (of course, we can also use Algorithm 1, +where the only difference is that the sample complexity will be higher). To compute the real +radius r˜o yielded from ˜o (since P may not be a real (α, β)-stable instance), we just need to +read the whole dataset P in one pass. Finally, we determine the final output based on the +ratio r˜o/rc. +Algorithm 6 Hybrid Approximation for MEB +Input: An instance P of MEB problem in Rd; two parameters 0 < ϵ, δ < 1. +1: View the input as a (P, δ/2) instance of MEB with outliers; apply the method of Corollary 2 to obtain a +bi-criteria (1 + ϵ2/2, 1 − δ/2)-approximation solution B(c, rc) on (P, δ/2). +2: Assume that the input P is an (α, β)-stable instance with α = ϵ2 and β = δ/2; then run Algorithm 2 to +obtain a candidate ball center ˜o. +3: Read the whole input dataset P in one-pass, and compute the radius r˜o = maxp∈P ||˜o − p||. +4: If the ratio r˜o +rc ≤ +1+ϵ +1−ϵ2/2, return the ball B(˜o, r˜o) and say “it is a (1 + ϵ)-radius approximation”. +5: Else, return the ball B(c, rc) and say “it is a (1 − δ)-covering approximation”. +Theorem 7. With constant success probability, Algorithm 6 returns either a (1 + ϵ)-radius +approximation or a (1−δ)-covering approximation, and the running time is O +�� +n+h(ϵ, δ) +� +·d +� +, +where h(ϵ, δ) = O +� +1 +1−δ/2 exp(O(1/ϵ2)) +� +. The algorithm only needs uniform sampling and a +single pass over the input data, and the space complexity in memory is O(h(ϵ, δ) · d). Moreover, +if the input data matrix (the n × d matrix representing the input P) has at most M ≪ nd +non-zeros entries, the total running time will be O +� +n + h(ϵ, δ) · d + M +� +. +Remark 7. In the following proof, we will see that when the algorithm returns a (1 − δ)- +covering approximation, the returned radius is not only ≤ Rad(P), but also at most +� +1 − +Θ(ϵ2) +� +· Rad(P) (see (52) and (54)). +Proof. We study the time and space complexities first. The method of Corollary 2 only needs +uniform samplings, and Step 2 of Algorithm 6 is a single pass over the input data. According to +3 The asymptotic notation ˜O(f) = O +� +f · polylog( +γ +η1δ(1−γ)) +� +. +23 + +Corollary 2, we know the space complexity is O(h(ϵ, δ)·d) with h(ϵ, δ) = O +� +1 +1−δ/2 exp(O(1/ϵ2)) +� +. +The total running time is O +�� +n + h(ϵ, δ) +� +· d +� +. Furthermore, we consider the case that the +input matrix is sparse. In Step 3, we need to compute the value r˜o = maxp∈P ||˜o − p||. For +each point p ∈ P, we know +||˜o − p||2 = ||˜o||2 + ||p||2 − 2⟨˜o, p⟩, +(50) +where ⟨˜o, p⟩ stands for their inner product. The value of ||˜o||2 can be obtained in O(d) time, and +if the input data matrix has at most M ≪ nd non-zeros entries, the complexity for computing +the values {||p||2 − 2⟨˜o, p⟩ | p ∈ P} is O(n + M). Overall, the complexity of Algorithm 6 is +O +� +n + h(ϵ, δ) · d + M +� +. +Now, we prove the solution quality. We let α = ϵ2 and β = δ/2, and consider the following +two cases. +Case 1: the instance P is (α, β)-stable. Then we directly have +r˜o ≤ (1 + ϵ) · Rad(P). +(51) +If r˜o +rc > +1+ϵ +1−ϵ2/2, together with (51), we have +rc < +� +1 − ϵ2/2 +� +· Rad(P). +(52) +Then we can return the ball B(c, rc) and say “it is a (1 − δ)-covering approximation”. On +the other hand, when r˜o +rc ≤ +1+ϵ +1−ϵ2/2, from (51) we can return the ball B(˜o, r˜o) and say “it is a +(1 + ϵ)-radius approximation”. +Case 2: P is not an (α, β)-stable instance. Then, from the definition of stability we know +the optimal radius of the instance (P, δ/2) is no larger than +(1 − ϵ2) · Rad(P). +(53) +So we have +rc < (1 + 1 +2ϵ2)(1 − ϵ2) · Rad(P) < +� +1 − ϵ2/2 +� +· Rad(P). +(54) +If r˜o +rc ≤ +1+ϵ +1−ϵ2/2, together with (54), it implies +r˜o < (1 + ϵ) · Rad(P). +(55) +Then we can return the ball B(˜o, r˜o) and say “it is a (1 + ϵ)-radius approximation”. On the +other hand, when r˜o +rc > +1+ϵ +1−ϵ2/2, from (54) we can return the ball B(c, rc) and say “it is a +(1 − δ)-covering approximation”. +Since the success probability of the method of Section 5.2 is constant, the overall success +probability of Algorithm 7 is constant as well. +⊓⊔ +More analysis on the result of Algorithm 6. We further consider an “inverse” question: +can we infer the stability degree of the given instance P from the output of Algorithm 6? In +Step 2, we assume that P is an (ϵ2, δ/2)-stable instance, but this may not be true in reality. +Recall the definition of “(α, β)-stable” in Definition 3. We know that there always exists a +value ˆα ∈ [0, 1) such that P is a (ˆα, δ/2)-stable. We can use “ˆα” to indicate the stability degree +of P, for the fixed “δ/2”. The following theorem shows that we can infer the value of ˆα through +Algorithm 6. +24 + +Theorem 8. If Algorithm 6 returns a (1 + ϵ)-radius approximation, then ˆα < ϵ; otherwise, +the algorithm returns a (1 − δ)-covering approximation and it implies ˆα > ϵ2 +2 . +In other words, the algorithm can distinguish the case ˆα ≥ ϵ (it must returns a (1 − δ)- +covering approximation) and the case ˆα ≤ ϵ2 +2 (it must returns a (1 + ϵ)-radius approximation); +but if ϵ2 +2 < ˆα < ϵ, the algorithm can return either a (1 − δ)-covering approximation or a +(1 + ϵ)-radius approximation. +Proof. Recall we set α = ϵ2 and β = δ/2 in Algorithm 6. First, we suppose the output is +a (1 + ϵ)-radius approximation. One possible case is the instance P is a real (α, β)-stable +instance, and then ˆα = α < ϵ. The other possible case is that P is not (α, β)-stable but the +ratio r˜o +rc ≤ +1+ϵ +1−ϵ2/2. Together with (49), we have +Rad(P) +r−δ/2 +≤ +r˜o +1 +1+ϵ2/2rc +≤ (1 + ϵ)(1 + ϵ2/2) +1 − ϵ2/2 +. +(56) +So ˆα = 1 − +r−δ/2 +Rad(P) ≤ 1 − +1−ϵ2/2 +(1+ϵ)(1+ϵ2/2) < ϵ. Overall, as long as the output is a (1 + ϵ)-radius +approximation, ˆα should be smaller than ϵ. +Then we suppose the output is a (1 − δ)-covering approximation. One possible case is the +instance P is not (α, β)-stable, and then ˆα > α = ϵ2. The other possible case is that P is +(α, β)-stable but the ratio r˜o +rc > +1+ϵ +1−ϵ2/2. Together with (51), we have +Rad(P) +r−δ/2 +≥ +1 +1+ϵr˜o +rc +> +1 +1 − ϵ2/2. +(57) +So ˆα = 1 − +r−δ/2 +Rad(P) > 1 − (1 − ϵ2/2) = ϵ2/2. Overall, as long as the output is a (1 − δ)-covering +approximation, ˆα > min{ϵ2, ϵ2/2} = ϵ2 +2 . +⊓⊔ +6 +Extension I: Hybrid Approximation for MEB with Outliers +In this section, we extend the idea of Section 5.3 to present a hybrid approximation algorithm +for the MEB with outliers problem (P, γ). First, we extend Definition 3 of MEB to MEB with +outliers. +Definition 6 ((α, β)-stable for MEB with Outliers). Let 0 < α, β < 1. Given an instance +(P, γ) of the MEB with outliers problem in Definition 4, (P, γ) is an (α, β)-stable instance if +(1) Rad(P \ Q) > (1 − α)Rad(Popt) for any Q ⊂ P with |Q| < +� +γ + β +� +n, and (2) there exists +a Q′ ⊂ P with |Q′| = ⌈(β + γ)n⌉ having Rad(P \ Q′) ≤ (1 − α)Rad(Popt). +Definition 6 directly implies the following claim. +Claim 2. If (P, γ) is an (α, β)-stable instance of the problem of MEB with outliers, the +corresponding Popt is an (α, ˜β)-stable instance of MEB with ˜β ≥ +β +1−γ . +Note that Definition 6 implicitly requires β < 1 − γ. So it implies the lower bound +β +1−γ of ˜β +in Claim 2 cannot be larger than 1. To see the correctness of Claim 2, we can use contradiction. +Suppose that there exists a subset P ′ ⊂ Popt such that |P ′| > (1 − +β +1−γ )|Popt| = (1 − γ − β)n +and Rad(P ′) ≤ (1 − α)Rad(Popt). Then, it is in contradiction to the fact that (P, γ) is an +(α, β)-stable instance of MEB with outliers. +To apply the idea of Section 5.3, a significant challenge is that the set Popt is mixed with +the outliers, and thus we cannot easily obtain a (1 + ϵ)-radius approximation as Algorithm 6. +Our starting point is still the sublinear time bi-criteria approximation algorithm proposed in +25 + +Section 5.2. Specifically, given any two small parameters 0 < ϵ, δ < 1, the algorithm returns a +set of candidate ball centers via the uniform-adaptive sampling procedure. We use Ξ to denote +this set. With constant probability, as least one candidate from Ξ, say s, satisfies the following +inequality: +��B +� +s, (1 + ϵ) · Rad(Popt) +� +∩ P +�� ≥ +� +1 − δ − γ +� +n. +(58) +Namely, it is a “(1 + ϵ, 1 − δ)-approximation”. To pick such a qualified candidate, it is possible +to estimate the size of B +� +s, (1 + ϵ) · Rad(Popt) +� +∩ P by using the uniform sampling based +technique “sandwich lemma” (instead of reading the whole dataset P). It is worth to note an +implicit fact about Theorem 5 of Section 5.2. Actually, in the proof it showed that among the +candidate set Ξ, there exists one solution s such that the ball B +� +s, (1 + ϵ) · Rad(Popt) +� +covers +at least +� +1 − δ − γ +� +n points from Popt (since the set T ⊂ Popt and the solution s is generated +from T (see Lemma 9)). So the solution s should satisfy +��B +� +s, (1 + ϵ) · Rad(Popt) +� +∩ Popt +�� ≥ +� +1 − δ − γ +� +n, +(59) +which is stronger than (58). But the sandwich lemma may ignore such a stronger solution, +since only selecting a solution satisfying (58) is already sufficient to guarantee a (1 + ϵ, 1 − δ)- +approximation. We introduce the following new algorithm for MEB with outliers based on this +observation. +The hybrid approximation algorithm. Let ϵ and δ be the two given parameters. First, +we apply the method of Section 5.2. But we do not directly input the couple (ϵ, δ) to the +bi-criteria approximation algorithm; instead, we use ( +1 +2(2 +√ +2+ +√ +3)2 ϵ2, δ) (we will explain why we +have the coefficient “ +1 +2(2 +√ +2+ +√ +3)2 ” in our analysis). That is, we compute a set Ξ of candidate +ball centers via the uniform-adaptive sampling of Section 5.2, and at least one center yields +a (1 + +1 +2(2 +√ +2+ +√ +3)2 ϵ2, 1 − δ)-approximation for the instance (P, γ). Then, for each candidate +q ∈ Ξ, we define two values: +rq = min +� +r > 0 | +��B(q, r) ∩ P +�� ≥ (1 − γ)n +� +; +(60) +r′ +q = min +� +r > 0 | +��B(q, r) ∩ P +�� ≥ +� +1 − δ − γ +� +n +� +. +(61) +We can compute these two values for all the candidates of Ξ by scanning the input P in one pass +(instead of using the sandwich lemma). We select the two points s1 = arg minq∈Ξ rq and s2 = +arg minq∈Ξ r′ +q (they may or may not be the same point). If the ratio rs1 +r′s2 ≤ +1+ϵ +1−ϵ2/ +� +2(2 +√ +2+ +√ +3)2�, +return the ball B(s1, rs1) and say “it is a (1 + ϵ)-radius approximation”; else, return the ball +B(s2, r′ +s2) and say “it is a (1 − δ)-covering approximation”. +Algorithm 7 Hybrid Approximation for MEB with Outliers +Input: An instance (P, γ) of MEB with outliers problem in Rd; two parameters 0 < ϵ, δ < 1. +1: Apply the uniform-adaptive sampling method of Section 5.2 to obtain a set Ξ of candidate ball centers, +where at least one center yields a (1 + +1 +2(2 +√ +2+ +√ +3)2 ϵ2, 1 − δ)-approximation for the instance (P, γ). +2: Read the whole input dataset P in one pass, and compute the values rq and r′ +q as the formulas (60) and +(61) for each q ∈ Ξ. +3: Let s1 = arg minq∈Ξ rq and s2 = arg minq∈Ξ r′ +q. +4: If the ratio +rs1 +r′s2 ≤ +1+ϵ +1−ϵ2/ +� +2(2 +√ +2+ +√ +3)2�, return the ball B(s1, rs1) and say “it is a (1+ϵ)-radius approximation”. +5: Else, return the ball B(s2, r′ +s2) and say “it is a (1 − δ)-covering approximation”. +26 + +Theorem 9. With constant success probability, Algorithm 7 returns either a (1 + ϵ)-radius +approximation or a (1 − δ)-covering approximation, and the running time is O(g(ϵ, δ, γ) · nd), +where g(ϵ, δ, γ) = O( +1 +1−γ ( γ+δ +δ )O(1/ϵ2)). The algorithm only needs uniform sampling and a single +pass over the input data, and the space complexity in memory is O(g(ϵ, δ, γ) · d). Moreover, +if the input data matrix (the n × d matrix representing the input P) has at most M ≪ nd +non-zeros entries, the total running time will be O +� +g(ϵ, δ, γ) · (n + d + M) +� +. +Remark 8. Similar to Theorem 7, we will see that when the algorithm returns a (1 − δ)- +covering approximation, the returned radius is at most +� +1 − Θ(ϵ2) +� +· Rad(Popt) (see (63) and +(64)). +Proof. (of Theorem 9) We study the time and space complexities first. The method of +Corollary 2 only needs uniform samplings, and Step 2 of Algorithm 7 is a single pass over the +input data. The size of Ξ is g(ϵ, δ, γ) = O( +1 +1−γ ( γ+δ +δ )O(1/ϵ2)) based on Corollary 2. Overall, the +space complexity is O(g(ϵ, δ, γ)·d). And the complexity for generating Ξ is O +� +|Ξ|·poly( 1 +ϵ, 1 +δ)d +� +which is sublinear in the input size nd. It is easy to see that the complexity of Step 2 dominates +the whole complexity. Therefore, the total running time is O(g(ϵ, δ, γ) · nd). Furthermore, we +consider the case that the input matrix is sparse. Similar to the proof of Theorem 7, we know +that the complexity of Algorithm 7 is O +� +g(ϵ, δ, γ) · (n + d + M) +� +if the input data matrix has +at most M ≪ nd non-zeros entries. +Now, we prove the solution quality. We let α = +1 +(2 +√ +2+ +√ +3)2 ϵ2 and β = (1 − γ)δ, and consider +the following two cases. +Case 1: the instance (P, γ) is (α, β)-stable (i.e., Popt is an (α, ˜β)-stable instance of MEB +with ˜β ≥ δ, according to Claim 2). Denote by o the optimal center of MEB(Popt). We suppose +one candidate ball center q0 of Ξ satisfies the formula (59). As a consequence, from Theorem 2, +we know that ||q0 − o|| ≤ (2 +√ +2 + +√ +3)√α · Rad(Popt) = ϵ · Rad(Popt). That is, +rs1 ≤ rq0 ≤ (1 + ϵ) · Rad(Popt). +(62) +If rs1 +r′s2 > +1+ϵ +1−ϵ2/ +� +2(2 +√ +2+ +√ +3)2�, together with (62), we have +r′ +s2 < +� +1 − ϵ2/ +� +2(2 +√ +2 + +√ +3)2�� +· Rad(Popt). +(63) +Then we can return the ball B(s2, r′ +s2) and say “it is a (1 − δ)-covering approximation”. On +the other hand, when rs1 +r′s2 ≤ +1+ϵ +1−ϵ2/ +� +2(2 +√ +2+ +√ +3)2�, from (62) we can return the ball B(s1, rs1) and +say “it is a (1 + ϵ)-radius approximation”. +Case 2: (P, γ) is not an (α, β)-stable instance. Then it implies +r′ +s2 < (1 + +1 +2(2 +√ +2 + +√ +3)2 ϵ2)(1 − +1 +(2 +√ +2 + +√ +3)2 ϵ2) · Rad(Popt) +< +� +1 − ϵ2/ +� +2(2 +√ +2 + +√ +3)2�� +· Rad(Popt). +(64) +If rs1 +r′s2 ≤ +1+ϵ +1−ϵ2/ +� +2(2 +√ +2+ +√ +3)2�, together with (64), it implies +rs1 < (1 + ϵ) · Rad(Popt). +(65) +Then we can return the ball B(s1, rs1) and say “it is a (1 + ϵ)-radius approximation”. On the +other hand, when rs1 +r′s2 > +1+ϵ +1−ϵ2/ +� +2(2 +√ +2+ +√ +3)2�, from (64) we can return the ball B(s2, r′ +s2) and say +“it is a (1 − δ)-covering approximation”. +Since the success probability of the method of Section 5.2 is constant, the overall success +probability of Algorithm 7 is constant as well. +⊓⊔ +27 + +We also have the following theorem for inferring the stability of the instance (P, γ), and +the proof is almost identical to the proof of Theorem 8. +Theorem 10. Suppose (P, γ) is a (ˆα, (1 − γ)δ)-stable instance. If Algorithm 7 returns a +(1 + ϵ)-radius approximation, then ˆα < ϵ; otherwise, the algorithm returns a (1 − δ)-covering +approximation and it implies ˆα > +ϵ2 +2(2 +√ +2+ +√ +3)2 . +7 +Extension II: Bi-criteria Approximations for MEX With Outliers +In this section, we extend Definition 4 for MEB with outliers and define a more general problem +called minimum enclosing “x” (MEX) with Outliers. Then we show that the ideas of +Lemma 7 and 8 can be generalized to deal with MEX with outliers problems, as long as the +shape “x” satisfies several properties. To describe a shape “x”, we need to clarify three +basic concepts: center, size, and distance function. +Let X be the set of specified shapes in Rd. We require that each shape x ∈ X is uniquely +determined by the following two components: “c(x)”, the center of x, and “s(x) ≥ 0”, the size +of x. For any two shapes x1, x2 ∈ X, x1 = x2 if and only if c(x1) = c(x2) and s(x1) = s(x2). +Moreover, given a center o0 and a value l0 ≥ 0, we use x(o0, l0) to denote the shape x with +c(x) = o0 and s(x) = l0. For different shapes, we have different definitions for the center +and size. For example, if x is a ball, c(x) and s(x) should be the ball center and the radius +respectively; given o0 ∈ Rd and l0 ≥ 0, x(o0, l0) should be the ball B(o0, l0). As a more +complicated example, consider the k-center clustering with outliers problem, which is to find +k balls to cover the input point set excluding a certain number of outliers and minimize the +maximum radius (w.l.o.g., we can assume that the k balls have the same radius). For this +problem, the shape “x” is a union of k balls in Rd; the center c(x) is the set of the k ball +centers and the size s(x) is the radius. +For any point p ∈ Rd and any shape x ∈ X, we also need to define a distance function +f(c(x), p) between the center c(x) and p. For example, if x is a ball, f(c(x), p) is simply equal +to ||p − c(x)||; if x is a union of k balls with the center c(x) = {c1, c2, · · · , ck}, the distance +should be min1≤j≤k ||p − cj||. Note that the distance function is only for ranking the points to +c(x), and not necessary to be non-negative (e.g., in Section 7.3, we define a distance function +f(c(x), p) ≤ 0 for SVM). By using this distance function, we can define the set “Q” and the +value “li” when generalizing Lemma 7 and 8 below. To guarantee their correctnesses, we also +require X to satisfy the following three properties. +Property 1. For any two shapes x1 ̸= x2 ∈ X, if c(x1) = c(x2), then +s(x1) ≤ s(x2) ⇐⇒ x1 is covered by x2, +(66) +where “x1 is covered by x2” means “for any point p ∈ Rd, p ∈ x1 ⇒ p ∈ x2”. +Property 2. Given any shape x ∈ X and any point p0 ∈ x, the set +{p | p ∈ Rd and f(c(x), p) ≤ f(c(x), p0)} ⊆ x. +(67) +Property 3. Given any shape center o0 and any point p0 ∈ Rd, let r0 = min{r | r ≥ 0, p0 ∈ +x(o0, r)}. Then p0 ∈ x(o0, r0) and p0 /∈ x(o0, r) for any r < r0. (Note: usually the value r0 is +just the distance from p0 to the shape center o0; but for some cases, such as the SVM problem +in Section 7.3, the shape size and distance function have different meanings). +Intuitively, Property 1 shows that s(x) defines an order of the shapes sharing the same +center c(x). Property 2 shows that the distance function f defines an order of the points to a +28 + +given shape center c(x). Property 3 shows that a center o0 and a point p0 can define a shape +just “touching” p0. We can take X = {all d-dimensional balls} as an example. For any two +concentric balls, the smaller one is always covered by the larger one (Property 1); if a point +p0 is inside a ball x, any point p having the distance ||p − c(x)|| ≤ ||p0 − c(x)|| should be +inside x too (Property 2); also, given a ball center o0 and a point p0, p0 ∈ B(o0, ||p0 − o0||) and +p0 /∈ B(o0, r) for any r < ||p0 − o0|| (Property 3). +Now, we introduce the formal definitions of the MEX with outliers problem and its +bi-criteria approximation. +Definition 7 (MEX with Outliers). Suppose the shape set X satisfies Property 1, 2, and +3. Given a set P of n points in Rd and a small parameter γ ∈ (0, 1), the MEX with outliers +problem is to find the smallest shape x ∈ X that covers (1 − γ)n points. Namely, the task is to +find a subset of P with size (1−γ)n such that its minimum enclosing shape of X is the smallest +among all possible choices of the subset. The obtained solution is denoted by MEX(P, γ). +Definition 8 (Bi-criteria Approximation). Given an instance (P, γ) for MEX with out- +liers and two small parameters 0 < ϵ, δ < 1, a (1+ϵ, 1−δ)-approximation of (P, γ) is a solution +x ∈ X that covers at least +� +1 − δ − γ +� +n points and has the size at most (1 + ϵ)s(xopt), where +xopt is the optimal solution. +It is easy to see that Definition 4 of MEB with outliers actually is a special case of Definition 7. +Similar to MEB with outliers, we still use Popt, where Popt ⊂ P and |Popt| = (1 − γ)n, to +denote the subset covered by the optimal solution of MEX with outliers. +Now, we provide the generalized versions of Lemma 7 and 8. Similar to the core-set +construction method in Section 2.1, we assume that there exists an iterative algorithm Γ to +compute MEX (without outliers); actually, this is an important prerequisite to design the +sub-linear time algorithms under our framework (we will discuss the iterative algorithms for +the MEX with outliers problems including flat fitting, k-center clustering, and SVM, in the +following subsections). In the i-th iteration of Γ, it maintains a shape center oi. Also, let Q be +the set of (δ + γ)n farthest points from P to oi with respect to the distance function f. First, +we need to define the value “li” by Q in the following claim. +Claim 3. There exists a value li ≥ 0 satisfying P \ x(oi, li) = Q. +Proof. The points of P can be ranked based on their distances to oi. Without loss of generality, +let P = {p1, p2, · · · , pn} with f(oi, p1) > f(oi, p2) > · · · > f(oi, pn) (for convenience, we assume +that any two distances are not equal; if there is a tie, we can arbitrarily decide their order to oi). +Then the set Q = {pj | 1 ≤ j ≤ (δ +γ)n}. Moreover, from Property 3, we know that each point +pj ∈ P corresponds to a value rj that pj ∈ x(oi, rj) and pj /∈ x(oi, r) for any r < rj. Denote +by xj the shape x(oi, rj). We select the point pj0 with j0 = (δ + γ)n + 1. From Property 2, we +know that pj ∈ xj0 for any j ≥ j0, i.e., (a) P \ Q ⊆ xj0. We also need to prove that pj /∈ xj0 +for any j < j0. Assume there exists some pj1 ∈ xj0 with j1 < j0. Then we have rj1 < rj0 and +thus pj0 /∈ xj1 (by Property 3). By Property 2, pj0 /∈ xj1 implies f(oi, pj0) > f(oi, pj1), which +is in contradiction to the fact f(oi, pj0) < f(oi, pj1). So we have (b) Q ∩ xj0 = ∅. +The above (a) and (b) imply that {P ∩ xj0, Q} is a partition of P, i.e., (P ∩ xj0) ∪ Q = P +and (P ∩ xj0) ∩ Q = ∅. So we know P \ xj0 = Q. Therefore, we can set the value li = rj0 and +then P \ x(oi, li) = Q. +⊓⊔ +Lemma 11 (Generalized Uniform-Adaptive Sampling). Let η1 ∈ (0, 1). If we sample +n′ = O( 1 +δ log 1 +η1 ) points independently and uniformly at random from P and let Q′ be the set +of farthest 3 +2(δ + γ)n′ points to oi from the sample, then, with probability at least 1 − η1, the +29 + +following holds +���Q′ ∩ +� +Popt ∩ Q +���� +|Q′| +≥ +δ +3(γ + δ). +(68) +Proof. Let A denote the set of sampled n′ points from P. Similar to (28), we have +���A ∩ +� +Popt ∩ Q +���� > 1 +2δn′ +and +���A ∩ Q +��� < 3 +2(δ + γ)n′ +(69) +with probability 1 − η1. Similar to (29), we have +A ∩ Q = {p ∈ A | f(oi, p) > f(oi, pj0)}, +(70) +where pj0 is the point selected in the proof of Claim 3. By using the same manner of Claim 3, +we also can select a point pj′ +0 ∈ A with +Q′ = {p ∈ A | f(oi, p) > f(oi, pj′ +0)}. +(71) +Then, we can prove +� +A ∩ +� +Popt ∩ Q +�� += +� +Q′ ∩ +� +Popt ∩ Q +�� +. +(72) +by using the same idea of (33). Hence, +���Q′ ∩ +� +Popt ∩ Q +���� +|Q′| += +���A ∩ +� +Popt ∩ Q +���� +|Q′| +≥ +δ +3(γ + δ), +(73) +where the final inequality comes from the first inequality of (69) and the fact |Q′| = 3 +2(δ + γ)n′. +⊓⊔ +Lemma 12 (Generalized Sandwich Lemma). Let η2 ∈ (0, 1) and assume δ < γ/3. li is +the value from Claim 3. We sample n′′ = O +� γ +δ2 log 1 +η2 +� +points independently and uniformly at +random from P and let q be the +� +(1 + δ/γ)2γn′′ + 1 +� +-th farthest one from the sampled points +to oi. If ˜li = min{r | r ≥ 0, q ∈ x(oi, r)} (similar to the way defining “r0” in Property 3), then, +with probability 1 − η2, the following holds +˜li ≤ li; +(74) +���P \ x(oi, ˜li) +��� ≤ (γ + 5δ)n. +(75) +Proof. Let B denote the set of sampled n′′ points from P. By using the same manner of +Claim 3, we know that there exists a value ˜l′ +i > 0 satisfying +���P \x(oi, ˜l′ +i) +��� = (γ+δ)2 +γ−δ γn. Similar to +the proof of Lemma 8, we can prove that ˜li ∈ [˜l′ +i, li]. Due to Property 1, we know that x(oi, ˜li) +is “sandwiched” by the two shapes x(oi, ˜l′ +i) and x(oi, li). Further, since x(oi, ˜l′ +i) is covered by +x(oi, ˜li), we have +���P \ x(oi, ˜li) +��� ≤ +���P \ x(oi, ˜l′ +i) +��� = (γ + δ)2 +γ − δ γn = (γ + 5δ)n, +(76) +where the last equality comes from the assumption δ < γ/3. So (74) and (75) are true. +⊓⊔ +By using Lemma 11 and Lemma 12, we study several applications in the following subsec- +tions. +30 + +7.1 +k-Center Clustering with Outliers +Let γ ∈ (0, 1). Given a set P of n points in Rd, the problem of k-center clustering with +outliers is to find k balls to cover (1 − γ)n points, and the maximum radius of the balls is +minimized (w.l.o.g., we can assume that the k balls have the same radius). Given an instance +(P, γ), let {C1, · · · , Ck} be the k clusters forming Popt (the subset of P yielding the optimal +solution), and ropt be the optimal radius; that is, each Cj is covered by an individual ball with +radius ropt. Similar to Section 5.2, we first introduce a linear time algorithm, and then show +how to modify it to be sublinear time by using Lemma 11 and 12. +Linear time algorithm. Our algorithm in Section 5.2.1 can be generalized to be a linear +time bi-criteria algorithm for the problem of k-center clustering with outliers, if k is assumed +to be a constant. Our idea is as follows. In Algorithm 4, we maintain a set T as the core-set +of Popt; here, we instead maintain k sets T1, T2, · · · , Tk as the core-sets of C1, C2, · · · , Ck, +respectively. Consequently, each Tj for 1 ≤ j ≤ k has an approximate MEB center oj +i in the +i-th round of Step 3, and we let Oi = {o1 +i , · · · , ok +i }. Initially, O0 and Tj for 1 ≤ j ≤ k are all +empty; we randomly select a point p ∈ P, and with probability 1 − γ, p ∈ Popt (w.l.o.g., we +assume p ∈ C1 and add it to T1; thus O1 = {p} after this step). We let Q be the set of farthest +t = (δ + γ)n points to Oi, and li be the (t + 1)-th largest distance from P to Oi (the distance +from a point p ∈ P to Oi is min1≤j≤k ||p − oj +i||). Then, we randomly select a point q ∈ Q, and +with probability +δ +γ+δ, q ∈ Popt (as (46) in Lemma 9). For ease of presentation, we assume +that q ∈ Popt happens and we have an “oracle” to guess which optimal cluster q belongs to, +say q ∈ Cjq; then, we add q to Tjq and update the approximate MEB center of Tjq. Since +each optimal cluster Cj for 1 ≤ j ≤ k has the core-set with size 2 +ϵ + 1 (by setting s = +ϵ +2+ϵ +in Theorem 1), after adding at most k( 2 +ϵ + 1) points, the distance li will be smaller than +(1 + ϵ)ropt. Consequently, a (1 + ϵ, 1 − δ)-approximation solution is obtained when i ≥ k( 2 +ϵ + 1). +Note that some “small” clusters could be missing from the above random sampling based +approach and therefore |Oi| could be less than k; however, it always can be guaranteed that +the total number of missing inliers is at most δn, i.e., a (1 + ϵ, 1 − δ)-approximation is always +guaranteed (otherwise, the ratio |Popt∩Q| +|Q| +> +δ +γ+δ and we can continue to sample a point from +Popt and then update Oi). +To remove the oracle for guessing the cluster containing q, we can enumerate all the possible +k cases; since we add k( 2 +ϵ + 1) points to T1, T2, · · · , Tk, it generates kk( 2 +ϵ +1) = 2k log k( 2 +ϵ +1) +solutions in total, and at least one yields a (1 + ϵ, 1 − δ)-approximation with probability +(1 − γ)( +δ +γ+δ)k( 2 +ϵ +1) (by the same manner for proving Theorem 5). +Theorem 11. Let (P, γ) be an instance of k-center clustering with outliers. Given two pa- +rameters ϵ, δ ∈ (0, 1), there exists an algorithm that outputs a (1 + ϵ, 1 − δ)-approximation with +probability (1 − γ)( +δ +γ+δ)k( 2 +ϵ +1). The running time is O(2k log k( 2 +ϵ +1)(n + 1 +ϵ5 )d). +If one repeatedly runs the algorithm O( +1 +1−γ ( γ+δ +δ )k( 2 +ϵ +1)) times, with constant probability, +the algorithm outputs a (1 + ϵ, 1 − δ)-approximation solution. +Similar to our discussion on the running time for MEB with outliers in Section 5.2.1, B˘adoiu +et al. [22] also achieved a linear time bi-criteria approximation for the k-center clustering with +outliers problem (see Section 4 in their paper). However, the hidden constant of their running +time is exponential in ( k +ϵδ)O(1) that is much larger than “k log k( 2 +ϵ + 1)” in Theorem 11. +Sublinear time algorithm. The linear time algorithm can be further improved to be +sublinear time; the idea is similar to that for designing sublinear time algorithm for MEB with +outliers in Section 5.2.2. First, we follow Definition 7 and define the shape set X, where each +x ∈ X is union of k balls in the space; the center c(x) should be the set of its k ball centers, say +c(x) = {o1 +x, o2 +x, · · · , ok +x}, and the size s(x) is the radius, i.e., x = ∪k +j=1B(oj +x, s(x)). Obviously, if +31 + +x is a feasible solution for the instance (P, γ), the size +���P ∩ (∪k +j=1B(oj +x, s(x))) +��� should be at +least (1 − γ)n. Also, define the distance function f(c(x), p) = min1≤j≤k ||p − oj +x||. It is easy +to verify that the shape set X satisfies Property 1, 2, and 3. From Lemma 11, we know that +it is possible to obtain a point in Popt ∩ Q with probability (1 − η1) +δ +3(γ+δ). Further, we can +estimate the value li and select the best candidate solution based on Lemma 12. Overall, we +have the following theorem. +Theorem 12. Let (P, γ) be an instance of k-center clustering with outliers. Given the parame- +ters ϵ, δ, η1, η2 ∈ (0, 1), there exists an algorithm that outputs a (1+ϵ, 1−δ)-approximation with +probability (1 − γ) +� +(1 − η1)(1 − η2) +δ +3(γ+δ) +�k( 2 +ϵ +1). The running time is ˜O(2k log k( 2 +ϵ +1)( γ +δ2 + 1 +ϵ5 )d). +If one repeatedly runs the algorithm N = O +� +1 +1−γ +� +1 +1−η1 ( 3(γ+δ) +δ +) +�k( 2 +ϵ +1)� +times with set- +ting η2 = O( +1 +2k log k( 2 +ϵ +1)N ), with constant probability, the algorithm outputs a (1 + ϵ, 1 − δ)- +approximation solution. +7.2 +Flat Fitting with Outliers +Let j be a fixed integer between 0 and d. Given a j-dimensional flat F and a point p ∈ Rd, we +define their distance, dist(F, p), to be the Euclidean distance from p to its projection onto F. +Let P be a set of n points in Rd. The problem of flat fitting is to find the j-dimensional flat +F that minimizes maxp∈P dist(F, p). It is easy to see that the MEB problem is the case j = 0 +of the flat fitting problem. Furthermore, given a parameter γ ∈ (0, 1), the flat fitting with +outliers problem is to find a subset P ′ ⊂ P with size (1 − γ)n such that maxp∈P ′ dist(F, p) +is minimized. Similar to MEB with outliers, we also use Popt to denote the optimal subset. +Before presenting our algorithms for flat fitting with outliers, we first introduce the linear time +algorithm from Har-Peled and Varadarajan [56] for the vanilla version (without outliers). +We start from the case j = 1, i.e., the flat F is a line in the space. Roughly speaking, +their algorithm is an iterative procedure to update the solution round by round, until it is +close enough to the optimal line lopt. There are two parts in the algorithm. (1) It picks an +arbitrary point p∆ ∈ P and let q∆ be the farthest point of P from p∆; it can be proved that +the line passing through p∆ and q∆, denoted as l0, is a good initial solution that yields a +4-approximation with respect to the objective function. (2) In each of the following rounds, the +algorithm updates the solution from li−1 to li where i ≥ 1 is the current number of rounds: let +pi be the farthest point of P from li−1 and let hi denote the 2-dimensional flat spanned by pi +and li−1; then the algorithm computes a set of O( 1 +ϵ8 log2 1 +ϵ) lines on hi, and picks one of them +as li via an “oracle”. They proved that the improvement from li−1 to li is significant enough; +thus, after running ν = O( 1 +ϵ3 log 1 +ϵ) rounds, it is able to achieve a (1 + ϵ)-approximation. To +remove the “oracle”, the algorithm can enumerate all the O( 1 +ϵ8 log2 1 +ϵ) lines on hi, and thus +the total running time is O +� +2 +1 +ϵ3 log2 1 +ϵ nd +� +. +Linear time algorithm. Now we consider to adapt the above algorithm to the case with +outliers, where in fact the idea is similar to the idea proposed in Section 5.2.1 for MEB with +outliers. For simplicity, we still use the same notations as above. Consider the part (1) first. If +we randomly pick a point p∆ from P, with probability 1 − γ, it belongs to Popt; further, we +randomly pick a point, denoted as q∆, from the set of (δ0 + γ)n farthest points of P from p∆, +where the value of δ0 will be determined below. Obviously, with probability +δ0 +γ+δ0 , q∆ ∈ Popt. +Denote by P0 = {p ∈ Popt | ||p − p∆|| ≤ ||q∆ − p∆||}. Then we have the following lemma. +Lemma 13. Denote by l0 the line passing through p∆ and q∆. Then, with probability (1 − +γ)( +δ0 +γ+δ0 ), +max +p∈P0 dist(l0, p) ≤ 4 max +p∈P0 dist(lopt, p) ≤ 4 max +p∈Popt dist(lopt, p). +(77) +32 + +Also, the size of P0 is at least +� +1 − (δ0 + γ) +� +n. +It is straightforward to obtain the size of P0. The inequality (77) directly comes from the +aforementioned result of [56], as long as p∆ and q∆ ∈ Popt. So we can use the line l0 as our initial +solution. Then, we can apply the same random sampling idea to select the point pi in the i-th +round. Namely, we randomly pick a point as pi from the set of (δ0+γ)n farthest points of P from +li. Moreover, we need to shrink the set Pi−1 to Pi = {p ∈ Pi−1 | dist(li−1, p) ≤ dist(li−1, pi)}. +Similar to Lemma 13, we can show that the improvement from li−1 to li is significant enough +with probability (1−γ)( +δ0 +γ+δ0 )i+1, and the size of Pi is at least +� +1−((i+1)δ0+γ) +� +n. After running +ν rounds, we obtain the line lν such that maxp∈Pν dist(lν, p) ≤ (1 + ϵ) maxp∈Popt dist(lopt, p), +and |Pν| ≥ +� +1 − ((ν + 1)δ0 + γ) +� +n. So if we set δ0 = +δ +ν+1 with a given δ ∈ (0, 1), the line lν will +be a bi-criteria (1 + ϵ, 1 − δ)-approximation of the instance (P, γ). By using the idea in [56], +we can extend the result to the case j > 1 with ν = eO(j2) +ϵ2j+1 log 1 +ϵ. We refer the reader to [56] for +more details. +Theorem 13. Let (P, γ) be an instance of j-dimensional flat fitting with outliers. Given two +parameters ϵ, δ ∈ (0, 1), there exists an algorithm that outputs a (1+ϵ, 1−δ)-approximation with +probability (1 − γ) +� 1 +2 +�g(j,ϵ) where g(j, ϵ) = poly(eO(j2), 1 +ϵj ). The running time is O(2g′(j,ϵ)nd) +where g′(j, ϵ) = poly(eO(j2), 1 +ϵj ). +If one repeatedly runs the algorithm 2g(j,ϵ) +1−γ +times, with constant probability, the algorithm +outputs a (1 + ϵ, 1 − δ)-approximation solution. +Sublinear time algorithm. We can view the flat fitting with outliers problem as an +MEX with outliers problem. Let r ≥ 0 and F be a j-dimensional flat. Then we can define a +j-dimensional “slab” SL(F, r) = {p ∈ Rd | dist(F, p) ≤ r}, where its “center” and “size” are +F and r respectively (e.g., a ball is a 0-dimensional slab); the distance function f(F, p) = +dist(F, p). It is easy to see that the shape set of slabs satisfies Property 1, 2, and 3. Furthermore, +finding the optimal flat is equivalent to finding the smallest slab covering (1 − γ)n points of P. +Therefore, by using Lemma 11 and 12, we achieve the following theorem. +Theorem 14. Let (P, γ) be an instance of j-dimensional flat fitting with outliers. Given +the parameters ϵ, δ, η1, η2 ∈ (0, 1), there exists an algorithm that outputs a (1 + ϵ, 1 − δ)- +approximation with probability (1−γ) +� +(1−η1)(1−η2) +δ +3(γ+δ) +�g(j,ϵ) where g(j, ϵ) = poly(eO(j2), 1 +ϵj ). +The running time is O(2g′(j,ϵ,δ,γ)d) where g′(j, ϵ) = poly(eO(j2), 1 +ϵj , 1 +δ, 1 +γ ). +If one repeatedly runs the algorithm N = O +� +1 +1−γ +� +1 +1−η1 ( 3(γ+δ) +δ +) +�g(j,ϵ)� +times with setting +η2 = O( +1 +2g(j,ϵ)N ), with constant probability, the algorithm outputs a (1 + ϵ, 1 − δ)-approximation +solution. +7.3 +One-class SVM with Outliers +In practice, datasets often contain outliers. The separating margin of SVM could be considerably +deteriorated by outliers. As mentioned in [40], most of existing techniques [88,93] for SVM +outliers removal are numerical approaches (e.g., adding some penalty item to the objective +function), and only can guarantee local optimums. Ding and Xu [40] modeled SVM with outliers +as a combinatorial optimization problem and provided an algorithm called “Random Gradient +Descent Tree”. We focus on one-class SVM with outliers first, and explain the extension for +two-class SVM with outliers in Section 7.4. Below is the definition of the one-class SVM with +outliers problem proposed in [40]. +Definition 9 (One-class SVM with Outliers). Given a set P of n points in Rd and a +small parameter γ ∈ (0, 1), the one-class SVM with outliers problem is to find a subset P ′ ⊂ P +33 + +Algorithm 8 Gilbert Algorithm [40,49] +Input: A point-set P in Rd, and N ∈ Z+. +Output: vi as an approximate solution of the polytope distance between the origin and P. +1. Initialize i = 1 and v1 to be the closest point in P to the origin o. +2. Iteratively perform the following steps until i = N. +(a) Find the point pi ∈ P whose orthogonal projection on the supporting line of segment ovi has the +closest distance to o (called the projection distance of pi), i.e., pi = arg minp∈P { ⟨p,vi⟩ +||vi|| }, where ⟨p, vi⟩ +is the inner product of p and vi (see Figure 6). +(b) Let vi+1 be the point on segment vipi closest to the origin o; update i = i + 1. +with size (1 − γ)n and a hyperplane H separating the origin o and P ′, such that the distance +between o and H is maximized. +o +xi +xi+1 +pi +pi |xi +vi +Vi+1 +vi +Fig. 6: An illustration of step 2 in Algorithm 8; pi |vi is the projection of pi on ovi. +Linear time algorithm. We briefly overview the algorithm of [40]. They also considered +the “bi-criteria approximation” with two small parameters ϵ, δ ∈ (0, 1): a hyperplane H +separates the origin o and a subset P ′ ⊂ P with size +� +1 − δ − γ +� +n, where the distance between +o and H is at least (1 − ϵ) of the optimum. The idea of [40] is based on the fact that the SVM +(without outliers) problem is equivalent to the polytope distance problem in computational +geometry [48]. +Let o be the origin and P be a given set of points in Rd. The polytope distance problem +is to find a point q inside the convex hull of P so that the distance ||q − o|| is minimized. +For an instance P of one-class SVM, it can be proved that the vector qopt − o, if qopt is +the optimal solution for the polytope distance between o and P, is the normal vector of the +optimal hyperplane. We refer the reader to [40,48] for more details. The polytope distance +problem can be efficiently solved by Gilbert Algorithm [46,49]. For completeness, we present it +in Algorithm 8. +Similar to the core-set construction method of MEB in Section 2.1, the algorithm also +greedily improves the current solution by selecting some point pi in each iteration. Let ρ be the +polytope distance between o and P, D = maxp,q∈P ||p − q||, and E = D2 +ρ2 . Given ϵ ∈ (0, 1), it +has been proved that a (1 − ϵ)-approximation of one-class SVM (i.e., a separating margin with +the width at least (1 − ϵ) of the optimum) can be achieved by running Algorithm 8 at most +2⌈2E/ϵ⌉ steps [29,48]. To handle outliers, the algorithm of [40] follows the similar intuition of +Section 5.2.1; it replaces the step of greedily selecting the point pi by randomly sampling a +point from a set Q, which contains the (δ + γ)n points having the smallest projection distances +(i.e., the values of the function ⟨p,vi⟩ +||vi|| in Step 2(a) of Algorithm 8). To achieve a (1 − ϵ, 1 − δ)- +approximation with constant success probability, the algorithm takes O +� +1 +1−γ (1 + γ +δ )z D2 +ϵρ2 nd +� +time, where z = O( D2 +ϵρ2 ). +Sublinear time algorithm. We define X to be the set of all the closed half-spaces not +covering the origin o in Rd; for each x ∈ X, let Hx be the hyperplane enclosing x and let hx +34 + +o +hx +Hx +Fig. 7: An illustration for Hx and hx. +be the projection of o on Hx (see Figure 7). We suppose that the given instance (P, γ) has +feasible solution. That is, there exists at least one half-space x ∈ X that the hyperplane Hx +separates the origin o and a subset P ′ with size (1 − γ)n. We define the center c(x) = +hx +||hx||; +since the MEX with outlier problem in Definition 7 is a minimization problem, we design +the size function s(x) = +1 +||hx||. Obviously, a (1 − ϵ)-approximation of the SVM with outliers +problem is equivalent to a +1 +1−ϵ-approximation with respect to the size function s(x). We also +define the distance function f(c(x), p) = −⟨p, +hx +||hx||⟩. It is easy to verify that the shape set X +satisfies Property 1, 2, and 3. +Recall that Algorithm 8 selects the point pi = arg minp∈P {⟨p,vi⟩ +||vi|| } in each iteration. Actually, +the vector +vi +||vi|| can be viewed as a shape center and pi is the farthest point to +vi +||vi|| based on +the distance function f(c(x), p). Moreover, the set Q mentioned in the previous linear time +algorithm actually is the set of the farthest (δ + γ)n points from P to +vi +||vi||. Consequently, we +can apply Lemma 11 to sample a point from Popt ∩ Q, and apply Lemma 12 to estimate the +value of li for each candidate solution +vi +||vi||. Overall, we can improve the running time of the +algorithm of [40] to be independent of n. +Theorem 15. Let (P, γ) be an instance of SVM with outliers. Given the parameters ϵ, δ, η1, η2 ∈ +(0, 1), there exists an algorithm that outputs a (1 − ϵ, 1 − δ)-approximation with probability +(1 − γ) +� +(1 − η1)(1 − η2) +δ +3(γ+δ) +�z where z = O( D2 +ϵρ2 ). The running time is ˜O( D2γ +δ2ϵ2ρ2 d). +If one repeatedly runs the algorithm N = O +� +1 +1−γ +� +1 +1−η1 (3 + 3γ +δ ) +�z� +times with setting +η2 = O( 1 +zN ), with constant probability, the algorithm outputs a (1 − ϵ, 1 − δ)-approximation +solution. +7.4 +Two-class SVM with Outliers +Below is the definition of the two-class SVM with outliers problem proposed in [40]. +Definition 10 (Two-class SVM with Outliers). Given two point sets P1 and P2 in Rd +and two small parameters γ1, γ2 ∈ (0, 1), the two-class SVM with outliers problem is to find two +subsets P ′ +1 ⊂ P1 and P ′ +2 ⊂ P2 with |P ′ +1| = (1 − γ1)|P1| and |P ′ +2| = (1 − γ2)|P2|, and a margin +separating P ′ +1 and P ′ +2, such that the width of the margin is maximized. +We use P opt +1 +and P opt +2 +, where |P opt +1 +| = (1 − γ1)|P1| and |P opt +2 +| = (1 − γ2)|P2|, to denote the +subsets of P1 and P2 which are separated by the optimal margin. The ordinary two-class SVM +(without outliers) problem is equivalent to computing the polytope distance between the origin +o and M(P1, P2), where M(P1, P2) is the Minkowski difference of P1 and P2 [48]. Note that it +is not necessary to compute the set M(P1, P2) explicitly. Instead, Algorithm 8 only needs to +select one point from M(P1, P2) in each iteration, and overall the running time is still linear +in the input size. To deal with two-class SVM with outliers, Ding and Xu [40] slightly modified +their algorithm for the case of one-class. In each iteration, it considers two subsets Q1 ⊂ P1 +and Q2 ⊂ P2, which respectively consist of points having the (δ + γ1)|P1| smallest projection +distances among all points in P1 and the (δ + γ2)|P2| largest projection distances among all +35 + +points in P2 on the vector vi; then, the algorithm randomly selects two points p1 +i ∈ Q1 and +p2 +i ∈ Q2, and their difference vector p2 +i −p1 +i will serve as the role of pi in Step 2(a) of Algorithm 8 +to update the current solution vi. This approach yields a (1 − ϵ, 1 − δ)-approximation in linear +time. +s⊥ +s⊥ +H⊥ +H⊥ +H⊤ +H⊤ +˜H⊥ +˜H⊥ +˜H⊤ +˜H⊤ +s⊤ +s⊤ +o +Fig. 8: An illustration for two-class SVM. The distances from o to H⊥ and H⊤ are s⊥ and +s⊤, respectively. The hyperplanes ˜H⊥ and ˜H⊤ are the estimations of H⊥ and H⊤, and the +distances from o to them are ˜s⊥ and ˜s⊤ respectively. +To improve the algorithm to be sublinear, we need several modifications on our previous +idea for the case of one-class. First, we change the distance function to be: +f(p, c) = +� +−⟨p, +hx +||hx||⟩ if p ∈ P1; +⟨p, +hx +||hx||⟩ +if p ∈ P2. +By using this new distance function, we can apply Lemma 11 to obtain the points p1 +i ∈ Q1∩P opt +1 +and p2 +i ∈ Q2 ∩ P opt +2 +separately in sublinear time. Given a vector (i.e., candidate center) +vi +||vi||, +assume H⊥ and H⊤ are the parallel hyperplanes orthogonal to +vi +||vi|| that the margin formed +by them separates P ′ +1 and P ′ +2, where P ′ +1 ⊂ P1 and P ′ +2 ⊂ P2 with |P ′ +1| = (1 − γ1)|P1| and +|P ′ +2| = (1 − γ2)|P2|. Without loss of generality, we assume that the origin o is inside the margin. +Suppose that the distances from o to H⊥ and H⊤ are s⊥ and s⊤, respectively. Then, we +obtain two shapes (closed half-spaces) x⊥ = (− vi +||vi||, 1 +s⊥ ) and x⊤ = ( vi +||vi||, 1 +s⊤ ) with P ′ +1 ⊂ x⊥ +and P ′ +2 ⊂ x⊤. Consequently, we can apply Lemma 12 twice to obtain two values +1 +˜s⊥ ≤ +1 +s⊥ and +1 +˜s⊤ ≤ +1 +s⊤ with +���P1 \ x(− vi +||vi||, 1 +˜s⊥ ) +��� ≤ (O(δ) + γ1)|P1| and +���P2 \ x( vi +||vi||, 1 +˜s⊤ ) +��� ≤ (O(δ) + γ2)|P2|. +Therefore, we can use the value ˜s⊥ + ˜s⊤ as an estimation of s⊥ + s⊤. See Figure 8 for an +illustration. Overall, we can achieve a (1 − ϵ, 1 − O(δ))-approximation in sublinear time. +8 +Future Work +Following our work, several interesting problems deserve to be studied in future. For example, +different from radius approximation, the current research on covering approximation of MEB is +still inadequate. In particular, can we provide a lower bound for the complexity of computing +covering approximate MEB, as the lower bound result for radius approximate MEB proved +by [30]? Also, is it possible to extend the stability notion to other geometric optimization +problems with more complicated structures? In Section 7, we only provide the bi-criteria +approximations for the MEX with outliers problems. So it is interesting to consider to extend +the stability notion to these geometric optimization problems, and then we can design the +hybrid approximation algorithms for them. +36 + +9 +Acknowledgements +The research of this work was supported in part by National Key R&D program of China +through grant 2021YFA1000900 and the Provincial NSF of Anhui through grant 2208085MF163. +The author also want to thank Prof. Jinhui Xu for his helpful comments on this draft. +References +1. P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan. Geometric approximation via coresets. Combinatorial +and Computational Geometry, 52:1–30, 2005. +2. P. K. Agarwal, S. Har-Peled, and H. Yu. Embeddings of surfaces, curves, and moving points in euclidean +space. 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Schuurmans. Robust support vector machine training via convex outlier +ablation. In AAAI, pages 536–542. AAAI Press, 2006. +94. H. Zarrabi-Zadeh and A. Mukhopadhyay. Streaming 1-center with outliers in high dimensions. In Proceedings +of the Canadian Conference on Computational Geometry (CCCG), pages 83–86, 2009. +A +Proof of Theorem 1 +To ensure the expected improvement in each iteration of the algorithm of [20], they showed +that the following two inequalities hold if the algorithm always selects the farthest point to +the current center of MEB(T): +ri+1 ≥ (1 + ϵ)Rad(P) − Li; +ri+1 ≥ +� +r2 +i + L2 +i , +(78) +where ri and ri+1 are the radii of MEB(T) in the i-th and (i + 1)-th iterations, respectively, +and Li is the shifting distance of the center of MEB(T) from the i-th to (i + 1)-th iteration. +However, we often compute only an approximate MEB(T) in each iteration. In the i-th +iteration, we let ci and oi denote the centers of the exact and the approximate MEB(T), +respectively. Suppose that ||ci − oi|| ≤ ξri, where ξ ∈ (0, +ϵ +1+ϵ) (we will see why this bound is +needed later). Note that we only compute oi rather than ci in each iteration. As a consequence, +we can only select the farthest point (say q) to oi. If ||q − oi|| ≤ (1 + ϵ)Rad(P), we are done +and a (1 + ϵ)-radius approximation of MEB is already obtained. Otherwise, we have +(1 + ϵ)Rad(P) < ||q − oi|| +≤ ||q − ci+1|| + ||ci+1 − ci|| + ||ci − oi|| +≤ ri+1 + Li + ξri +(79) +by the triangle inequality. In other words, we should replace the first inequality of (78) by +ri+1 > (1 + ϵ)Rad(P) − Li − ξri. Also, the second inequality of (78) still holds since it depends +only on the property of the exact MEB (see Lemma 2.1 in [20]). Thus, we have +ri+1 ≥ max +�� +r2 +i + L2 +i , (1 + ϵ)Rad(P) − Li − ξri +� +. +(80) +40 + +Similar to the analysis in [20], we let λi = +ri +(1+ϵ)Rad(P). Because ri is the radius of MEB(T) +and T ⊂ P, we know ri ≤ Rad(P) and then λi ≤ 1/(1 + ϵ). By simple calculation, we know +that when Li = +� +(1+ϵ)Rad(P)−ξri +�2 +−r2 +i +2 +� +(1+ϵ)Rad(P)−ξri +� +the lower bound of ri+1 in (80) achieves the minimum +value. Plugging this value of Li into (80), we have +λ2 +i+1 ≥ λ2 +i + +� +(1 − ξλi)2 − λ2 +i +�2 +4(1 − ξλi)2 +. +(81) +To simplify inequality (81), we consider the function g(x) = (1−x)2−λ2 +i +1−x +, where 0 < x < ξ. Its +derivative g′(x) = −1 − +λ2 +i +(1−x)2 is always negative, thus we have +g(x) ≥ g(ξ) = (1 − ξ)2 − λ2 +i +1 − ξ +. +(82) +Because ξ < +ϵ +1+ϵ and λi ≤ 1/(1 + ϵ), we know that the right-hand side of (82) is always +non-negative. Using (82), inequality (81) can be simplified to +λ2 +i+1 ≥ λ2 +i + 1 +4 +� +g(ξ) +�2 += λ2 +i + +� +(1 − ξ)2 − λ2 +i +�2 +4(1 − ξ)2 +. +(83) +(83) can be further rewritten as +� λi+1 +1 − ξ +�2 +≥ 1 +4 +� +1 + ( λi +1 − ξ )2�2 +=⇒ λi+1 +1 − ξ ≥ 1 +2 +� +1 + ( λi +1 − ξ )2� +. +(84) +Now, we can apply a similar transformation of λi which was used in [20]. Let γi = +1 +1− λi +1−ξ +. +We know γi > 1 (note 0 ≤ λi ≤ +1 +1+ϵ and ξ < +ϵ +1+ϵ). Then, (84) implies that +γi+1 ≥ +γi +1 − +1 +2γi += γi +� +1 + 1 +2γi ++ ( 1 +2γi +)2 + · · · +� +> γi + 1 +2, +(85) +where the equation comes from the fact that γi > 1 and thus +1 +2γi ∈ (0, 1 +2). Note that λ0 = 0 +and thus γ0 = 1. As a consequence, we have γi > 1 + i +2. In addition, since λi ≤ +1 +1+ϵ, that is, +γi ≤ +1 +1− +1 +(1+ϵ)(1−ξ) , we have +i < +2 +ϵ − ξ − ϵξ = +2 +(1 − 1+ϵ +ϵ ξ)ϵ. +(86) +Consequently, we obtain the theorem. +B +Lemma 2.2 in [22] +Lemma 14 ( [22]). Let B(c, r) be a minimum enclosing ball of a point set P ⊂ Rd, then any +closed half-space that contains c, must also contain at least a point from P that is at distance +r from c. +41 + diff --git a/_dE1T4oBgHgl3EQfDAIA/content/tmp_files/load_file.txt b/_dE1T4oBgHgl3EQfDAIA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7006b4863852f2b9da2a5ffd958e7ef63c5fe3ff --- /dev/null +++ b/_dE1T4oBgHgl3EQfDAIA/content/tmp_files/load_file.txt @@ -0,0 +1,2124 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf,len=2123 +page_content='Sublinear Time Algorithms for Several Geometric Optimization (With Outliers) Problems In Machine Learning⋆ Hu Ding School of Computer Science and Engineering, University of Science and Technology of China He Fei, China huding@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='cn Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In this paper, we study several important geometric optimization problems arising in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we revisit the Minimum Enclosing Ball (MEB) problem in Euclidean space Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The problem has been extensively studied before, but real-world machine learning tasks often need to handle large-scale datasets so that we cannot even afford linear time algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Motivated by the recent studies on beyond worst-case analysis, we introduce the notion of stability for MEB, which is natural and easy to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Roughly speaking, an instance of MEB is stable, if the radius of the resulting ball cannot be significantly reduced by removing a small fraction of the input points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Under the stability assumption, we present two sampling algorithms for computing radius-approximate MEB with sample complexities independent of the number of input points n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In particular, the second algorithm has the sample complexity even independent of the dimensionality d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We also consider the general case without the stability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We present a hybrid algorithm that can output either a radius-approximate MEB or a covering-approximate MEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our algorithm improves the running time and the number of passes for the previous sublinear MEB algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our method relies on two novel techniques, the Uniform-Adaptive Sampling method and Sandwich Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Furthermore, we observe that these two techniques can be generalized to design sublinear time algorithms for a broader range of geometric optimization problems with outliers in high dimensions, including MEB with outliers, one-class and two-class linear SVMs with outliers, k-center clustering with outliers, and flat fitting with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our proposed algorithms also work fine for kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1 Introduction Many real-world machine learning tasks can be formulated as geometric optimization problems in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We start with a fundamental geometric optimization problem, Minimum Enclosing Ball (MEB), which has attracted a lot of attentions in past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a set P of n points in Euclidean space Rd, where d could be quite high, the problem of MEB is to find a ball with minimum radius to cover all the points in P [20,45,69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' MEB finds several important applications in machine learning [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For example, the popular classification model Support Vector Machine (SVM) can be formulated as an MEB problem in high dimensional space, if all the mapped points have the same norm by using kernel method, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the popular radial basis function kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' this SVM is called “Core Vector Machine (CVM)” which is currently one of the most important SVM training methods for large-scale data sets, since it was proposed in 2005 [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Hence fast MEB algorithms can be used to speed up its training procedure [29,30,90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Recently, MEB has also been studied for preserving privacy [44,77] and quantum cryptography [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Usually, we consider the approximate solutions of MEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If a ball covers all the n points but has a radius larger than the optimal one, we call it a “radius-approximate solution”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' if a ball has the radius no larger than the optimal one but covers less than n points, we call it a “covering-approximate solution” instead (the formal definitions are shown in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In the era of big data, the dataset could be so large that we cannot even afford linear time algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' This motivates us to ask the following questions: ⋆ Part of this work has appeared in [37,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='02870v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='DS] 7 Jan 2023 Is it possible to develop approximation algorithms for MEB that run in sublinear time in the input size?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' And how about other high-dimensional geometric optimization problems arising in machine learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is common to assume that the input data is represented by a n × d matrix, and any algorithm having complexity o(nd) can be considered as a sublinear time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In practice, data items are usually represented as sparse vectors in Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' so it can be fast to perform the operations, like distance computing, even though the dimensionality d is high (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', if each vector has s ≪ d non-zero entries, the time for computing the distance is O(s) rather than O(d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' see the concluding remarks of [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, the number of input points n is often much larger than the dimensionality d in many real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, we are interested in designing the algorithms that have complexities sublinear in n (or linear in n but with small factor before it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 Our Main Ideas and Results Our idea for designing sublinear time MEB algorithms is inspired by the recent studies on optimization with respect to stable instances, under the umbrella of beyond worst-case analysis [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For example, several recent works introduced the notion of stability for problems like clustering and max-cut [8,13,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In this paper, we give the notion of “stability” for MEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Roughly speaking, an instance of MEB is stable, if the radius of the resulting ball cannot be significantly reduced by removing a small fraction of the input points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the radius cannot be reduced by 10% if only 1% of the points are removed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The rationale behind this notion is quite natural: if the given instance is not stable, the small fraction of points causing significant reduction in the radius should be viewed as outliers (or we may need multiple balls to cover the input points as the k-center clustering problem [51,61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To the best of our knowledge, this is the first study on MEB from the perspective of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We prove an important implication of the stability assumption: informally speaking, if an instance of MEB is stable, its center should reveal a certain extent of robustness in the space (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Using this implication, we propose two sampling algorithms for computing (1 + ϵ)-radius approximate MEB with sublinear time complexities (Section 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' in particular, our second algorithm has the sample size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the number of sampled points) independent of the number of input points n and dimensionality d (to the best of our knowledge, this is the first algorithm achieving (1 + ϵ)-radius approximation with such a sublinear complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, we have an interesting observation: the ideas developed under the stability assumption can even help us to solve the general instance without the stability assumption, if we relax the requirement slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We introduce a hybrid approach that can output either a radius-approximate MEB or a covering-approximate MEB, depending upon whether the input instance is sufficiently stable1 (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, a byproduct is that we can infer the stability degree of the given instance from the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is worth noting that the simple uniform sampling idea based on VC-dimension [58,92] can only yield a “bi-criteria” approximation, which has errors on both the radius and the number of covered points (see the discussion on our first sampling algorithm in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Comparing with the sublinear time MEB algorithm proposed by Clarkson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [30], we reduce the total running time from ˜O(ϵ−2n + ϵ−1d + M) to O(n + h(ϵ, δ) · d + M), where M is the number of non-zero entries in the input n × d matrix and h(ϵ, δ) is a factor depending on the pre-specified radius error bound ϵ and covering error bound δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, our improvement is significant if n ≫ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The only tradeoff is that we allow a covering approximation for unstable instance (given the lower bound proved by [30], it is quite unlikely to reduce the term ϵ−2n if we keep restricting the output to be (1 + ϵ)-radius approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, our algorithm only needs uniform sampling and a single pass 1 We do not need to explicitly know whether the instance is stable or not, when running our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2 over the data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' on the other hand, the algorithm of [30] needs ˜O(ϵ−1) passes (the details are shown in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In addition to the stability idea, our method also relies on two key techniques, the novel “Uniform-Adaptive Sampling” method and “Sandwich Lemma”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Roughly speaking, the Uniform-Adaptive Sampling method can help us to bound the error induced in each “randomized greedy selection” step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' the Sandwich Lemma enables us to estimate the objective value of each candidate and select the best one in sublinear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Results Quality Time Number of passes Extendibility for MEB with outliers Clarkson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [30] (1 + ϵ)-rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ˜O(ϵ−2n + ϵ−1d + M) ˜O(ϵ−1) N/A Core-sets methods [20,29,69,79] (1 + ϵ)-rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' roughly O(ϵ−1nd) or O(ϵ−1(n + d + M)) if M = o(nd) O(ϵ−1) bi-criteria approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [22] Numerical method [84] (1 + ϵ)-rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ˜O(ϵ−1/2nd) or ˜O(ϵ−1/2(n + d + M)) if M = o(nd) O(ϵ−1/2) N/A Numerical method [6] (1 + ϵ)-rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ˜O(nd + n √ d/√ϵ) ˜O(d + � d/ϵ) N/A Streaming algorithm [4,25] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='22-rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' O(nd/ϵ5) one pass N/A This paper stable instance (1 + ϵ)-rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' O(C1 · d) (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2) uniform sampling N/A general instance (1 + ϵ)-rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' or (1 − δ)-cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' O � (n + C2)d � or O(n + C2 · d + M) if M = o(nd) (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='3) uniform sampling plus a single pass (1 + ϵ)-rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' or (1 − δ)-cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 6) Table 1: The existing and our results for computing MEB in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In the table, “rad.” and “cov.” stand for “radius approximation” and “covering approximation”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' “M” is the number of non-zero entries in the input n × d matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The factor C1 depends on ϵ and the stability degree of the given instance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' the factor C2 depends on ϵ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Finally, we present several extensions of our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In practice, we may assume the presence of outliers in given datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In particular, as the rapid development of machine learning, the field of adversarial machine learning has attracted a great amount of attentions [17,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' A small set of outliers could be added by some adversarial attacker to make the model severely deviate and cause unexpected error (the seminal paper [16] on poisoning attacks against SVM has just received the ICML2022 Test of Time award).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To defend such poisoning attacks, we often design robust algorithms that are resilient against outliers [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, the presence of outliers makes the problem not only non-convex but also highly combinatorial in high dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' for example, if m of the input n data items are outliers (m < n), we have to consider an exponentially large number �n m � of different possible cases when optimizing the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So we consider to design sublinear time algorithms for the following problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' MEB with outliers is a natural generalization of the MEB problem, where the goal is to find the minimum ball covering at least a certain fraction of input points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We can apply MEB with outliers to solve many practical problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', outlier recognition) in data mining and data analysis [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We define the stability for MEB with outliers, and propose the sublinear time approximation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our algorithm is the first sublinear time algorithm for the MEB with outliers problem (comparing with the previous linear time algorithms like [22]), to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 3 Other enclosing with outliers problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Besides MEB with outliers, we observe that our proposed techniques can be used to solve a broader range of enclosing with outliers problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We define a general optimization problem called minimum enclosing “x” (MEX) with Outliers, where the “x” could be a specified kind of shape (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the shape is a ball for MEB with outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We prove that it is possible to generalize the Uniform-Adaptive Sampling method and Sandwich Lemma to adapt the shape “x”, as long as it satisfies several properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In particular we focus on the MEX with outlier problems including flat fitting, k-center clustering, and SVM with outliers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' a common characteristic of these problems is that each of them has an iterative algorithm based on greedy selection for its vanilla version (without outliers) that is similar to the MEB algorithm of [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Though these problems have been widely studied before, the research in terms of their sublinear time algorithms is till quite limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Because the geometric optimization problems studied in this paper are motivated from machine learning applications, we also take into account the kernels [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our proposed algorithms only need to conduct the basic operations, like computing the distance and inner product, on the data items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, they also work fine for kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2, we summarize the previous results that are related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Section 2, we present the important definitions and briefly introduce the coreset construction method for MEB from [20] (which will be used in our following algorithms and analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Section 3, we prove the implication of MEB stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Further, in Section 4 we propose two sublinear time MEB algorithms for stable instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Section 5, we propose two key techniques, Uniform-Adaptive sampling and Sandwich lemma, and then present our sublinear time algorithm for general MEB without the stability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Section 6, we extend the idea of Section 5 to the MEB with outliers problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Finally, we present the generalized Uniform-Adaptive sampling and Sandwich lemma, together with the applications in several enclosing with outliers problems (including flat fitting, k-center clustering, and SVM with outliers) in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 Previous Work The works most related to ours are [7,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Clarkson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [30] developed an elegant perceptron framework for solving several optimization problems arising in machine learning, such as MEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a set of n points in Rd represented as an n × d matrix with M non-zero entries, their framework can compute the MEB in ˜O( n ϵ2 + d ϵ ) time 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that the parameter “ϵ” is an additive error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the resulting radius is r + ϵ if r is the radius of the optimal MEB) which can be converted into a relative error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', (1 + ϵ)r) in O(M) preprocessing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, if M = o(nd), the running time is still sublinear in the input size nd (please see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The framework of [30] also inspires the sublinear time algorithms for training SVMs [60] and approximating Semidefinite Programs [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Hayashi and Yoshida [59] presented a sampling- based method for minimizing quadratic functions of which the MEB objective is a special case, but it yields a large additive error O(ϵn2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [7] studied the following property testing problem: given a set of n points in some metric space, determine whether the instance is (k, b)-clusterable, where an instance is called (k, b)-clusterable if it can be covered by k balls with radius (or diameter) b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' They proposed several sampling algorithms to answer the question “approximately”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Specifically, they distinguish between the case that the instance is (k, b)-clusterable and the case that it is ϵ-far away from (k, b′)-clusterable, where ϵ ∈ (0, 1) and b′ ≥ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' “ϵ-far” means that more than ϵn points should be removed so that it becomes (k, b′)-clusterable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that their method cannot yield a single criterion radius-approximation or covering-approximation algorithm for the MEB 2 The asymptotic notation ˜O(f) = O � f · polylog( nd ϵ ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4 problem, since it will introduce unavoidable errors on the radius and the number of covered points due to the relaxation of “ϵ-far”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' But it is possible to convert it into a “bi-criteria” approximation, where it allows approximations on both the radius and the number of uncovered outliers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', discard more than the pre-specified number of outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' MEB and core-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' A core-set is a small set of points that approximates the struc- ture/shape of a much larger point set [1, 43, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The core-set idea has also been used to compute approximate MEB in high dimensional space [22,67,69,79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' B˘adoiu and Clarkson [20] showed that it is possible to find a core-set of size ⌈2/ϵ⌉ that yields a (1+ϵ)-radius approximate MEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Several other methods can yield even lower core-set sizes, such as [21,67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In fact, the algorithm for computing the core-set of MEB is a Frank-Wolfe algorithm [46], which has been systematically studied by Clarkson [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Other MEB algorithms that do not rely on core-sets include [6,45,84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Agarwal and Sharathkumar [4] presented a streaming (1+ √ 3 2 + ϵ)-radius ap- proximation algorithm for computing MEB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' later, Chan and Pathak [25] proved that the same algorithm actually yields an approximation ratio less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Very recently, Cohen-Addad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [31] proposed the sublinear time algorithm for computing high dimensional power means (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', geometric median and mean points) by using core-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' MEB with outliers and k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The MEB with outliers problem can be viewed as the case k = 1 of the k-center clustering with outliers problem [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' B˘adoiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [22] extended their core-set idea to the problems of MEB and k-center clustering with outliers, and achieved linear time bi-criteria approximation algorithms (if k is assumed to be a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [62] and Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [41] respectively showed that simple uniform sampling approach can yield bi-criteria approximation of k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Several algorithms for the low dimensional MEB with outliers have also been developed [5, 42, 54, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' There also exist a number of works on streaming MEB and k-center clustering with outliers [24, 28, 72, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Other related topics include robust optimization [14], robust fitting [3,57], and optimization with uncertainty [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' SVM with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given two point sets P1 and P2 in Rd, the problem of Support Vector Machine (SVM) is to find the largest margin to separate P1 and P2 (if they are separable) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' SVM can be formulated as a quadratic programming problem, and a number of efficient techniques have been developed in the past, such as the soft margin SVM [32,81], ν-SVM [33,86], and Core-SVM [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' There also exist a number of works on designing robust algorithms for SVM with outliers [40,88,93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Flat fitting with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given an integer j ≥ 0 and a set of points in Rd, the flat fitting problem is to find a j-dimensional flat having the smallest maximum distance to the input points [55];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' obviously, the MEB problem is a special case with j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In high dimensions, Har-Peled and Varadarajan [56] provided a linear time algorithm if j is assumed to be fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' their running time was further reduced by Panigrahy [79] based on a core-set approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' There also exist several methods considering flat fitting with outliers but only for low-dimensional case [3,57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Optimizations under stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Bilu and Linial [18] showed that the Max-Cut problem becomes easier if the given instance is stable with respect to perturbation on edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Ostrovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [78] proposed a separation condition for k-means clustering which refers to the scenario where the clustering cost of k-means is significantly lower than that of (k − 1)-means for a given instance, and demonstrated the effectiveness of the Lloyd heuristic [70] under the separation condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Balcan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [13] introduced the concept of approximation-stability for finding the ground-truth of k-median and k-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Awasthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [8] introduced another notion of clustering stability and gave a PTAS for k-median and k-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' More clustering algorithms under stability assumption were studied in [9–12,68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 5 Sublinear time algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Besides the aforementioned sublinear MEB algorithm [30], a number of sublinear time algorithms have been studied for the problems like clustering [35,63, 64,73,74] and property testing [15,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' More detailed discussion on sublinear time algorithms can be found in the survey papers [34,83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2 Definitions and Preliminaries We describe and analyze our algorithms in the unit-cost RAM model [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose the input is represented by an n × d matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', n points in Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As mentioned in [30], it is common to assume that each entry of the matrix can be recovered in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We let |A| denote the number of points of a given point set A in Rd, and ||x − y|| denote the Euclidean distance between two points x and y in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We use B(c, r) to denote the ball centered at a point c with radius r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Below, we give the definitions for MEB and the notion of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To keep the structure of our paper more compact, we place other necessary definitions for our extensions to Section 5, Section 6, and Section 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 1 (Minimum Enclosing Ball (MEB)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a set P of n points in Rd, the MEB problem is to find a ball with minimum radius to cover all the points in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The resulting ball and its radius are denoted by MEB(P) and Rad(P), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 2 (Radius Approximation and Covering Approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let 0 < ϵ, δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' A ball B(c, r) is called a (1 + ϵ)-radius approximation of MEB(P), if the ball covers all points in P and has radius r ≤ (1 + ϵ)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' On the other hand, the ball is called a (1 − δ)- covering approximation of MEB(P), if it covers at least (1 − δ)n points in P and has radius r ≤ Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Both radius approximation and covering approximation are single-criterion approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' When ϵ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', δ) approaches to 0, the (1 + ϵ)-radius approximation (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', (1 − δ)-covering approximation) will approach to MEB(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The “covering approximation” seems to be similar to “MEB with outliers”, but actually they are quite different (see Definition 4 in Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 3 ((α, β)-stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a set P of n points in Rd with two parameters α and β in (0, 1), P is an (α, β)-stable instance if (1) Rad(P \\ Q) > (1 − α)Rad(P) for any Q ⊂ P with |Q| < βn, and (2) there exists a Q′ ⊂ P with |Q′| = ⌈βn⌉ having Rad(P \\ Q′) ≤ (1 − α)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The intuition of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Actually, β can be viewed as a function of α, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For example, given an α > 0, there always exists a β ≥ 1 n such that P is an (α, β)-stable instance (β ≥ 1 n because we must remove at least one point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The property of stability indicates that Rad(P) cannot be significantly reduced unless removing a large enough fraction of points from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For a fixed α, the larger β is, the more stable P should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similarly, for a fixed β, the smaller α is, the more stable P should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Actually, our stability assumption is quite reasonable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For example, if the radius can be reduced considerably (say by α = 10%) after removing only a very small fraction (say β = 1%) of points, it is natural to view the small fraction of points as outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To better understand the notion of stability in high dimensions, we consider the following two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Example (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose that the distribution of P is uniform and dense inside MEB(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let α ∈ (0, 1) be a fixed number, and we study the corresponding β of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we want the radius of the remaining (1 − β)n points to be as small as possible, intuitively we should remove the outermost βn points (since P is uniform and dense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let Q′ denote the set of outermost βn points that has Rad(P \\ Q′) ≤ (1 − α)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we have |P\\Q′| |P| ≈ V ol � MEB(P\\Q′) � V ol � MEB(P) � = 6 (Rad(P\\Q′))d (Rad(P))d ≤ (1 − α)d, where V ol(·) is the volume function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' That is, 1 − β ≤ (1 − α)d and it implies limd→∞ β = 1 when α is fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' that means P tends to be very stable as d increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Example (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consider a regular d-dimensional simplex P containing d + 1 points where each pair of points have the pairwise distance equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is not hard to obtain Rad(P) = � d 2(1+d), and we denote it by rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we remove β(d + 1) points from P, namely it becomes a regular d′-dimensional simplex with d′ = (1 − β)(d + 1) − 1, the new radius rd′ = � d′ 2(1+d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To achieve rd′ rd ≤ 1 − α with a fixed α, it is easy to see that 1 − β should be no larger than 1 1+(2α−α2)d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' this implies limd→∞ β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to example (i), the instance P tends to be very stable as d increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In practice, it is difficult to know the exact value of β for a fixed α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, the value of β only affects the sample sizes in our proposed algorithms in Section 4, and thus only assuming a reasonable lower bound β0 < β is already sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Section 5, we also consider the general case without the stability assumption, where the proposed algorithm does not even need to input β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 A More Careful Analysis for Core-set Construction in [20] We first briefly introduce the core-set construction for MEB, since it will be used in our proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let 0 < ϵ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The algorithm in [20] yields an MEB core-set of size 2/ϵ (for convenience, we always assume that 2/ϵ is an integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' But there is a small issue in their paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The analysis assumes that the exact MEB of the core-set is computed in each iteration, but in fact one may only compute an approximate MEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, an immediate question is whether the quality is still guaranteed with such a change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [69] fixed this issue, and showed that computing a (1 + O(ϵ2))-approximate MEB for the core-set in each iteration still guarantees a core-set with size O(1/ϵ), where the hidden constant is larger than 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Clarkson [29] showed that the greedy core-set construction algorithm of MEB, as a special case of the Frank-Wolfe algorithm, yields a core-set with size slightly larger than 4/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that there exist several other methods yielding even lower core-set size [21,67], but their construction algorithms are more complicated and thus not applicable to our problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Below we show that it is possible to guarantee a core-set of [20] with the size being arbitrarily close to 2/ϵ, even if we only compute an approximate MEB in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' This improves the core-set sizes of [29,69], and the new analysis is also interesting in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For the sake of completeness, we first briefly introduce the idea of the core-set construction algorithm in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a point set P ⊂ Rd, the algorithm is a simple iterative procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Initially, it selects an arbitrary point from P and places it into an initially empty set T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In each of the following 2/ϵ iterations, the algorithm updates the center of MEB(T) and adds to T the farthest point from the current center of MEB(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Finally, the center of MEB(T) induces a (1 + ϵ)-approximation for MEB(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The selected set of 2/ϵ points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', T) is called the core-set of MEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To ensure the expected improvement in each iteration, they [20] showed that the following two inequalities hold if the algorithm always selects the farthest point to the current center of MEB(T): ri+1 ≥ (1 + ϵ)Rad(P) − Li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ri+1 ≥ � r2 i + L2 i , (1) where ri and ri+1 are the radii of MEB(T) in the i-th and (i + 1)-th iterations, respectively, and Li is the shifting distance of the center of MEB(T) from the i-th to (i + 1)-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As mentioned earlier, we often compute only an approximate MEB(T) in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In the i-th iteration, we let ci and oi denote the centers of the exact and the approximate MEB(T), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose that ||ci − oi|| ≤ ξri, where ξ ∈ (0, ϵ 1+ϵ) (we will see why this 7 q oioi cici ci+1 ci+1 ≤ ri+1 ≤ ri+1 > (1 + ϵ)Rad(P) > (1 + ϵ)Rad(P) = Li = Li ≤ ξri ≤ ξri Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1: An illustration of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' bound is needed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Using another algorithm proposed in [20], one can obtain the point oi in O( 1 ξ2 |T|d) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that we only compute oi rather than ci in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Hence we can only select the farthest point (say q) to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If ||q − oi|| ≤ (1 + ϵ)Rad(P), we are done and a (1 + ϵ)-approximation of MEB is already obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Otherwise, we have (1 + ϵ)Rad(P) < ||q − oi|| ≤ ||q − ci+1|| + ||ci+1 − ci|| + ||ci − oi|| ≤ ri+1 + Li + ξri (2) by the triangle inequality (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In other words, we should replace the first inequality of (1) by “ri+1 > (1 + ϵ)Rad(P) − Li − ξri”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, the second inequality of (1) still holds since it depends only on the property of the exact MEB (see [20, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, we have ri+1 ≥ max �� r2 i + L2 i , (1 + ϵ)Rad(P) − Li − ξri � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (3) This leads to the following theorem whose proof can be found in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In the core-set construction algorithm of [20], if one computes an approximate MEB for T in each iteration and the resulting center oi has the distance to ci less than ξri = s ϵ 1+ϵri for some s ∈ (0, 1), the final core-set size is bounded by z = 2 (1−s)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, the bound could be arbitrarily close to 2/ϵ when s is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We can simply set s to be any constant in (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' for instance, if s = 1/3, the core-set size will be bounded by z = 3/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Since |T| ≤ z in each iteration, the total running time is O � z � |P|d + 1 ξ2 zd �� = O � 1 ϵ � |P| + 1 ϵ3 � d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We also want to emphasize a simple observation on the above core-set construction procedure, which will be used in our algorithms and analyses later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The algorithm always selects the farthest point to oi in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, this is actually not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As long as the selected point has distance at least (1 + ϵ)Rad(P), the result presented in Theorem 1 is still true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If no such a point exists (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', P \\ B � oi, (1 + ϵ)Rad(P) � = ∅), a (1 + ϵ)-radius approximate MEB (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the ball B � oi, (1 + ϵ)Rad(P) � ) has been already obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Remark 4 (kernels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If each point p ∈ P is mapped to ψ(p) in RD by some kernel function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', as the CVM [90]), where D could be +∞, we can still run the core-set algorithm of [20], since the algorithm only needs to compute the distances and the center oi is always a convex combination of T in each iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' instead of returning an explicit center, the algorithm will output the coefficients of the convex combination for the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' And similarly, our Algorithm 2 presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 also works fine for kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 3 Implication of the Stability Property In this section, we show an important implication of the stability property of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 8 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Assume ϵ, ϵ′, β0 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let P be an (ϵ2, β)-stable instance of the MEB problem with β > β0, and o be the center of its MEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let ˜o be a given point in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Assume the number r ≤ (1 + ϵ′2)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If the ball B � ˜o, r � covers at least (1 − β0)n points from P, the following holds ||˜o − o|| < (2 √ 2ϵ + √ 3ϵ′)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (4) Theorem 2 indicates that if a ball covers a large enough subset of P and its radius is bounded, its center should be close to the center of MEB(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let P ′ = B � ˜o, r � ∩ P, and assume o′ is the center of MEB(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To bound the distance between ˜o and o, we bridge them by the point o′ (since ||˜o − o|| ≤ ||˜o − o′|| + ||o′ − o||).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The following are two key lemmas for proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The distance ||o′ − o|| ≤ √ 2ϵRad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We consider two cases: MEB(P ′) is totally covered by MEB(P) and otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For the first case (see Figure 2(a)), it is easy to see that ||o′ − o|| ≤ Rad(P) − (1 − ϵ2)Rad(P) = ϵ2Rad(P) < √ 2ϵRad(P), (5) where the first inequality comes from the fact that MEB(P ′) has radius at least (1−ϵ2)Rad(P) (Definition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, we can focus on the second case below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let a be any point located on the intersection of the two spheres of MEB(P ′) and MEB(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we have the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='" # $ $" % oH oH !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='" # !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='" ˜o˜o # $ (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2: (a) The case MEB(P ′) ⊂ MEB(P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (b) an illustration under the assumption ∠ao′o < π/2 in the proof of Claim 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (c) the angle ∠ao′o ≥ π/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (d) an illustration of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The angle ∠ao′o ≥ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose that ∠ao′o < π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that ∠aoo′ is always smaller than π/2 since ||o − a|| = Rad(P) ≥ Rad(P ′) = ||o′ − a||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, o and o′ are separated by the hyperplane H that is orthogonal to the segment o′o and passes through the point a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' See Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Now we show that P ′ can be covered by a ball smaller than MEB(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let oH be the point H ∩ o′o, and t (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', t′) be the point collinear with o and o′ on the right side of the sphere of MEB(P ′) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', left side of the sphere of MEB(P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' see Figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, we have ||t − oH|| + ||oH − o′|| = ||t − o′|| = ||a − o′|| < ||o′ − oH|| + ||oH − a|| =⇒ ||t − oH|| < ||oH − a||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (6) Similarly, we have ||t′ − oH|| < ||oH − a||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consequently, MEB(P) ∩ MEB(P ′) is covered by the ball B(oH, ||oH −a||) (the “red dotted” ball in Figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Further, because P ′ is covered by MEB(P) ∩ MEB(P ′) and ||oH − a|| < ||o′ − a|| = Rad(P ′), P ′ is covered by the ball B(oH, ||oH − a||) that is smaller than MEB(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' This contradicts to the fact that MEB(P ′) is the minimum enclosing ball of P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, the claim ∠ao′o ≥ π/2 is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ 9 Given Claim 1, we know that ||o′ − o|| ≤ �� Rad(P) �2 − � Rad(P ′) �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' See Figure 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, Definition 3 implies that Rad(P ′) ≥ (1 − ϵ2)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, we have ||o′ − o|| ≤ �� Rad(P) �2 − � (1 − ϵ2)Rad(P) �2 ≤ √ 2ϵRad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (7) ⊓⊔ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The distance ||˜o − o′|| < ( √ 2ϵ + √ 3ϵ′)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let L be the hyperplane orthogonal to the segment ˜oo′ and passing through the center o′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose ˜o is located on the left side of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, there always exists a point b ∈ P ′ located on the right closed semi-sphere of MEB(P ′) divided by L (this result is from [22, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' for completeness, we state the lemma in Section B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' See Figure 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' That is, the angle ∠bo′˜o ≥ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As a consequence, we have ||˜o − o′|| ≤ � ||˜o − b||2 − ||b − o′||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (8) Moreover, since ||˜o − b|| ≤ r ≤ (1 + ϵ′2)Rad(P) and ||b − o′|| = Rad(P ′) ≥ (1 − ϵ2)Rad(P), (8) implies that ||˜o−o′|| ≤ � (1 + ϵ′2)2 − (1 − ϵ2)2Rad(P), where this upper bound is equal to � 2ϵ′2 + ϵ′4 + 2ϵ2 − ϵ4Rad(P) < � 3ϵ′2 + 2ϵ2Rad(P) < ( √ 2ϵ + √ 3ϵ′)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (9) ⊓⊔ By triangle inequality, Lemmas 1 and 2, we immediately have ||˜o − o|| ≤ ||˜o − o′|| + ||o′ − o|| < (2 √ 2ϵ + √ 3ϵ′)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (10) This completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4 Sublinear Time Algorithms for MEB under Stability Assumption Suppose ϵ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We assume that the given instance P is an (ϵ2, β)-stable instance where β is larger than a given lower bound β0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', β > β0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Using Theorem 2, we present two different sublinear time sampling algorithms for computing MEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Following most of the articles on sublinear time algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', [35,73,74]), in each sampling step of our algorithms, we always take the sample independently and uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 The First Algorithm r c Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 3: An illustration for the first sampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The red points are the samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' we expand B(c, r) slightly and the larger ball is a radius-approximate MEB of the whole input point set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 10 The first algorithm is based on the theory of VC dimension and ϵ-nets [58,92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Roughly speaking, we compute an approximate MEB of a small random sample (say, B(c, r)), and expand the ball slightly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' then we prove that this expanded ball is an approximate MEB of the whole data set (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our key idea is to show that B(c, r) covers at least (1 − β0)n points and therefore c is close to the optimal center by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As emphasized in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1, our result is a single-criterion approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If simply applying the uniform sample idea without the stability assumption (as the ideas in [41,62]), it will yield a bi-criteria approximation where the ball has to cover less than n points for achieving the desired bounded radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Algorithm 1 MEB Algorithm I Input: Two parameters 0 < ϵ, η < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' an (ϵ2, β)-stable instance P of MEB problem in Rd, where β is larger than a given lower bound β0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1: Sample a set S of Θ( 1 β0 · max{log 1 η , d log d β0 }) points from P uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2: Apply any approximate MEB algorithm (such as the core-set based algorithm [20]) to compute a (1 + ϵ2)- radius approximate MEB of S, and let the obtained ball be B(c, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 3: Output the ball B � c, 1+(2 √ 2+ √ 3)ϵ 1−ϵ2 r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' With probability 1 − η, Algorithm 1 returns a λ-radius approximate MEB of P, where λ = � 1 + (2 √ 2 + √ 3)ϵ � (1 + ϵ2) 1 − ϵ2 = 1 + O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (11) Before proving Theorem 3, we prove the following lemma first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let S be a set of Θ( 1 β0 · max{log 1 η, d log d β0 }) points sampled randomly and inde- pendently from a given point set P ⊂ Rd, and B be any ball covering S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, with probability 1 − η, |B ∩ P| ≥ (1 − β0)|P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consider the range space Σ = (P, Φ) where each range φ ∈ Φ is the complement of a ball in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In a range space, a subset Y ⊂ P is a β0-net if for any φ ∈ Φ, |P ∩ φ| |P| ≥ β0 =⇒ Y ∩ φ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (12) The size |S| = Θ( 1 β0 ·max{log 1 η, d log d β0 }), and from [58,92] we know that S is a β0-net of P with probability 1− η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, if |B ∩ P| < (1− β0)|P|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', |P \\B| > β0|P|, we have S ∩ � P \\B � ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' This contradicts to the fact that S is covered by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consequently, |B ∩ P| ≥ (1 − β0)|P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (of Theorem 3) Denote by o the center of MEB(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Since S ⊂ P and B(c, r) is a (1 + ϵ2)-radius approximate MEB of S, we know that r ≤ (1 + ϵ2)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, Lemma 3 implies that |B(c, r) ∩ P| ≥ (1 − β0)|P| with probability 1 − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose it is true and let P ′ = B(c, r) ∩ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, we have the distance ||c − o|| ≤ (2 √ 2 + √ 3)ϵRad(P) (13) via Theorem 2 (we set ϵ′ = ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For simplicity, we use x to denote (2 √ 2 + √ 3)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The inequality (13) implies that the point set P is covered by the ball B(c, (1 + x)Rad(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that we cannot directly return B(c, (1 + x)Rad(P)) as the final result, since we do not know the value of Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, we have to estimate the radius (1 + x)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 11 Since P ′ is covered by B(c, r) and |P ′| ≥ (1 − β0)|P|, r should be at least (1 − ϵ2)Rad(P) due to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Hence, we have 1 + x 1 − ϵ2 r ≥ (1 + x)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (14) That is, P is covered by the ball B(c, 1+x 1−ϵ2 r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, the radius 1 + x 1 − ϵ2 r ≤ 1 + x 1 − ϵ2 (1 + ϵ2)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (15) This means the ball B(c, 1+x 1−ϵ2 r) is a λ-radius approximate MEB of P, where λ = (1 + ϵ2) 1 + x 1 − ϵ2 = � 1 + (2 √ 2 + √ 3)ϵ � (1 + ϵ2) 1 − ϵ2 (16) and λ = 1 + O(ϵ) if ϵ is a fixed small number in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Running time of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For simplicity, we assume log 1 η < d log d β0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we use the core-set based algorithm [20] to compute B(c, r) (see Remark 3), the running time of Algorithm 1 is O � 1 ϵ2 (|S|d + 1 ϵ6 d) � = O � d2 ϵ2β0 log d β0 + d ϵ8 � = ˜O(d2) where the hidden factor depends on ϵ and β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If the dimensionality d is too high, the random projection technique Johnson- Lindenstrauss (JL) transform [36] can be used to approximately preserve the radius of enclosing ball [2,66,87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, it is not useful for reducing the time complexity of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we apply the JL-transform on the sampled Θ( d β0 log d β0 ) points in Step 1, the JL-transform step itself already takes Ω( d2 β0 log d β0 ) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 The Second Algorithm Our first algorithm in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 is simple, but has a sample size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the number of sampled points) depending on the dimensionality d, while the second algorithm has a sample size independent of both n and d (it is particularly important when a kernel function is applied, because the new dimension could be very large or even +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We briefly overview our idea first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' High level idea of the second algorithm: Recall our Remark 3 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we know the value of (1 + ϵ)Rad(P), we can perform almost the same core-set construction procedure described in Theorem 1 to achieve an approximate center of MEB(P), where the only difference is that we add a point with distance at least (1 + ϵ)Rad(P) to oi in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In this way, we avoid selecting the farthest point to oi, since this operation will inevitably have a linear time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To implement our strategy in sublinear time, we need to determine the value of (1+ϵ)Rad(P) first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We propose Lemma 4 below to estimate the range of Rad(P), and then perform a binary search on the range to determine the value of (1 + ϵ)Rad(P) approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Based on the stability property, we observe that the core-set construction procedure can serve as an “oracle” to help us to guess the value of (1 + ϵ)Rad(P) (see Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let h > 0 be a candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We add a point with distance at least h to oi in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We prove that the procedure cannot continue for more than z iterations if h ≥ (1 + ϵ)Rad(P), and will continue more than z iterations with constant probability if h < (1 − ϵ)Rad(P), where z is the size of core-set described in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, during the core-set construction, we add the points to the core-set via random sampling, rather than a deterministic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' A minor issue here is that we need to replace ϵ by ϵ2 in Theorem 1, so as to achieve the overall (1 + O(ϵ))-radius approximation in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 12 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a parameter η ∈ (0, 1), one selects an arbitrary point p1 ∈ P and takes a sample Q ⊂ P with |Q| = 1 β0 log 1 η uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let p2 = arg maxp∈Q ||p − p1||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, with probability 1 − η, Rad(P) ∈ [1 2||p1 − p2||, 1 1 − ϵ2 ||p1 − p2||].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, the lower bound of Rad(P) is obvious since ||p1 − p2|| is always no larger than 2Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, we consider the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let B(p1, l) be the ball covering exactly (1−β0)n points of P, and thus l ≥ (1 − ϵ2)Rad(P) according to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To complete our proof, we also need the following folklore lemma presented in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [39] Let N be a set of elements, and N′ be a subset of N with size |N′| = τ |N| for some τ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given η ∈ (0, 1), if one randomly samples ln 1/η ln 1/(1−τ) ≤ 1 τ ln 1 η elements from N, then with probability at least 1 − η, the sample contains at least one element of N′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' l p1 p1 p2 p2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4: An illustration of Lemma 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' the red points are the sampled set Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Lemma 5, let N and N′ be the point set P and the subset P \\ B(p1, l), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We know that Q contains at least one point from N′ according to Lemma 5 (by setting τ = β0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Namely, Q contains at least one point outside B(p1, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, because p2 = arg maxp∈Q ||p − p1||, we have ||p1 − p2|| ≥ l ≥ (1 − ϵ2)Rad(P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', Rad(P) ≤ 1 1−ϵ2 ||p1 − p2|| (see Figure 4 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Algorithm 3 serves as a subroutine in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Algorithm 3, we simply set z = 3 ϵ2 with s = 1/3 as described in Theorem 1 (as mentioned before, we replace ϵ by ϵ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' we compute oi having distance less than s ϵ2 1+ϵ2 Rad(T) to the center of MEB(T) in Step 2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Algorithm 2 MEB Algorithm II Input: Two parameters 0 < ϵ, η0 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' an (ϵ2, β)-stable instance P of MEB problem in Rd, where β is larger than a given lower bound β0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Set the interval [a, b] for Rad(P) that is obtained by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1: Among the set {(1−ϵ2)a, (1+ϵ2)(1−ϵ2)a, · · · , (1+ϵ2)w(1−ϵ2)a = (1+ϵ2)b} where w = ⌈log1+ϵ2 2 (1−ϵ2)2 ⌉+1 = O( 1 ϵ2 ), perform binary search for the value h by using Algorithm 3 with z = 3 ϵ2 and η = η0 2 log w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2: Suppose that Algorithm 3 returns “no” when h = (1 + ϵ2)i0(1 − ϵ2)a and returns “yes” when h = (1 + ϵ2)i0+1(1 − ϵ2)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 3: Run Algorithm 3 again with h = (1 + ϵ2)i0+2a, z = 3 ϵ2 , and η = η0/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' let ˜o be the obtained ball center of T when the loop stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4: Return the ball B(˜o, r), where r = 1+(2 √ 2+ 2 √ 6 √ 1−ϵ2 )ϵ 1+ϵ2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 13 Algorithm 3 Oracle for testing h Input: An instance P, a parameter η ∈ (0, 1), h > 0, and a positive integer z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1: Initially, arbitrarily select a point p ∈ P and let T = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2: i = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' repeat the following steps: (1) Compute an approximate MEB of T and let the ball center be oi as described in Theorem 1 (replace ϵ by ϵ2 and set s = 1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (2) Sample a set Q ⊂ P with |Q| = 1 β0 log z η uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (3) Select the point q ∈ Q that is farthest to oi, and add it to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (4) If ||q − oi|| < h, stop the loop and output “yes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (5) i = i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' if i > z, stop the loop and output “no”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' With probability 1 − η0, Algorithm 2 returns a λ-radius approximate MEB of P, where λ = (1 + x1)(1 + x2) 1 + ϵ2 = 1 + O(ϵ) with x1 = O � ϵ2 1 − ϵ2 � , x2 = O � ϵ √ 1 − ϵ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (18) The running time is ˜O � ( 1 ϵ2β0 + 1 ϵ8 )d � , where ˜O(f) = O(f · polylog( 1 ϵ, 1 η0 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Before proving Theorem 4, we provide Lemma 6 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If h ≥ (1 + ϵ2)Rad(P), Algorithm 3 returns “yes”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' else if h < (1 − ϵ2)Rad(P), Algorithm 3 returns “no” with probability at least 1 − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we assume that h ≥ (1 + ϵ2)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Recall the remark following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we always add a point q with distance at least h ≥ (1 + ϵ2)Rad(P) to oi, the loop 2(1)-(5) cannot continue more than z iterations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', Algorithm 3 will return “yes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Now, we consider the case h < (1 − ϵ2)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to the proof of Lemma 4, we consider the ball B(oi, l) covering exactly (1 − β0)n points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' According to Definition 3, we know that l ≥ (1 − ϵ2)Rad(P) > h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, with probability 1 − η/z, the sample Q contains at least one point outside B(oi, l) due to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By taking the union bound, with probability (1 − η/z)z ≥ 1 − η, ||q − oi|| is always larger than h and eventually Algorithm 3 will return “no”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (of Theorem 4) Since Algorithm 3 returns “no” when h = (1 + ϵ2)i0(1 − ϵ2)a and returns “yes” when h = (1 + ϵ2)i0+1(1 − ϵ2)a, from Lemma 6 we know that (1 + ϵ2)i0(1 − ϵ2)a < (1 + ϵ2)Rad(P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (19) (1 + ϵ2)i0+1(1 − ϵ2)a ≥ (1 − ϵ2)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (20) The above inequalities together imply that (1 + ϵ2)3 1 − ϵ2 Rad(P) > (1 + ϵ2)i0+2a ≥ (1 + ϵ2)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (21) Thus, when running Algorithm 3 with h = (1 + ϵ2)i0+2a in Step 3, the algorithm returns “yes” (by the right hand-side of (21)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, consider the ball B(˜o, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We claim that |P\\B(˜o, h)| < β0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Otherwise, the sample Q contains at least one point outside B(˜o, h) with probability 1 − η/z in Step 2(2) of Algorithm 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the loop will continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, it contradicts to the fact that the algorithm returns “yes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let P ′ = P ∩ B(˜o, h), and then |P ′| ≥ (1 − β0)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, the left hand-side of (21) indicates that h = (1 + ϵ2)i0+2a < (1 + 8ϵ2 1 − ϵ2 )Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (22) 14 Now, we can apply Theorem 2, where we set “ϵ′” to be “ � 8ϵ2 1−ϵ2 ” in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let o be the center of MEB(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consequently, we have ||˜o − o|| < (2 √ 2 + 2 √ 6/ � 1 − ϵ2)ϵ · Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (23) For simplicity, we let x1 = 8ϵ2 1−ϵ2 and x2 = (2 √ 2+2 √ 6/ √ 1 − ϵ2)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Hence, h ≤ (1+x1)Rad(P) and ||˜o−o|| ≤ x2Rad(P) in (22) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' From (23), we know that P ⊂ B(˜o, (1+x2)Rad(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' From the right hand-side of (21), we know that (1 + x2)Rad(P) ≤ 1+x2 1+ϵ2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, we have P ⊂ B � ˜o, 1+x2 1+ϵ2 h � where 1+x2 1+ϵ2 h = 1+(2 √ 2+ 2 √ 6 √ 1−ϵ2 )ϵ 1+ϵ2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, the radius 1 + x2 1 + ϵ2 h ≤ ���� by (22) (1 + x2)(1 + x1) 1 + ϵ2 Rad(P) = λ · Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (24) Thus B � ˜o, 1+x2 1+ϵ2 h � is a λ-radius approximate MEB of P, and λ = 1 + O(ϵ) if ϵ is a fixed small number in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Success probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The success probability of Algorithm 3 is 1 − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Algorithm 2, we set η = η0 2 log w in Step 1 and η = η0/2 in Step 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We take the union bound and the success probability of Algorithm 2 is (1 − η0 2 log w)log w(1 − η0/2) > 1 − η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As the subroutine, Algorithm 3 runs in O(z( 1 β0 (log z η)d + 1 ϵ6 d)) time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Algorithm 2 calls the subroutine O � log( 1 ϵ2 ) � times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that z = O( 1 ϵ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, the total running time is ˜O � ( 1 ϵ2β0 + 1 ϵ8 )d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ 5 Sublinear Time Algorithm for General MEB In Section 4, we propose the sublinear time algorithms under the stability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Specif- ically, we assume that the given instance is (ϵ2, β)-stable and β is larger than a reasonable known lower bound β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, when β0’s value is unknown, we cannot not determine the sample size for the algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' or we may only know a trivial lower bound, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', 1 n, and then the sample size could be too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So in this section we consider the general case without the stability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' High-level idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' An interesting observation is that the ideas developed for stable instance can even help us to develop a hybrid approach for MEB when the stability assumption does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we “suppose” the input instance is (α, β)-stable where “α” and “β” are designed based on the pre-specified radius error bound ϵ and covering error bound δ, and compute a “potential” (1 + ϵ)-radius approximation (say a ball B1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' then we compute a “potential” (1−δ)-covering approximation (say a ball B2), where the definition of “covering approximation” is given in Definition 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' finally, we determine the final output based on the ratio of their radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Specifically, we set a threshold τ that is determined by the given radius error bound ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If the ratio is no larger than τ, we can infer that B1 is a “true” (1 + ϵ)-radius approximation and return it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' otherwise, we return B2 that is a “true” (1 − δ)-covering approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, for the latter case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', returning a (1 − δ)-covering approximation), we will show that our proposed algorithm yields a radius not only being strictly smaller than Rad(P), but also having a gap of Θ(ϵ2) · Rad(P) to Rad(P) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the returned radius is at most � 1 − Θ(ϵ2) � Rad(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our algorithm only needs uniform sampling and a single pass over the input data, where the space complexity in memory is O(d) (the hidden factor depends on ϵ and δ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' if the input data matrix is sparse (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', M = o(nd)), the time complexity is sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Before presenting our algorithms, we need to show the formal definitions for the problem of MEB with outliers first, since it will be used for computing the (1 − δ)-covering approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 15 Definition 4 (MEB with Outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a set P of n points in Rd and a small parameter γ ∈ [0, 1), the MEB with outliers problem is to find the smallest ball that covers (1−γ)n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Namely, the task is to find a subset of P with size (1 − γ)n such that the resulting MEB is the smallest among all possible choices of the subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The obtained ball is denoted by MEB(P, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For convenience, we use Popt to denote the optimal subset of P with respect to MEB(P, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Namely, Popt = argQ min � Rad(Q) | Q ⊂ P, |Q| = (1 − γ)n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' From Definition 4, we can see that the main challenge is to determine the subset of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to Definition 2, we also define the radius approximation and covering approximation for MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 5 (Radius Approximation and Covering Approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let 0 < ϵ, δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' A ball B(c, r) is called a (1 + ϵ)-radius approximation of MEB(P, γ), if the ball covers (1 − γ)n points of P and has radius r ≤ (1 + ϵ)Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' On the other hand, the ball is called a (1 − δ)-covering approximation of MEB(P, γ), if it covers at least (1 − δ − γ)n points in P and has radius r ≤ Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' A bi-criteria (1 + ϵ, 1 − δ)-approximation is a ball that covers at least � 1 − δ − γ � n points and has radius at most (1 + ϵ)Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we introduce two random sampling techniques in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1, which are the keys for designing the sublinear bi-criteria approximation algorithm for MEB with outliers in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Based on the bi-criteria approximation of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2, we can solve the general MEB problem in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 Two Key Lemmas for Handling Outliers To shed some light on our ideas, consider using the core-set construction method in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 to compute a bi-criteria (1+ϵ, 1−δ)-approximation for an instance (P, γ) of MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let oi be the obtained ball center in the current iteration, and Q be the set of (δ +γ)n farthest points to oi from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' A key step for updating oi is finding a point in the set Popt ∩ Q (the formal analysis is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Actually, this can be done by performing a random sampling from Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, it requires to compute the set Q in advance, which takes an Ω(nd) time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To keep the running time to be sublinear, we need to find a point from Popt ∩ Q by a more sophisticated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Since Popt is mixed with outliers in the set Q, simple uniform sampling cannot realize our goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To solve this issue, we propose a “two level” sampling procedure which is called “Uniform-Adaptive Sampling”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Roughly speaking, we take a random sample A of size n′ first (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the uniform sampling step), and then randomly select a point from Q′, the set of the farthest 3 2(δ + γ)n′ points from A to oi (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the adaptive sampling step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' According to Lemma 7, with probability at least (1 − η1) δ 3(δ+γ), the selected point belongs to Popt ∩ Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' more importantly, the sample size n′ is independent of n and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The key to prove Lemma 7 is to show that the size of the intersection Q′ ∩ � Popt ∩ Q � is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By setting an appropriate value for n′, we can prove a lower bound of |Q′ ∩ � Popt ∩Q � |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Lemma 7 (Uniform-Adaptive Sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let η1 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we sample n′ = O( 1 δ log 1 η1 ) points independently and uniformly at random from P and let Q′ be the set of farthest 3 2(δ+γ)n′ points to oi from the sample, then, with probability at least 1 − η1, the following holds ���Q′ ∩ � Popt ∩ Q ���� |Q′| ≥ δ 3(δ + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let A denote the set of sampled n′ points from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we know |Q| = (δ + γ)n and |Popt ∩ Q| ≥ δn (since there are at most γn outliers in Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For ease of presentation, let 16 λ = |Popt∩Q| n ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let {xi | 1 ≤ i ≤ n′} be n′ independent random variables with xi = 1 if the i-th sampled point of A belongs to Popt ∩ Q, and xi = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, E[xi] = λ for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let σ be a small parameter in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By using the Chernoff bound, we have Pr � �n′ i=1 xi /∈ (1 ± σ)λn′� ≤ e−O(σ2λn′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' That is, Pr � |A ∩ � Popt ∩ Q � | ∈ (1 ± σ)λn′� ≥ 1 − e−O(σ2λn′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (26) Similarly, we have Pr � |A ∩ Q| ∈ (1 ± σ)(δ + γ)n′� ≥ 1 − e−O(σ2(δ+γ)n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (27) Note that n′ = O( 1 δ log 1 η1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By setting σ < 1/2 for (26) and (27), we have ���A ∩ � Popt ∩ Q ���� > 1 2δn′ and ���A ∩ Q ��� < 3 2(δ + γ)n′ (28) with probability 1 − η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that Q contains all the farthest (δ + γ)n points to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Denote by li the � (δ + γ)n + 1 � th largest distance from P to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we have A ∩ Q = {p ∈ A | ||p − oi|| > li}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (29) Also, since Q′ is the set of the farthest 3 2(δ + γ)n′ points to oi from A, there exists some l′ i > 0 such that Q′ = {p ∈ A | ||p − oi|| > l′ i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (30) (29) and (30) together imply that either (A∩Q) ⊆ Q′ or Q′ ⊆ (A∩Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Since ��A∩Q �� < 3 2(δ+γ)n′ and |Q′| = 3 2(δ + γ)n′, we know � A ∩ Q � ⊆ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, � A ∩ � Popt ∩ Q �� = � Popt ∩ � A ∩ Q �� ⊆ Q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (31) Also, it is obvious that � A ∩ � Popt ∩ Q �� ⊆ � Popt ∩ Q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (32) The above (31) and (32) together imply � A ∩ � Popt ∩ Q �� ⊆ � Q′ ∩ � Popt ∩ Q �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (33) Moreover, since Q′ ⊆ A, we have � Q′ ∩ � Popt ∩ Q �� ⊆ � A ∩ � Popt ∩ Q �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (34) Consequently, (33) and (34) together imply Q′ ∩ � Popt ∩ Q � = A ∩ � Popt ∩ Q � and hence ���Q′ ∩ � Popt ∩ Q ���� |Q′| = ���A ∩ � Popt ∩ Q ���� |Q′| ≥ δ 3(δ + γ), (35) where the inequality comes from the first inequality of (28) and the fact |Q′| = 3 2(δ + γ)n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ 17 The random sampling method is not always guaranteed to succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To boost the overall success probability, we have to repeatedly run the algorithm multiple times and each time the algorithm will generate a candidate solution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the ball center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consequently we have to select the best one as our final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' With a slight abuse of notation, we still use oi to denote a candidate ball center;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' since our goal is to achieve a (1 + ϵ, 1 − δ)-approximation, we need to compute the � (δ + γ)n + 1 � th largest distance from P to oi, which is denoted as li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' A straightforward way is to compute the value “li” in linear time for each candidate and return the one having the smallest li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In this section, we propose the “Sandwich Lemma” to estimate li in sublinear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let B be the set of n′′ sampled points from P in Lemma 8, and ˜li be the � (1+δ/γ)2γn′′ +1 � th largest distance from B to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we can prove the inequalities (37) and (38) of Lemma 8, then they can imply that ˜li is a qualified estimation of li: if B(oi, li) is a (1 + ϵ, 1 − δ)-approximation, the ball B(oi, ˜li) should be a (1 + ϵ, 1 − O(δ))-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The key idea is to prove that the ball B(oi, ˜li) is “sandwiched” by two balls B(oi, ˜l′ i) and B(oi, li), where ˜l′ i is a carefully designed value satisfying (i) ˜l′ i ≤ ˜li ≤ li and (ii) ���P \\ B(oi, ˜l′ i) ��� ≤ (γ + O(δ))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (36) See Figure 5 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' These two conditions of ˜l′ i can imply the inequalities (37) and (38) of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to Lemma 7, the sample size n′′ is also independent of n and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ˜l′ i˜l′ i lili ˜li˜li Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 5: The red points are the sampled n′′ points in Lemma 8, and the � (1 + δ/γ)2γn′′ + 1 � th farthest point is in the ring bounded by the spheres B(oi, ˜l′ i) and B(oi, li).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Lemma 8 (Sandwich Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let η2 ∈ (0, 1) and assume δ < γ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we sample n′′ = O � γ δ2 log 1 η2 � points independently and uniformly at random from P and let ˜li be the � (1 + δ/γ)2γn′′ + 1 � th largest distance from the sample to oi, then, with probability 1 − η2, the following holds ˜li ≤ li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (37) ���P \\ B(oi, ˜li) ��� ≤ (γ + 5δ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (38) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let B denote the set of sampled n′′ points from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For simplicity, let t = (δ + γ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Assume ˜l′ i > 0 is the value such that ���P \\ B(oi, ˜l′ i) ��� = (γ+δ)2 γ−δ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Recall that li is the � t + 1 � th largest distance from P to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Since (δ + γ)n < (γ+δ)2 γ−δ n, it is easy to know ˜l′ i ≤ li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Below, we aim to prove that the � (1 + δ/γ)2γn′′ + 1 � th farthest point from B is in the ring bounded by the spheres B(oi, ˜l′ i) and B(oi, li) (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 18 Note the size |B| = n′′ = O � γ δ2 log 1 η2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Again, using the Chernoff bound (let σ = δ/2) and the same idea for proving (28), we have ���B \\ B(oi, ˜l′ i) ��� ≥ (1 − δ 2γ )(γ + δ)2 γ − δ n′′ > (1 − δ γ )(γ + δ)2 γ − δ n′′ = (1 + δ/γ)2γn′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (39) ���B ∩ Q �� ≤ (1 + δ 2γ ) t nn′′ < (1 + δ/γ) t nn′′ = (1 + δ/γ)2γn′′, (40) with probability 1 − η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose that (39) and (40) both hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Recall that ˜li is the � (1 + δ/γ)2γn′′ + 1 � th largest distance from the sampled points B to oi, so ���B \\ B(oi, ˜li) ��� = (1 + δ/γ)2γn′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Together with (39), we have ���B \\ B(oi, ˜li) ��� ≤ ���B \\ B(oi, ˜l′ i) ���, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', ˜li ≥ ˜l′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (41) The inequality (40) implies that the � (1 + δ/γ)2γn′′ + 1 � th farthest point (say qx) from B to oi is not in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, we claim that B(oi, ˜li) ∩ Q = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Otherwise, let qy ∈ B(oi, ˜li) ∩ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we have ||qy − oi|| ≤ ˜li = ||qx − oi||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (42) Note that Q is the set of farthest t points to oi of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So qx /∈ Q implies ||qx − oi|| < min q∈Q ||q − oi|| ≤ ||qy − oi|| (43) which is in contradiction to (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, B(oi, ˜li) ∩ Q = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Further, since B(oi, li) excludes exactly the farthest t points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', Q), “B(oi, ˜li) ∩ Q = ∅” implies ˜li ≤ li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (44) Overall, we have ˜li ∈ [˜l′ i, li] from (41) and (44), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the � (1 + δ/γ)2γn′′ + 1 � th farthest point from B locates in the ring bounded by the spheres B(oi, ˜l′ i) and B(oi, li) as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, ˜li ≥ ˜l′ i implies ���P \\ B(oi, ˜li) ��� ≤ ���P \\ B(oi, ˜l′ i) ��� = (γ + δ)2 γ − δ n < (γ + 5δ)n, (45) where the last equality comes from the assumption δ < γ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So (37) and (38) are true in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Actually our proposed Uniform-Adaptive Sampling method and Sandwich lemma are quite generic, and we will show that they can be generalized to solve a broader range of enclosing with outliers problems in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 Sublinear Time Algorithm for Bi-criteria Approximation In this section, we propose a sublinear time algorithm for computing a bi-criteria (1 + ϵ, 1 − δ)- approximation for the input instance (P, γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' that is, the returned ball covers at least � 1−δ−γ � n points and has radius at most (1 + ϵ)Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Recall the remark following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As long as the selected point has a distance to the center of MEB(T) larger than (1 + ϵ) times the optimal radius, the expected improvement will always be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Following this observation, we investigate the following approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose we run the core-set construction procedure decribed in Theorem 1 (we should 19 replace P by Popt in our following analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In the i-th step, we add an arbitrary point from Popt \\ B(oi, (1 + ϵ)Rad(Popt)) to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We know that a (1 + ϵ)-approximation is obtained after at most 2 (1−s)ϵ steps, that is, Popt ⊂ B � oi, (1 + ϵ)Rad(Popt) � for some i ≤ 2 (1−s)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, we need to solve two key issues for realizing the above approach: (i) how to determine the value of Rad(Popt) and (ii) how to correctly select a point from Popt \\ B(oi, (1 + ϵ)Rad(Popt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Actually, we can implicitly avoid the first issue via replacing (1+ϵ)Rad(Popt) by the t-th largest distance from the points of P to oi, where we set t = (δ + γ)n for guaranteeing a (1 + ϵ, 1 − δ)-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For the second issue, we randomly select one point from the farthest t points of P to oi, and show that it belongs to Popt \\ B(oi, (1 + ϵ)Rad(Popt)) with a certain probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Based on the above idea, we present a sublinear time (1+ϵ, 1−δ)-approximation algorithm in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To better understand the algorithm, we show a linear time algorithm first (Algorithm 4 in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that B˘adoiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [22] also presented a (1 + ϵ, 1 − δ)- approximation algorithm but with a higher complexity, and please see our detailed analysis on the running time at the end of Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' More importantly, we can improve the running time of Algorithm 4 to be sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For this purpose, we need to avoid computing the farthest t points to oi, since this operation will take linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, Algorithm 4 generates a set of candidates for the solution and we need to select the best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' This process also costs linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By using the techniques proposed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1, we can solve these issues and develop a sublinear time algorithm that has the sample complexity independent of n and d, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 A Linear Time Algorithm In this section, we present our linear time (1 + ϵ, 1 − δ)- approximation algorithm for MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Algorithm 4 (1 + ϵ, 1 − δ)-approximation Algorithm for MEB with Outliers Input: A point set P with n points in Rd, the fraction of outliers γ ∈ (0, 1), and the parameters 0 < ϵ, δ < 1, z ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1: Let t = (δ + γ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2: Initially, randomly select a point p ∈ P and let T = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 3: i = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' repeat the following steps until i > z: (1) Denote by ci the exact center of MEB(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Compute the approximate center oi with a distance to ci of less than ξRad(T) = s ϵ 1+ϵRad(T) as described in Theorem 1, where s is set to be ϵ 2+ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (2) Let Q be the set of farthest t points from P to oi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' denote by li the (t + 1)-th largest distance from P to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (3) Randomly select a point q ∈ Q, and add it to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (4) i = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4: Output the ball B(oˆi, lˆi) where ˆi = argi min{li | 1 ≤ i ≤ z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If the input parameter z = 2 (1−s)ϵ (we assume it is an integer for convenience), then with probability (1 − γ)( δ γ+δ)z, Algorithm 4 outputs a (1 + ϵ, 1 − δ)-approximation for the MEB with outliers problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Before proving Theorem 5, we present the following two lemmas first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' With probability (1 − γ)( δ γ+δ)z, after running z rounds in Step 3 of Algorithm 4, the obtained set T ⊂ Popt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Initially, because |Popt|/|P| = 1 − γ, the first selected point in Step 2 belongs to Popt with probability 1 − γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In each of the z rounds in Step 3, the selected point belongs to Popt 20 with probability δ γ+δ, since |Popt ∩ Q| |Q| = 1 − |Q \\ Popt| |Q| ≥ 1 − |P \\ Popt| |Q| = 1 − γn (δ + γ)n = δ δ + γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (46) Therefore, with probability (1 − γ)( δ γ+δ)z the whole set T ⊂ Popt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In the i-th round of Step 3 for 1 ≤ i ≤ z, at least one of the following two events happens: (1) oi is the ball center of a (1+ϵ, 1−δ)-approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (2) ri+1 > (1+ϵ)Rad(Popt)− ||ci − ci+1|| − ξri, where ri is the exact radius of MEB(T) is the i-th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If li ≤ (1 + ϵ)Rad(Popt), then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' That is, the ball B(oi, li) covers (1 − δ − γ)n points with radius li ≤ (1 + ϵ)Rad(Popt) (the first event happens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Otherwise, li > (1 + ϵ)Rad(Popt) and we consider the second event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let q be the point added to T in the i-th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Using the triangle inequality, we have ||oi − q|| ≤ ||oi − ci|| + ||ci − ci+1|| + |ci+1 − q|| ≤ ξri + ||ci − ci+1|| + ri+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (47) Since li > (1+ϵ)Rad(Popt) and q lies outside of B(oi, li), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e, ||oi −q|| ≥ li > (1+ϵ)Rad(Popt), (47) implies that the second event happens and the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (of Theorem 5) Suppose that the first event of Lemma 10 never happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As a consequence, we obtain a series of inequalities for each pair of radii ri+1 and ri, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', ri+1 > (1 + ϵ)Rad(Popt) − ||ci − ci+1|| − ξri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Assume that T ⊂ Popt in Lemma 9, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', each time the algorithm correctly adds a point from Popt to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Using the almost identical idea for proving Theorem 1 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1, we know that a (1 + ϵ)-approximate MEB of Popt is obtained after at most z rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The success probability directly comes from Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Overall, we obtain Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Theorem 5 directly implies the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If one repeatedly runs Algorithm 4 O( 1 1−γ (1 + γ δ )z) times, with constant prob- ability, the algorithm outputs a (1 + ϵ, 1 − δ)-approximation for the problem of MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Theorem 5, we set z = 2 (1−s)ϵ and s ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To keep z small, according to Theorem 1, we set s = ϵ 2+ϵ so that z = 2 ϵ + 1 (only larger than the lower bound 2 ϵ by 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For each round of Step 3, we need to compute an approximate center oi that has a distance to the exact one less than ξri = s ϵ 1+ϵri = O(ϵ2)ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Using the algorithm proposed in [20], this can be done in O( 1 ξ2 |T|d) = O( 1 ϵ5 d) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, the set Q can be obtained in linear time by the algorithm in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In total, the time complexity for obtaining a (1 + ϵ, 1 − δ)-approximation in Corollary 1 is O �C ϵ (n + 1 ϵ5 )d � , (48) where C = O( 1 1−γ (1 + γ δ ) 2 ϵ +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As mentioned before, B˘adoiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [22] also proposed a linear time bi-criteria approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, the hidden constant of their running time is exponential in Θ( 1 ϵδ) that is much larger than 2 ϵ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 Improvement on Running Time In this section, we show that the running time of Algorithm 4 can be further improved to be independent of the number of points n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we observe that it is not necessary to compute the set Q of the farthest t points in Step 3(2) of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Actually, as long as the selected point q is part of Popt ∩ Q in Step 3(3), a (1 + ϵ, 1 − δ)-approximation is still guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The Uniform-Adaptive Sampling procedure proposed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 can help us to obtain a point q ∈ Popt ∩ Q without computing the set Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, in Lemma 8, we show that the radius of each candidate solution can be estimated via random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Overall, we achieve a sublinear time algorithm (Algorithm 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Following the analysis in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1, we set s = ϵ 2+ϵ so that z = 2 (1−s)ϵ = 2 ϵ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We present the results in Theorem 6 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Comparing with Theorem 5, we have an extra (1−η1)(1−η2) in the success probability in Theorem 6, due to the probabilities from Lemmas 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Another minor issue is that the covering approximation error is increased from δ to 5δ when applying Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Actually this issue can be easily solved by replacing δ by δ/5 in the parameters n′, t′, n′′, and t′′, and the asymptotic complexity does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Algorithm 5 Sublinear Time (1 + ϵ, 1 − δ)-approximation Algorithm for MEB with Outliers Input: A point set P with n points in Rd, the fraction of outliers γ ∈ (0, 1), and the parameters ϵ, η1, η2 ∈ (0, 1), δ ∈ (0, 1/3γ), and z ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1: Let n′ = O( 1 δ log 1 η1 ), n′′ = O � γ δ2 log 1 η2 � , t′ = 3 2(δ/5 + γ)n′, and t′′ = (1 + δ 5γ )2γn′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2: Initially, randomly select a point p ∈ P and let T = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 3: i = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' repeat the following steps until j = z: (1) Compute the approximate MEB center oi of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (2) Sample n′ points uniformly at random from P, and let Q′ be the set of farthest t′ points to oi from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (3) Randomly select a point q ∈ Q′, and add it to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (4) Sample n′′ points uniformly at random from P, and let ˜li be the (t′′ + 1)-th largest distance from the sampled points to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (5) i = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4: Output the ball B(oˆi, ˜lˆi) where ˆi = argi min{˜li | 1 ≤ i ≤ z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If the input parameter z = 2 ϵ + 1, then with probability (1 − γ) � (1 − η1)(1 − η2) δ/5 3(γ+δ/5) �z, Algorithm 5 outputs a (1 + ϵ, 1 − δ)-approximation for the problem of MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To boost the success probability in Theorem 6, we need to repeatedly run Algorithm 5 and output the best candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, we need to be careful on setting the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The success probability in Theorem 6 consists of two parts, P1 = (1 − γ) � (1 − η1) δ/5 3(γ+δ/5) �z and P2 = (1 − η2)z, where P1 indicates the probability that {o1, · · · , oz} contains a qualified candidate, and P2 indicates the success probability of Lemma 8 over all the z rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, if we run Algorithm 5 N = O( 1 P1 ) times, with constant probability (by taking the union bound), the set of all the generated candidates contains at least one that yields a (1 + ϵ, 1 − δ)- approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' moreover, to guarantee that we can correctly estimate the resulting radii of all the candidates via the Sandwich Lemma with constant probability, we need to set η2 = O( 1 zN ) (because there are O(zN) candidates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If one repeatedly runs Algorithm 5 N = O � 1 1−γ � 1 1−η1 (3 + 3γ δ/5) �z� times with setting η2 = O( 1 zN ), with constant probability, the algorithm outputs a (1+ϵ, 1−δ)-approximation for the problem of MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 22 The calculation of running time is similar to (48) in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We just replace n by max{n′, n′′} = O � γ δ2 log 1 η2 � = O � γ δ2 log(zN) � = ˜O � γ δ2ϵ � 3, and change the value of C to be O � 1 1−γ � 1 1−η1 (3 + 3γ δ/5) � 2 ϵ +1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So the total running time is independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='3 General MEB Problem Now we consider solving the general MEB problem without the stability assumption in this Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let 0 < ϵ, δ < 1 be two given parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we view the input P as an instance (P, δ/2) of MEB with outliers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', γ = δ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, we apply the algorithm of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 to obtain a bi-criteria (1+ϵ2/2, 1−δ/2)-approximation solution B(c, rc) (we replace the “ϵ” by ϵ2/2 and replace the “δ” by δ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The obtained ball B(c, rc) covers at least (1−δ/2−δ/2)n = (1−δ)n points of P, and the radius rc ≤ (1 + 1 2ϵ2) · r−δ/2, (49) where r−δ/2 stands for the radius of the smallest ball that covers at least (1 − δ/2)n points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Second, we assume that the input P is an (α, β)-stable instance with α = ϵ2 and β = δ/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' then run Algorithm 2 to obtain a candidate ball center ˜o (of course, we can also use Algorithm 1, where the only difference is that the sample complexity will be higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To compute the real radius r˜o yielded from ˜o (since P may not be a real (α, β)-stable instance), we just need to read the whole dataset P in one pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Finally, we determine the final output based on the ratio r˜o/rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Algorithm 6 Hybrid Approximation for MEB Input: An instance P of MEB problem in Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' two parameters 0 < ϵ, δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1: View the input as a (P, δ/2) instance of MEB with outliers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' apply the method of Corollary 2 to obtain a bi-criteria (1 + ϵ2/2, 1 − δ/2)-approximation solution B(c, rc) on (P, δ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2: Assume that the input P is an (α, β)-stable instance with α = ϵ2 and β = δ/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' then run Algorithm 2 to obtain a candidate ball center ˜o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 3: Read the whole input dataset P in one-pass, and compute the radius r˜o = maxp∈P ||˜o − p||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4: If the ratio r˜o rc ≤ 1+ϵ 1−ϵ2/2, return the ball B(˜o, r˜o) and say “it is a (1 + ϵ)-radius approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 5: Else, return the ball B(c, rc) and say “it is a (1 − δ)-covering approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' With constant success probability, Algorithm 6 returns either a (1 + ϵ)-radius approximation or a (1−δ)-covering approximation, and the running time is O �� n+h(ϵ, δ) � d � , where h(ϵ, δ) = O � 1 1−δ/2 exp(O(1/ϵ2)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The algorithm only needs uniform sampling and a single pass over the input data, and the space complexity in memory is O(h(ϵ, δ) · d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, if the input data matrix (the n × d matrix representing the input P) has at most M ≪ nd non-zeros entries, the total running time will be O � n + h(ϵ, δ) · d + M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In the following proof, we will see that when the algorithm returns a (1 − δ)- covering approximation, the returned radius is not only ≤ Rad(P), but also at most � 1 − Θ(ϵ2) � Rad(P) (see (52) and (54)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We study the time and space complexities first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The method of Corollary 2 only needs uniform samplings, and Step 2 of Algorithm 6 is a single pass over the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' According to 3 The asymptotic notation ˜O(f) = O � f · polylog( γ η1δ(1−γ)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 23 Corollary 2, we know the space complexity is O(h(ϵ, δ)·d) with h(ϵ, δ) = O � 1 1−δ/2 exp(O(1/ϵ2)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The total running time is O �� n + h(ϵ, δ) � d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Furthermore, we consider the case that the input matrix is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Step 3, we need to compute the value r˜o = maxp∈P ||˜o − p||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For each point p ∈ P, we know ||˜o − p||2 = ||˜o||2 + ||p||2 − 2⟨˜o, p⟩, (50) where ⟨˜o, p⟩ stands for their inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The value of ||˜o||2 can be obtained in O(d) time, and if the input data matrix has at most M ≪ nd non-zeros entries, the complexity for computing the values {||p||2 − 2⟨˜o, p⟩ | p ∈ P} is O(n + M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Overall, the complexity of Algorithm 6 is O � n + h(ϵ, δ) · d + M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Now, we prove the solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We let α = ϵ2 and β = δ/2, and consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Case 1: the instance P is (α, β)-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we directly have r˜o ≤ (1 + ϵ) · Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (51) If r˜o rc > 1+ϵ 1−ϵ2/2, together with (51), we have rc < � 1 − ϵ2/2 � Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (52) Then we can return the ball B(c, rc) and say “it is a (1 − δ)-covering approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' On the other hand, when r˜o rc ≤ 1+ϵ 1−ϵ2/2, from (51) we can return the ball B(˜o, r˜o) and say “it is a (1 + ϵ)-radius approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Case 2: P is not an (α, β)-stable instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, from the definition of stability we know the optimal radius of the instance (P, δ/2) is no larger than (1 − ϵ2) · Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (53) So we have rc < (1 + 1 2ϵ2)(1 − ϵ2) · Rad(P) < � 1 − ϵ2/2 � Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (54) If r˜o rc ≤ 1+ϵ 1−ϵ2/2, together with (54), it implies r˜o < (1 + ϵ) · Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (55) Then we can return the ball B(˜o, r˜o) and say “it is a (1 + ϵ)-radius approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' On the other hand, when r˜o rc > 1+ϵ 1−ϵ2/2, from (54) we can return the ball B(c, rc) and say “it is a (1 − δ)-covering approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Since the success probability of the method of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 is constant, the overall success probability of Algorithm 7 is constant as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ More analysis on the result of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We further consider an “inverse” question: can we infer the stability degree of the given instance P from the output of Algorithm 6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Step 2, we assume that P is an (ϵ2, δ/2)-stable instance, but this may not be true in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Recall the definition of “(α, β)-stable” in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We know that there always exists a value ˆα ∈ [0, 1) such that P is a (ˆα, δ/2)-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We can use “ˆα” to indicate the stability degree of P, for the fixed “δ/2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The following theorem shows that we can infer the value of ˆα through Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 24 Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If Algorithm 6 returns a (1 + ϵ)-radius approximation, then ˆα < ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' otherwise, the algorithm returns a (1 − δ)-covering approximation and it implies ˆα > ϵ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In other words, the algorithm can distinguish the case ˆα ≥ ϵ (it must returns a (1 − δ)- covering approximation) and the case ˆα ≤ ϵ2 2 (it must returns a (1 + ϵ)-radius approximation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' but if ϵ2 2 < ˆα < ϵ, the algorithm can return either a (1 − δ)-covering approximation or a (1 + ϵ)-radius approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Recall we set α = ϵ2 and β = δ/2 in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we suppose the output is a (1 + ϵ)-radius approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' One possible case is the instance P is a real (α, β)-stable instance, and then ˆα = α < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The other possible case is that P is not (α, β)-stable but the ratio r˜o rc ≤ 1+ϵ 1−ϵ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Together with (49), we have Rad(P) r−δ/2 ≤ r˜o 1 1+ϵ2/2rc ≤ (1 + ϵ)(1 + ϵ2/2) 1 − ϵ2/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (56) So ˆα = 1 − r−δ/2 Rad(P) ≤ 1 − 1−ϵ2/2 (1+ϵ)(1+ϵ2/2) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Overall, as long as the output is a (1 + ϵ)-radius approximation, ˆα should be smaller than ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we suppose the output is a (1 − δ)-covering approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' One possible case is the instance P is not (α, β)-stable, and then ˆα > α = ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The other possible case is that P is (α, β)-stable but the ratio r˜o rc > 1+ϵ 1−ϵ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Together with (51), we have Rad(P) r−δ/2 ≥ 1 1+ϵr˜o rc > 1 1 − ϵ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (57) So ˆα = 1 − r−δ/2 Rad(P) > 1 − (1 − ϵ2/2) = ϵ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Overall, as long as the output is a (1 − δ)-covering approximation, ˆα > min{ϵ2, ϵ2/2} = ϵ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ 6 Extension I: Hybrid Approximation for MEB with Outliers In this section, we extend the idea of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='3 to present a hybrid approximation algorithm for the MEB with outliers problem (P, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we extend Definition 3 of MEB to MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 6 ((α, β)-stable for MEB with Outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let 0 < α, β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given an instance (P, γ) of the MEB with outliers problem in Definition 4, (P, γ) is an (α, β)-stable instance if (1) Rad(P \\ Q) > (1 − α)Rad(Popt) for any Q ⊂ P with |Q| < � γ + β � n, and (2) there exists a Q′ ⊂ P with |Q′| = ⌈(β + γ)n⌉ having Rad(P \\ Q′) ≤ (1 − α)Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 6 directly implies the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If (P, γ) is an (α, β)-stable instance of the problem of MEB with outliers, the corresponding Popt is an (α, ˜β)-stable instance of MEB with ˜β ≥ β 1−γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that Definition 6 implicitly requires β < 1 − γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So it implies the lower bound β 1−γ of ˜β in Claim 2 cannot be larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To see the correctness of Claim 2, we can use contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose that there exists a subset P ′ ⊂ Popt such that |P ′| > (1 − β 1−γ )|Popt| = (1 − γ − β)n and Rad(P ′) ≤ (1 − α)Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, it is in contradiction to the fact that (P, γ) is an (α, β)-stable instance of MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To apply the idea of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='3, a significant challenge is that the set Popt is mixed with the outliers, and thus we cannot easily obtain a (1 + ϵ)-radius approximation as Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our starting point is still the sublinear time bi-criteria approximation algorithm proposed in 25 Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Specifically, given any two small parameters 0 < ϵ, δ < 1, the algorithm returns a set of candidate ball centers via the uniform-adaptive sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We use Ξ to denote this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' With constant probability, as least one candidate from Ξ, say s, satisfies the following inequality: ��B � s, (1 + ϵ) · Rad(Popt) � ∩ P �� ≥ � 1 − δ − γ � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (58) Namely, it is a “(1 + ϵ, 1 − δ)-approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To pick such a qualified candidate, it is possible to estimate the size of B � s, (1 + ϵ) · Rad(Popt) � ∩ P by using the uniform sampling based technique “sandwich lemma” (instead of reading the whole dataset P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is worth to note an implicit fact about Theorem 5 of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Actually, in the proof it showed that among the candidate set Ξ, there exists one solution s such that the ball B � s, (1 + ϵ) · Rad(Popt) � covers at least � 1 − δ − γ � n points from Popt (since the set T ⊂ Popt and the solution s is generated from T (see Lemma 9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So the solution s should satisfy ��B � s, (1 + ϵ) · Rad(Popt) � ∩ Popt �� ≥ � 1 − δ − γ � n, (59) which is stronger than (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' But the sandwich lemma may ignore such a stronger solution, since only selecting a solution satisfying (58) is already sufficient to guarantee a (1 + ϵ, 1 − δ)- approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We introduce the following new algorithm for MEB with outliers based on this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The hybrid approximation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let ϵ and δ be the two given parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we apply the method of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' But we do not directly input the couple (ϵ, δ) to the bi-criteria approximation algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' instead, we use ( 1 2(2 √ 2+ √ 3)2 ϵ2, δ) (we will explain why we have the coefficient “ 1 2(2 √ 2+ √ 3)2 ” in our analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' That is, we compute a set Ξ of candidate ball centers via the uniform-adaptive sampling of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2, and at least one center yields a (1 + 1 2(2 √ 2+ √ 3)2 ϵ2, 1 − δ)-approximation for the instance (P, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, for each candidate q ∈ Ξ, we define two values: rq = min � r > 0 | ��B(q, r) ∩ P �� ≥ (1 − γ)n � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (60) r′ q = min � r > 0 | ��B(q, r) ∩ P �� ≥ � 1 − δ − γ � n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (61) We can compute these two values for all the candidates of Ξ by scanning the input P in one pass (instead of using the sandwich lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We select the two points s1 = arg minq∈Ξ rq and s2 = arg minq∈Ξ r′ q (they may or may not be the same point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If the ratio rs1 r′s2 ≤ 1+ϵ 1−ϵ2/ � 2(2 √ 2+ √ 3)2�, return the ball B(s1, rs1) and say “it is a (1 + ϵ)-radius approximation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' else, return the ball B(s2, r′ s2) and say “it is a (1 − δ)-covering approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Algorithm 7 Hybrid Approximation for MEB with Outliers Input: An instance (P, γ) of MEB with outliers problem in Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' two parameters 0 < ϵ, δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1: Apply the uniform-adaptive sampling method of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 to obtain a set Ξ of candidate ball centers, where at least one center yields a (1 + 1 2(2 √ 2+ √ 3)2 ϵ2, 1 − δ)-approximation for the instance (P, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2: Read the whole input dataset P in one pass, and compute the values rq and r′ q as the formulas (60) and (61) for each q ∈ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 3: Let s1 = arg minq∈Ξ rq and s2 = arg minq∈Ξ r′ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 4: If the ratio rs1 r′s2 ≤ 1+ϵ 1−ϵ2/ � 2(2 √ 2+ √ 3)2�, return the ball B(s1, rs1) and say “it is a (1+ϵ)-radius approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 5: Else, return the ball B(s2, r′ s2) and say “it is a (1 − δ)-covering approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 26 Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' With constant success probability, Algorithm 7 returns either a (1 + ϵ)-radius approximation or a (1 − δ)-covering approximation, and the running time is O(g(ϵ, δ, γ) · nd), where g(ϵ, δ, γ) = O( 1 1−γ ( γ+δ δ )O(1/ϵ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The algorithm only needs uniform sampling and a single pass over the input data, and the space complexity in memory is O(g(ϵ, δ, γ) · d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, if the input data matrix (the n × d matrix representing the input P) has at most M ≪ nd non-zeros entries, the total running time will be O � g(ϵ, δ, γ) · (n + d + M) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to Theorem 7, we will see that when the algorithm returns a (1 − δ)- covering approximation, the returned radius is at most � 1 − Θ(ϵ2) � Rad(Popt) (see (63) and (64)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (of Theorem 9) We study the time and space complexities first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The method of Corollary 2 only needs uniform samplings, and Step 2 of Algorithm 7 is a single pass over the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The size of Ξ is g(ϵ, δ, γ) = O( 1 1−γ ( γ+δ δ )O(1/ϵ2)) based on Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Overall, the space complexity is O(g(ϵ, δ, γ)·d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' And the complexity for generating Ξ is O � |Ξ|·poly( 1 ϵ, 1 δ)d � which is sublinear in the input size nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is easy to see that the complexity of Step 2 dominates the whole complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, the total running time is O(g(ϵ, δ, γ) · nd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Furthermore, we consider the case that the input matrix is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to the proof of Theorem 7, we know that the complexity of Algorithm 7 is O � g(ϵ, δ, γ) · (n + d + M) � if the input data matrix has at most M ≪ nd non-zeros entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Now, we prove the solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We let α = 1 (2 √ 2+ √ 3)2 ϵ2 and β = (1 − γ)δ, and consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Case 1: the instance (P, γ) is (α, β)-stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', Popt is an (α, ˜β)-stable instance of MEB with ˜β ≥ δ, according to Claim 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Denote by o the optimal center of MEB(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We suppose one candidate ball center q0 of Ξ satisfies the formula (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As a consequence, from Theorem 2, we know that ||q0 − o|| ≤ (2 √ 2 + √ 3)√α · Rad(Popt) = ϵ · Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' That is, rs1 ≤ rq0 ≤ (1 + ϵ) · Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (62) If rs1 r′s2 > 1+ϵ 1−ϵ2/ � 2(2 √ 2+ √ 3)2�, together with (62), we have r′ s2 < � 1 − ϵ2/ � 2(2 √ 2 + √ 3)2�� Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (63) Then we can return the ball B(s2, r′ s2) and say “it is a (1 − δ)-covering approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' On the other hand, when rs1 r′s2 ≤ 1+ϵ 1−ϵ2/ � 2(2 √ 2+ √ 3)2�, from (62) we can return the ball B(s1, rs1) and say “it is a (1 + ϵ)-radius approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Case 2: (P, γ) is not an (α, β)-stable instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then it implies r′ s2 < (1 + 1 2(2 √ 2 + √ 3)2 ϵ2)(1 − 1 (2 √ 2 + √ 3)2 ϵ2) · Rad(Popt) < � 1 − ϵ2/ � 2(2 √ 2 + √ 3)2�� Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (64) If rs1 r′s2 ≤ 1+ϵ 1−ϵ2/ � 2(2 √ 2+ √ 3)2�, together with (64), it implies rs1 < (1 + ϵ) · Rad(Popt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (65) Then we can return the ball B(s1, rs1) and say “it is a (1 + ϵ)-radius approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' On the other hand, when rs1 r′s2 > 1+ϵ 1−ϵ2/ � 2(2 √ 2+ √ 3)2�, from (64) we can return the ball B(s2, r′ s2) and say “it is a (1 − δ)-covering approximation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Since the success probability of the method of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 is constant, the overall success probability of Algorithm 7 is constant as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ 27 We also have the following theorem for inferring the stability of the instance (P, γ), and the proof is almost identical to the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose (P, γ) is a (ˆα, (1 − γ)δ)-stable instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If Algorithm 7 returns a (1 + ϵ)-radius approximation, then ˆα < ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' otherwise, the algorithm returns a (1 − δ)-covering approximation and it implies ˆα > ϵ2 2(2 √ 2+ √ 3)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 7 Extension II: Bi-criteria Approximations for MEX With Outliers In this section, we extend Definition 4 for MEB with outliers and define a more general problem called minimum enclosing “x” (MEX) with Outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we show that the ideas of Lemma 7 and 8 can be generalized to deal with MEX with outliers problems, as long as the shape “x” satisfies several properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To describe a shape “x”, we need to clarify three basic concepts: center, size, and distance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let X be the set of specified shapes in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We require that each shape x ∈ X is uniquely determined by the following two components: “c(x)”, the center of x, and “s(x) ≥ 0”, the size of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For any two shapes x1, x2 ∈ X, x1 = x2 if and only if c(x1) = c(x2) and s(x1) = s(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, given a center o0 and a value l0 ≥ 0, we use x(o0, l0) to denote the shape x with c(x) = o0 and s(x) = l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For different shapes, we have different definitions for the center and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For example, if x is a ball, c(x) and s(x) should be the ball center and the radius respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' given o0 ∈ Rd and l0 ≥ 0, x(o0, l0) should be the ball B(o0, l0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As a more complicated example, consider the k-center clustering with outliers problem, which is to find k balls to cover the input point set excluding a certain number of outliers and minimize the maximum radius (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', we can assume that the k balls have the same radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For this problem, the shape “x” is a union of k balls in Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' the center c(x) is the set of the k ball centers and the size s(x) is the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For any point p ∈ Rd and any shape x ∈ X, we also need to define a distance function f(c(x), p) between the center c(x) and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For example, if x is a ball, f(c(x), p) is simply equal to ||p − c(x)||;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' if x is a union of k balls with the center c(x) = {c1, c2, · · · , ck}, the distance should be min1≤j≤k ||p − cj||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that the distance function is only for ranking the points to c(x), and not necessary to be non-negative (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='3, we define a distance function f(c(x), p) ≤ 0 for SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By using this distance function, we can define the set “Q” and the value “li” when generalizing Lemma 7 and 8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To guarantee their correctnesses, we also require X to satisfy the following three properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For any two shapes x1 ̸= x2 ∈ X, if c(x1) = c(x2), then s(x1) ≤ s(x2) ⇐⇒ x1 is covered by x2, (66) where “x1 is covered by x2” means “for any point p ∈ Rd, p ∈ x1 ⇒ p ∈ x2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given any shape x ∈ X and any point p0 ∈ x, the set {p | p ∈ Rd and f(c(x), p) ≤ f(c(x), p0)} ⊆ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (67) Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given any shape center o0 and any point p0 ∈ Rd, let r0 = min{r | r ≥ 0, p0 ∈ x(o0, r)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then p0 ∈ x(o0, r0) and p0 /∈ x(o0, r) for any r < r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (Note: usually the value r0 is just the distance from p0 to the shape center o0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' but for some cases, such as the SVM problem in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='3, the shape size and distance function have different meanings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Intuitively, Property 1 shows that s(x) defines an order of the shapes sharing the same center c(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Property 2 shows that the distance function f defines an order of the points to a 28 given shape center c(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Property 3 shows that a center o0 and a point p0 can define a shape just “touching” p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We can take X = {all d-dimensional balls} as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For any two concentric balls, the smaller one is always covered by the larger one (Property 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' if a point p0 is inside a ball x, any point p having the distance ||p − c(x)|| ≤ ||p0 − c(x)|| should be inside x too (Property 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' also, given a ball center o0 and a point p0, p0 ∈ B(o0, ||p0 − o0||) and p0 /∈ B(o0, r) for any r < ||p0 − o0|| (Property 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Now, we introduce the formal definitions of the MEX with outliers problem and its bi-criteria approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 7 (MEX with Outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose the shape set X satisfies Property 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a set P of n points in Rd and a small parameter γ ∈ (0, 1), the MEX with outliers problem is to find the smallest shape x ∈ X that covers (1 − γ)n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Namely, the task is to find a subset of P with size (1−γ)n such that its minimum enclosing shape of X is the smallest among all possible choices of the subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The obtained solution is denoted by MEX(P, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 8 (Bi-criteria Approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given an instance (P, γ) for MEX with out- liers and two small parameters 0 < ϵ, δ < 1, a (1+ϵ, 1−δ)-approximation of (P, γ) is a solution x ∈ X that covers at least � 1 − δ − γ � n points and has the size at most (1 + ϵ)s(xopt), where xopt is the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is easy to see that Definition 4 of MEB with outliers actually is a special case of Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to MEB with outliers, we still use Popt, where Popt ⊂ P and |Popt| = (1 − γ)n, to denote the subset covered by the optimal solution of MEX with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Now, we provide the generalized versions of Lemma 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to the core-set construction method in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1, we assume that there exists an iterative algorithm Γ to compute MEX (without outliers);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' actually, this is an important prerequisite to design the sub-linear time algorithms under our framework (we will discuss the iterative algorithms for the MEX with outliers problems including flat fitting, k-center clustering, and SVM, in the following subsections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In the i-th iteration of Γ, it maintains a shape center oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, let Q be the set of (δ + γ)n farthest points from P to oi with respect to the distance function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we need to define the value “li” by Q in the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' There exists a value li ≥ 0 satisfying P \\ x(oi, li) = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The points of P can be ranked based on their distances to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Without loss of generality, let P = {p1, p2, · · · , pn} with f(oi, p1) > f(oi, p2) > · · · > f(oi, pn) (for convenience, we assume that any two distances are not equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' if there is a tie, we can arbitrarily decide their order to oi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then the set Q = {pj | 1 ≤ j ≤ (δ +γ)n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, from Property 3, we know that each point pj ∈ P corresponds to a value rj that pj ∈ x(oi, rj) and pj /∈ x(oi, r) for any r < rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Denote by xj the shape x(oi, rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We select the point pj0 with j0 = (δ + γ)n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' From Property 2, we know that pj ∈ xj0 for any j ≥ j0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', (a) P \\ Q ⊆ xj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We also need to prove that pj /∈ xj0 for any j < j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Assume there exists some pj1 ∈ xj0 with j1 < j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we have rj1 < rj0 and thus pj0 /∈ xj1 (by Property 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By Property 2, pj0 /∈ xj1 implies f(oi, pj0) > f(oi, pj1), which is in contradiction to the fact f(oi, pj0) < f(oi, pj1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So we have (b) Q ∩ xj0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The above (a) and (b) imply that {P ∩ xj0, Q} is a partition of P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', (P ∩ xj0) ∪ Q = P and (P ∩ xj0) ∩ Q = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So we know P \\ xj0 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, we can set the value li = rj0 and then P \\ x(oi, li) = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Lemma 11 (Generalized Uniform-Adaptive Sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let η1 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we sample n′ = O( 1 δ log 1 η1 ) points independently and uniformly at random from P and let Q′ be the set of farthest 3 2(δ + γ)n′ points to oi from the sample, then, with probability at least 1 − η1, the 29 following holds ���Q′ ∩ � Popt ∩ Q ���� |Q′| ≥ δ 3(γ + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (68) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let A denote the set of sampled n′ points from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to (28), we have ���A ∩ � Popt ∩ Q ���� > 1 2δn′ and ���A ∩ Q ��� < 3 2(δ + γ)n′ (69) with probability 1 − η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to (29), we have A ∩ Q = {p ∈ A | f(oi, p) > f(oi, pj0)}, (70) where pj0 is the point selected in the proof of Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By using the same manner of Claim 3, we also can select a point pj′ 0 ∈ A with Q′ = {p ∈ A | f(oi, p) > f(oi, pj′ 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (71) Then, we can prove � A ∩ � Popt ∩ Q �� = � Q′ ∩ � Popt ∩ Q �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (72) by using the same idea of (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Hence, ���Q′ ∩ � Popt ∩ Q ���� |Q′| = ���A ∩ � Popt ∩ Q ���� |Q′| ≥ δ 3(γ + δ), (73) where the final inequality comes from the first inequality of (69) and the fact |Q′| = 3 2(δ + γ)n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ Lemma 12 (Generalized Sandwich Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let η2 ∈ (0, 1) and assume δ < γ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' li is the value from Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We sample n′′ = O � γ δ2 log 1 η2 � points independently and uniformly at random from P and let q be the � (1 + δ/γ)2γn′′ + 1 � th farthest one from the sampled points to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If ˜li = min{r | r ≥ 0, q ∈ x(oi, r)} (similar to the way defining “r0” in Property 3), then, with probability 1 − η2, the following holds ˜li ≤ li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (74) ���P \\ x(oi, ˜li) ��� ≤ (γ + 5δ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (75) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let B denote the set of sampled n′′ points from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By using the same manner of Claim 3, we know that there exists a value ˜l′ i > 0 satisfying ���P \\x(oi, ˜l′ i) ��� = (γ+δ)2 γ−δ γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to the proof of Lemma 8, we can prove that ˜li ∈ [˜l′ i, li].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Due to Property 1, we know that x(oi, ˜li) is “sandwiched” by the two shapes x(oi, ˜l′ i) and x(oi, li).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Further, since x(oi, ˜l′ i) is covered by x(oi, ˜li), we have ���P \\ x(oi, ˜li) ��� ≤ ���P \\ x(oi, ˜l′ i) ��� = (γ + δ)2 γ − δ γn = (γ + 5δ)n, (76) where the last equality comes from the assumption δ < γ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So (74) and (75) are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⊓⊔ By using Lemma 11 and Lemma 12, we study several applications in the following subsec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 k-Center Clustering with Outliers Let γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a set P of n points in Rd, the problem of k-center clustering with outliers is to find k balls to cover (1 − γ)n points, and the maximum radius of the balls is minimized (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', we can assume that the k balls have the same radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given an instance (P, γ), let {C1, · · · , Ck} be the k clusters forming Popt (the subset of P yielding the optimal solution), and ropt be the optimal radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' that is, each Cj is covered by an individual ball with radius ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2, we first introduce a linear time algorithm, and then show how to modify it to be sublinear time by using Lemma 11 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Linear time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our algorithm in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 can be generalized to be a linear time bi-criteria algorithm for the problem of k-center clustering with outliers, if k is assumed to be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Our idea is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Algorithm 4, we maintain a set T as the core-set of Popt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' here, we instead maintain k sets T1, T2, · · · , Tk as the core-sets of C1, C2, · · · , Ck, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consequently, each Tj for 1 ≤ j ≤ k has an approximate MEB center oj i in the i-th round of Step 3, and we let Oi = {o1 i , · · · , ok i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Initially, O0 and Tj for 1 ≤ j ≤ k are all empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' we randomly select a point p ∈ P, and with probability 1 − γ, p ∈ Popt (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', we assume p ∈ C1 and add it to T1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' thus O1 = {p} after this step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We let Q be the set of farthest t = (δ + γ)n points to Oi, and li be the (t + 1)-th largest distance from P to Oi (the distance from a point p ∈ P to Oi is min1≤j≤k ||p − oj i||).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, we randomly select a point q ∈ Q, and with probability δ γ+δ, q ∈ Popt (as (46) in Lemma 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For ease of presentation, we assume that q ∈ Popt happens and we have an “oracle” to guess which optimal cluster q belongs to, say q ∈ Cjq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' then, we add q to Tjq and update the approximate MEB center of Tjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Since each optimal cluster Cj for 1 ≤ j ≤ k has the core-set with size 2 ϵ + 1 (by setting s = ϵ 2+ϵ in Theorem 1), after adding at most k( 2 ϵ + 1) points, the distance li will be smaller than (1 + ϵ)ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consequently, a (1 + ϵ, 1 − δ)-approximation solution is obtained when i ≥ k( 2 ϵ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that some “small” clusters could be missing from the above random sampling based approach and therefore |Oi| could be less than k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' however, it always can be guaranteed that the total number of missing inliers is at most δn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', a (1 + ϵ, 1 − δ)-approximation is always guaranteed (otherwise, the ratio |Popt∩Q| |Q| > δ γ+δ and we can continue to sample a point from Popt and then update Oi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To remove the oracle for guessing the cluster containing q, we can enumerate all the possible k cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' since we add k( 2 ϵ + 1) points to T1, T2, · · · , Tk, it generates kk( 2 ϵ +1) = 2k log k( 2 ϵ +1) solutions in total, and at least one yields a (1 + ϵ, 1 − δ)-approximation with probability (1 − γ)( δ γ+δ)k( 2 ϵ +1) (by the same manner for proving Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let (P, γ) be an instance of k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given two pa- rameters ϵ, δ ∈ (0, 1), there exists an algorithm that outputs a (1 + ϵ, 1 − δ)-approximation with probability (1 − γ)( δ γ+δ)k( 2 ϵ +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The running time is O(2k log k( 2 ϵ +1)(n + 1 ϵ5 )d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If one repeatedly runs the algorithm O( 1 1−γ ( γ+δ δ )k( 2 ϵ +1)) times, with constant probability, the algorithm outputs a (1 + ϵ, 1 − δ)-approximation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to our discussion on the running time for MEB with outliers in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1, B˘adoiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' [22] also achieved a linear time bi-criteria approximation for the k-center clustering with outliers problem (see Section 4 in their paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, the hidden constant of their running time is exponential in ( k ϵδ)O(1) that is much larger than “k log k( 2 ϵ + 1)” in Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Sublinear time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The linear time algorithm can be further improved to be sublinear time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' the idea is similar to that for designing sublinear time algorithm for MEB with outliers in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we follow Definition 7 and define the shape set X, where each x ∈ X is union of k balls in the space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' the center c(x) should be the set of its k ball centers, say c(x) = {o1 x, o2 x, · · · , ok x}, and the size s(x) is the radius, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', x = ∪k j=1B(oj x, s(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Obviously, if 31 x is a feasible solution for the instance (P, γ), the size ���P ∩ (∪k j=1B(oj x, s(x))) ��� should be at least (1 − γ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, define the distance function f(c(x), p) = min1≤j≤k ||p − oj x||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is easy to verify that the shape set X satisfies Property 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' From Lemma 11, we know that it is possible to obtain a point in Popt ∩ Q with probability (1 − η1) δ 3(γ+δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Further, we can estimate the value li and select the best candidate solution based on Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Overall, we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let (P, γ) be an instance of k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given the parame- ters ϵ, δ, η1, η2 ∈ (0, 1), there exists an algorithm that outputs a (1+ϵ, 1−δ)-approximation with probability (1 − γ) � (1 − η1)(1 − η2) δ 3(γ+δ) �k( 2 ϵ +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The running time is ˜O(2k log k( 2 ϵ +1)( γ δ2 + 1 ϵ5 )d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If one repeatedly runs the algorithm N = O � 1 1−γ � 1 1−η1 ( 3(γ+δ) δ ) �k( 2 ϵ +1)� times with set- ting η2 = O( 1 2k log k( 2 ϵ +1)N ), with constant probability, the algorithm outputs a (1 + ϵ, 1 − δ)- approximation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 Flat Fitting with Outliers Let j be a fixed integer between 0 and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a j-dimensional flat F and a point p ∈ Rd, we define their distance, dist(F, p), to be the Euclidean distance from p to its projection onto F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let P be a set of n points in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The problem of flat fitting is to find the j-dimensional flat F that minimizes maxp∈P dist(F, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is easy to see that the MEB problem is the case j = 0 of the flat fitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Furthermore, given a parameter γ ∈ (0, 1), the flat fitting with outliers problem is to find a subset P ′ ⊂ P with size (1 − γ)n such that maxp∈P ′ dist(F, p) is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to MEB with outliers, we also use Popt to denote the optimal subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Before presenting our algorithms for flat fitting with outliers, we first introduce the linear time algorithm from Har-Peled and Varadarajan [56] for the vanilla version (without outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We start from the case j = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the flat F is a line in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Roughly speaking, their algorithm is an iterative procedure to update the solution round by round, until it is close enough to the optimal line lopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' There are two parts in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (1) It picks an arbitrary point p∆ ∈ P and let q∆ be the farthest point of P from p∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' it can be proved that the line passing through p∆ and q∆, denoted as l0, is a good initial solution that yields a 4-approximation with respect to the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (2) In each of the following rounds, the algorithm updates the solution from li−1 to li where i ≥ 1 is the current number of rounds: let pi be the farthest point of P from li−1 and let hi denote the 2-dimensional flat spanned by pi and li−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' then the algorithm computes a set of O( 1 ϵ8 log2 1 ϵ) lines on hi, and picks one of them as li via an “oracle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' They proved that the improvement from li−1 to li is significant enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' thus, after running ν = O( 1 ϵ3 log 1 ϵ) rounds, it is able to achieve a (1 + ϵ)-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To remove the “oracle”, the algorithm can enumerate all the O( 1 ϵ8 log2 1 ϵ) lines on hi, and thus the total running time is O � 2 1 ϵ3 log2 1 ϵ nd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Linear time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Now we consider to adapt the above algorithm to the case with outliers, where in fact the idea is similar to the idea proposed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 for MEB with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For simplicity, we still use the same notations as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consider the part (1) first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If we randomly pick a point p∆ from P, with probability 1 − γ, it belongs to Popt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' further, we randomly pick a point, denoted as q∆, from the set of (δ0 + γ)n farthest points of P from p∆, where the value of δ0 will be determined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Obviously, with probability δ0 γ+δ0 , q∆ ∈ Popt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Denote by P0 = {p ∈ Popt | ||p − p∆|| ≤ ||q∆ − p∆||}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Denote by l0 the line passing through p∆ and q∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, with probability (1 − γ)( δ0 γ+δ0 ), max p∈P0 dist(l0, p) ≤ 4 max p∈P0 dist(lopt, p) ≤ 4 max p∈Popt dist(lopt, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (77) 32 Also, the size of P0 is at least � 1 − (δ0 + γ) � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is straightforward to obtain the size of P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The inequality (77) directly comes from the aforementioned result of [56], as long as p∆ and q∆ ∈ Popt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So we can use the line l0 as our initial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, we can apply the same random sampling idea to select the point pi in the i-th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Namely, we randomly pick a point as pi from the set of (δ0+γ)n farthest points of P from li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, we need to shrink the set Pi−1 to Pi = {p ∈ Pi−1 | dist(li−1, p) ≤ dist(li−1, pi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to Lemma 13, we can show that the improvement from li−1 to li is significant enough with probability (1−γ)( δ0 γ+δ0 )i+1, and the size of Pi is at least � 1−((i+1)δ0+γ) � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' After running ν rounds, we obtain the line lν such that maxp∈Pν dist(lν, p) ≤ (1 + ϵ) maxp∈Popt dist(lopt, p), and |Pν| ≥ � 1 − ((ν + 1)δ0 + γ) � n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So if we set δ0 = δ ν+1 with a given δ ∈ (0, 1), the line lν will be a bi-criteria (1 + ϵ, 1 − δ)-approximation of the instance (P, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By using the idea in [56], we can extend the result to the case j > 1 with ν = eO(j2) ϵ2j+1 log 1 ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We refer the reader to [56] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let (P, γ) be an instance of j-dimensional flat fitting with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given two parameters ϵ, δ ∈ (0, 1), there exists an algorithm that outputs a (1+ϵ, 1−δ)-approximation with probability (1 − γ) � 1 2 �g(j,ϵ) where g(j, ϵ) = poly(eO(j2), 1 ϵj ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The running time is O(2g′(j,ϵ)nd) where g′(j, ϵ) = poly(eO(j2), 1 ϵj ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If one repeatedly runs the algorithm 2g(j,ϵ) 1−γ times, with constant probability, the algorithm outputs a (1 + ϵ, 1 − δ)-approximation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Sublinear time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We can view the flat fitting with outliers problem as an MEX with outliers problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let r ≥ 0 and F be a j-dimensional flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then we can define a j-dimensional “slab” SL(F, r) = {p ∈ Rd | dist(F, p) ≤ r}, where its “center” and “size” are F and r respectively (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', a ball is a 0-dimensional slab);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' the distance function f(F, p) = dist(F, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is easy to see that the shape set of slabs satisfies Property 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Furthermore, finding the optimal flat is equivalent to finding the smallest slab covering (1 − γ)n points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, by using Lemma 11 and 12, we achieve the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let (P, γ) be an instance of j-dimensional flat fitting with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given the parameters ϵ, δ, η1, η2 ∈ (0, 1), there exists an algorithm that outputs a (1 + ϵ, 1 − δ)- approximation with probability (1−γ) � (1−η1)(1−η2) δ 3(γ+δ) �g(j,ϵ) where g(j, ϵ) = poly(eO(j2), 1 ϵj ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The running time is O(2g′(j,ϵ,δ,γ)d) where g′(j, ϵ) = poly(eO(j2), 1 ϵj , 1 δ, 1 γ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If one repeatedly runs the algorithm N = O � 1 1−γ � 1 1−η1 ( 3(γ+δ) δ ) �g(j,ϵ)� times with setting η2 = O( 1 2g(j,ϵ)N ), with constant probability, the algorithm outputs a (1 + ϵ, 1 − δ)-approximation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='3 One-class SVM with Outliers In practice, datasets often contain outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The separating margin of SVM could be considerably deteriorated by outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As mentioned in [40], most of existing techniques [88,93] for SVM outliers removal are numerical approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', adding some penalty item to the objective function), and only can guarantee local optimums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Ding and Xu [40] modeled SVM with outliers as a combinatorial optimization problem and provided an algorithm called “Random Gradient Descent Tree”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We focus on one-class SVM with outliers first, and explain the extension for two-class SVM with outliers in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Below is the definition of the one-class SVM with outliers problem proposed in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 9 (One-class SVM with Outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a set P of n points in Rd and a small parameter γ ∈ (0, 1), the one-class SVM with outliers problem is to find a subset P ′ ⊂ P 33 Algorithm 8 Gilbert Algorithm [40,49] Input: A point-set P in Rd, and N ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Output: vi as an approximate solution of the polytope distance between the origin and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Initialize i = 1 and v1 to be the closest point in P to the origin o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Iteratively perform the following steps until i = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (a) Find the point pi ∈ P whose orthogonal projection on the supporting line of segment ovi has the closest distance to o (called the projection distance of pi), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', pi = arg minp∈P { ⟨p,vi⟩ ||vi|| }, where ⟨p, vi⟩ is the inner product of p and vi (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (b) Let vi+1 be the point on segment vipi closest to the origin o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' update i = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' with size (1 − γ)n and a hyperplane H separating the origin o and P ′, such that the distance between o and H is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' o xi xi+1 pi pi |xi vi Vi+1 vi Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 6: An illustration of step 2 in Algorithm 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' pi |vi is the projection of pi on ovi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Linear time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We briefly overview the algorithm of [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' They also considered the “bi-criteria approximation” with two small parameters ϵ, δ ∈ (0, 1): a hyperplane H separates the origin o and a subset P ′ ⊂ P with size � 1 − δ − γ � n, where the distance between o and H is at least (1 − ϵ) of the optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The idea of [40] is based on the fact that the SVM (without outliers) problem is equivalent to the polytope distance problem in computational geometry [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let o be the origin and P be a given set of points in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The polytope distance problem is to find a point q inside the convex hull of P so that the distance ||q − o|| is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For an instance P of one-class SVM, it can be proved that the vector qopt − o, if qopt is the optimal solution for the polytope distance between o and P, is the normal vector of the optimal hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We refer the reader to [40,48] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The polytope distance problem can be efficiently solved by Gilbert Algorithm [46,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For completeness, we present it in Algorithm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Similar to the core-set construction method of MEB in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1, the algorithm also greedily improves the current solution by selecting some point pi in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let ρ be the polytope distance between o and P, D = maxp,q∈P ||p − q||, and E = D2 ρ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given ϵ ∈ (0, 1), it has been proved that a (1 − ϵ)-approximation of one-class SVM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', a separating margin with the width at least (1 − ϵ) of the optimum) can be achieved by running Algorithm 8 at most 2⌈2E/ϵ⌉ steps [29,48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To handle outliers, the algorithm of [40] follows the similar intuition of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' it replaces the step of greedily selecting the point pi by randomly sampling a point from a set Q, which contains the (δ + γ)n points having the smallest projection distances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', the values of the function ⟨p,vi⟩ ||vi|| in Step 2(a) of Algorithm 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To achieve a (1 − ϵ, 1 − δ)- approximation with constant success probability, the algorithm takes O � 1 1−γ (1 + γ δ )z D2 ϵρ2 nd � time, where z = O( D2 ϵρ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Sublinear time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We define X to be the set of all the closed half-spaces not covering the origin o in Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' for each x ∈ X, let Hx be the hyperplane enclosing x and let hx 34 o hx Hx Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 7: An illustration for Hx and hx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' be the projection of o on Hx (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We suppose that the given instance (P, γ) has feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' That is, there exists at least one half-space x ∈ X that the hyperplane Hx separates the origin o and a subset P ′ with size (1 − γ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We define the center c(x) = hx ||hx||;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' since the MEX with outlier problem in Definition 7 is a minimization problem, we design the size function s(x) = 1 ||hx||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Obviously, a (1 − ϵ)-approximation of the SVM with outliers problem is equivalent to a 1 1−ϵ-approximation with respect to the size function s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We also define the distance function f(c(x), p) = −⟨p, hx ||hx||⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' It is easy to verify that the shape set X satisfies Property 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Recall that Algorithm 8 selects the point pi = arg minp∈P {⟨p,vi⟩ ||vi|| } in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Actually, the vector vi ||vi|| can be viewed as a shape center and pi is the farthest point to vi ||vi|| based on the distance function f(c(x), p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Moreover, the set Q mentioned in the previous linear time algorithm actually is the set of the farthest (δ + γ)n points from P to vi ||vi||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consequently, we can apply Lemma 11 to sample a point from Popt ∩ Q, and apply Lemma 12 to estimate the value of li for each candidate solution vi ||vi||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Overall, we can improve the running time of the algorithm of [40] to be independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let (P, γ) be an instance of SVM with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given the parameters ϵ, δ, η1, η2 ∈ (0, 1), there exists an algorithm that outputs a (1 − ϵ, 1 − δ)-approximation with probability (1 − γ) � (1 − η1)(1 − η2) δ 3(γ+δ) �z where z = O( D2 ϵρ2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The running time is ˜O( D2γ δ2ϵ2ρ2 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If one repeatedly runs the algorithm N = O � 1 1−γ � 1 1−η1 (3 + 3γ δ ) �z� times with setting η2 = O( 1 zN ), with constant probability, the algorithm outputs a (1 − ϵ, 1 − δ)-approximation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='4 Two-class SVM with Outliers Below is the definition of the two-class SVM with outliers problem proposed in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Definition 10 (Two-class SVM with Outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given two point sets P1 and P2 in Rd and two small parameters γ1, γ2 ∈ (0, 1), the two-class SVM with outliers problem is to find two subsets P ′ 1 ⊂ P1 and P ′ 2 ⊂ P2 with |P ′ 1| = (1 − γ1)|P1| and |P ′ 2| = (1 − γ2)|P2|, and a margin separating P ′ 1 and P ′ 2, such that the width of the margin is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We use P opt 1 and P opt 2 , where |P opt 1 | = (1 − γ1)|P1| and |P opt 2 | = (1 − γ2)|P2|, to denote the subsets of P1 and P2 which are separated by the optimal margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The ordinary two-class SVM (without outliers) problem is equivalent to computing the polytope distance between the origin o and M(P1, P2), where M(P1, P2) is the Minkowski difference of P1 and P2 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that it is not necessary to compute the set M(P1, P2) explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Instead, Algorithm 8 only needs to select one point from M(P1, P2) in each iteration, and overall the running time is still linear in the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To deal with two-class SVM with outliers, Ding and Xu [40] slightly modified their algorithm for the case of one-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In each iteration, it considers two subsets Q1 ⊂ P1 and Q2 ⊂ P2, which respectively consist of points having the (δ + γ1)|P1| smallest projection distances among all points in P1 and the (δ + γ2)|P2| largest projection distances among all 35 points in P2 on the vector vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' then, the algorithm randomly selects two points p1 i ∈ Q1 and p2 i ∈ Q2, and their difference vector p2 i −p1 i will serve as the role of pi in Step 2(a) of Algorithm 8 to update the current solution vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' This approach yields a (1 − ϵ, 1 − δ)-approximation in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' s⊥ s⊥ H⊥ H⊥ H⊤ H⊤ ˜H⊥ ˜H⊥ ˜H⊤ ˜H⊤ s⊤ s⊤ o Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 8: An illustration for two-class SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The distances from o to H⊥ and H⊤ are s⊥ and s⊤, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' The hyperplanes ˜H⊥ and ˜H⊤ are the estimations of H⊥ and H⊤, and the distances from o to them are ˜s⊥ and ˜s⊤ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' To improve the algorithm to be sublinear, we need several modifications on our previous idea for the case of one-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' First, we change the distance function to be: f(p, c) = � −⟨p, hx ||hx||⟩ if p ∈ P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ⟨p, hx ||hx||⟩ if p ∈ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By using this new distance function, we can apply Lemma 11 to obtain the points p1 i ∈ Q1∩P opt 1 and p2 i ∈ Q2 ∩ P opt 2 separately in sublinear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Given a vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=', candidate center) vi ||vi||, assume H⊥ and H⊤ are the parallel hyperplanes orthogonal to vi ||vi|| that the margin formed by them separates P ′ 1 and P ′ 2, where P ′ 1 ⊂ P1 and P ′ 2 ⊂ P2 with |P ′ 1| = (1 − γ1)|P1| and |P ′ 2| = (1 − γ2)|P2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Without loss of generality, we assume that the origin o is inside the margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose that the distances from o to H⊥ and H⊤ are s⊥ and s⊤, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, we obtain two shapes (closed half-spaces) x⊥ = (− vi ||vi||, 1 s⊥ ) and x⊤ = ( vi ||vi||, 1 s⊤ ) with P ′ 1 ⊂ x⊥ and P ′ 2 ⊂ x⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Consequently, we can apply Lemma 12 twice to obtain two values 1 ˜s⊥ ≤ 1 s⊥ and 1 ˜s⊤ ≤ 1 s⊤ with ���P1 \\ x(− vi ||vi||, 1 ˜s⊥ ) ��� ≤ (O(δ) + γ1)|P1| and ���P2 \\ x( vi ||vi||, 1 ˜s⊤ ) ��� ≤ (O(δ) + γ2)|P2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Therefore, we can use the value ˜s⊥ + ˜s⊤ as an estimation of s⊥ + s⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' See Figure 8 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Overall, we can achieve a (1 − ϵ, 1 − O(δ))-approximation in sublinear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 8 Future Work Following our work, several interesting problems deserve to be studied in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' For example, different from radius approximation, the current research on covering approximation of MEB is still inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In particular, can we provide a lower bound for the complexity of computing covering approximate MEB, as the lower bound result for radius approximate MEB proved by [30]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, is it possible to extend the stability notion to other geometric optimization problems with more complicated structures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Section 7, we only provide the bi-criteria approximations for the MEX with outliers problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' So it is interesting to consider to extend the stability notion to these geometric optimization problems, and then we can design the hybrid approximation algorithms for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 36 9 Acknowledgements The research of this work was supported in part by National Key R&D program of China through grant 2021YFA1000900 and the Provincial NSF of Anhui through grant 2208085MF163.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In Proceedings of the Canadian Conference on Computational Geometry (CCCG), pages 83–86, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' A Proof of Theorem 1 To ensure the expected improvement in each iteration of the algorithm of [20], they showed that the following two inequalities hold if the algorithm always selects the farthest point to the current center of MEB(T): ri+1 ≥ (1 + ϵ)Rad(P) − Li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' ri+1 ≥ � r2 i + L2 i , (78) where ri and ri+1 are the radii of MEB(T) in the i-th and (i + 1)-th iterations, respectively, and Li is the shifting distance of the center of MEB(T) from the i-th to (i + 1)-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' However, we often compute only an approximate MEB(T) in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In the i-th iteration, we let ci and oi denote the centers of the exact and the approximate MEB(T), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Suppose that ||ci − oi|| ≤ ξri, where ξ ∈ (0, ϵ 1+ϵ) (we will see why this bound is needed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that we only compute oi rather than ci in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As a consequence, we can only select the farthest point (say q) to oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' If ||q − oi|| ≤ (1 + ϵ)Rad(P), we are done and a (1 + ϵ)-radius approximation of MEB is already obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Otherwise, we have (1 + ϵ)Rad(P) < ||q − oi|| ≤ ||q − ci+1|| + ||ci+1 − ci|| + ||ci − oi|| ≤ ri+1 + Li + ξri (79) by the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In other words, we should replace the first inequality of (78) by ri+1 > (1 + ϵ)Rad(P) − Li − ξri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Also, the second inequality of (78) still holds since it depends only on the property of the exact MEB (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='1 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Thus, we have ri+1 ≥ max �� r2 i + L2 i , (1 + ϵ)Rad(P) − Li − ξri � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (80) 40 Similar to the analysis in [20], we let λi = ri (1+ϵ)Rad(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Because ri is the radius of MEB(T) and T ⊂ P, we know ri ≤ Rad(P) and then λi ≤ 1/(1 + ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' By simple calculation, we know that when Li = � (1+ϵ)Rad(P)−ξri �2 −r2 i 2 � (1+ϵ)Rad(P)−ξri � the lower bound of ri+1 in (80) achieves the minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Plugging this value of Li into (80), we have λ2 i+1 ≥ λ2 i + � (1 − ξλi)2 − λ2 i �2 4(1 − ξλi)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (81) To simplify inequality (81), we consider the function g(x) = (1−x)2−λ2 i 1−x , where 0 < x < ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Its derivative g′(x) = −1 − λ2 i (1−x)2 is always negative, thus we have g(x) ≥ g(ξ) = (1 − ξ)2 − λ2 i 1 − ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (82) Because ξ < ϵ 1+ϵ and λi ≤ 1/(1 + ϵ), we know that the right-hand side of (82) is always non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Using (82), inequality (81) can be simplified to λ2 i+1 ≥ λ2 i + 1 4 � g(ξ) �2 = λ2 i + � (1 − ξ)2 − λ2 i �2 4(1 − ξ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (83) (83) can be further rewritten as � λi+1 1 − ξ �2 ≥ 1 4 � 1 + ( λi 1 − ξ )2�2 =⇒ λi+1 1 − ξ ≥ 1 2 � 1 + ( λi 1 − ξ )2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (84) Now, we can apply a similar transformation of λi which was used in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let γi = 1 1− λi 1−ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' We know γi > 1 (note 0 ≤ λi ≤ 1 1+ϵ and ξ < ϵ 1+ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Then, (84) implies that γi+1 ≥ γi 1 − 1 2γi = γi � 1 + 1 2γi + ( 1 2γi )2 + · · · � > γi + 1 2, (85) where the equation comes from the fact that γi > 1 and thus 1 2γi ∈ (0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Note that λ0 = 0 and thus γ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' As a consequence, we have γi > 1 + i 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' In addition, since λi ≤ 1 1+ϵ, that is, γi ≤ 1 1− 1 (1+ϵ)(1−ξ) , we have i < 2 ϵ − ξ − ϵξ = 2 (1 − 1+ϵ ϵ ξ)ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' (86) Consequently, we obtain the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' B Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content='2 in [22] Lemma 14 ( [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' Let B(c, r) be a minimum enclosing ball of a point set P ⊂ Rd, then any closed half-space that contains c, must also contain at least a point from P that is at distance r from c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} +page_content=' 41' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dE1T4oBgHgl3EQfDAIA/content/2301.02870v1.pdf'} diff --git a/_dFJT4oBgHgl3EQfqSzu/content/tmp_files/2301.11604v1.pdf.txt b/_dFJT4oBgHgl3EQfqSzu/content/tmp_files/2301.11604v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1e745b8ae29d9c2c0fe008f3a67a65e7b3c78a4 --- /dev/null +++ b/_dFJT4oBgHgl3EQfqSzu/content/tmp_files/2301.11604v1.pdf.txt @@ -0,0 +1,1426 @@ +A critical look at deep neural network for dynamic system modeling +Jinming Zhoua, Yucai Zhua,∗ +aState Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China +Abstract +Neural network models become increasingly popular as dynamic modeling tools in the control community. They have many +appealing features including nonlinear structures, being able to approximate any functions. While most researchers hold optimistic +attitudes towards such models, this paper questions the capability of (deep) neural networks for the modeling of dynamic systems +using input-output data. For the identification of linear time-invariant (LTI) dynamic systems, two representative neural network +models, Long Short-Term Memory (LSTM) and Cascade Foward Neural Network (CFNN) are compared to the standard Prediction +Error Method (PEM) of system identification. In the comparison, four essential aspects of system identification are considered, +then several possible defects and neglected issues of neural network based modeling are pointed out. Detailed simulation studies +are performed to verify these defects: for the LTI system, both LSTM and CFNN fail to deliver consistent models even in noise-free +cases; and they give worse results than PEM in noisy cases. +Keywords: Deep neural network, Model structure, Error criteria, Consistency, Model validation, System identification. +1. Introduction +Process modeling is fundamental to most process applica- +tions, from control, optimization to fault diagnosis and soft sen- +sor. An accurate model that well reflects the process behavior is +essential for success of all these applications. Neural network +and deep learning models have become increasingly popular in +the control community, inspired by their tremendous success +in, e.g., Computer Vision (CV), Natural Language Processing +(NLP) [1, 2]. Unlike system identification theory that starts +from the well-established theory for linear systems [3, 4], neu- +ral networks have inherently nonlinear structures. Moreover, +they are so-called universal approximators capable of approxi- +mating any functions within any degree of accuracy [5, 6]. Nu- +merous papers have been published for the development of neu- +ral network based methods for various applications in systems +and control [7, 8, 9]. +The input and output spaces of a neural network are very gen- +eral: the input could be signals, images, etc., while the output +could be numerical values or classifications. Among the many +possibilities, this paper only focuses on the case where the neu- +ral network is used to learn the dynamic relations between pro- +cess inputs and outputs. Such model can be used, e.g., as the +internal model in model predictive control (MPC), for residual +generation in FDI, or as a soft sensor. +According to some state-of-the-art review papers [7, 8, 9, +10, 11], three representative network structures are Feedfoward +Neural Network (FNN), Recurrent Neural Network (RNN), and +Convolution Neural Network (CNN). FNN is a classical struc- +ture and its use in process industry can be dated back to 20 years +∗Corresponding author +Email addresses: zhoujinming@zju.edu.cn (Jinming Zhou), +zhuyucai@zju.edu.cn (Yucai Zhu) +ago [12, 13]. It still receives great research interests nowadays +in nonlinear modeling and MPC applications [10, 14, 15, 16]. +For dynamic modeling, lagged input and output signals should +be fed into FNN to manually create dynamics[14], while RNN +has naturally a dynamic structure which can be written in a non- +linear state-space form [1], thus it is very popular in process +control research where dynamic modeling is desirable. As re- +ported [1], most successful applications of RNN use the LSTM +structure, which can successfully handle the gradient vanish- +ing/exploding problem of original RNN. LSTM has already +been extensively used in FDI [17], MPC [15], soft sensor [18]. +AutoEncoder (AE) is another popular RNN structure whose ap- +plications are mainly in FDI and soft sensor. CNN is most well- +known for its success in computer vision, it can also be used in +local dynamic modeling and frequency domain modeling [18]. +Great efforts have been put into research on neural networks +and deep learning methods with the hope that they can pro- +mote developments of process control industry. However, un- +like in the fields of CV and NLP, very few successful appli- +cations based on neural networks are reported in process in- +dustries. What can the user really benefits from a neural net- +work based modeling, compared to linear system identification +and simple nonlinear models? Most papers investigating neu- +ral networks tend to compare only the performance between +several network structures and ignore this important question. +Only recently, such comparison studies are carried out by sev- +eral researchers. In [19], based on a 660MW boiler dataset, the +model quality of LSTM is compared to those of simple (lin- +ear) statistical models and the results show that LSTM gives +the worst performance. Based on the Silverbox dataset, in [14], +LSTM gives worse result than FNN; in [20], the authors find +that some network structures can act as noise amplifiers that +deteriorate the model quality. The two papers reveal a common +Preprint +January 30, 2023 +arXiv:2301.11604v1 [cs.LG] 27 Jan 2023 + +phenomenon: a complex model structure mathematically capa- +ble of learning arbitrary systems well can fail in practice, even +if the real system is not complex at all. +This paper further investigates the deficiencies of neural net- +work based modeling. Besides LSTM that has been criticized +in [19, 14], it will be shown that the CFNN suggested in [14, 10] +also has problems that may hinder its use in modeling dynamic +systems. Instead of considering modeling of nonlinear system +as in [14, 10], this paper returns to the most fundamental prob- +lem: modeling of LTI systems. If a method cannot model such +system well, it certainly cannot handle more complex nonlin- +ear systems. It will be shown that, although a CFNN contains +both linear part and nonlinear part, it fails to identify an LTI +system without noise with consistency. Increasing the number +of hidden units or hidden layers do not improve the situation. +These findings will be discussed and explained through four es- +sential aspects of system identification, model structure, error +criterion, estimation properties and model validation. Simula- +tion studies will also be presented to support the declarations. +The rest of the paper unfolds as follows. +Section 2 +gives background knowledge about linear system identification, +LSTM and CFNN; Section 3 discusses and compares the three +models from a system identification perspective, points out po- +tential problems for neural network based modeling; Section +4 contains detailed simulation studies of a LTI system and a +Hammerstein system; Section 5 gives conclusions. +Notations +q denotes the forward time shifter. σ(·) denotes activation +function (vector) in neural networks, such as sigmoid function, +Rectified Linear Unit (ReLU) function. +Details about these +functions can be found in [1]. ∗ is the Hadamard (element- +wise) product. N+ denotes positive natural number. Φz denotes +power spectrum of signal {z(t)}. Var[·] is the mathematical vari- +ance operator. ‘With probability α’ is abbreviated as w.p. α. +Euclidean norm and Frobenius norm are denoted as ∥ · ∥ and +∥ · ∥F. +2. Background +Consider a general Single-Input Single-Output (SISO) LTI +system [3]: +S : y(t) = G0(q)u(t) +�������������� +:=y0(t) ++ H0(q)e0(t) +���������������� +:=v0(t) +, +(1) +where u(t) and y(t) denote system input and output signals, +v(t) denotes the disturbance. G0(q) denotes the system trans- +fer function, +G0(q) = B0(q) +A0(q) = +b0 +0 + · · · + bnb +0 q−nb +1 + a1 +0q−1 + · · · + ana +0 q−na . +(2) +which is assumed stable. For simplicity assume na = nb = +n. e0(t) is a zero-mean white noise sequence with variance λ2 +0. +H0(q) is assumed to be stable and inversely stable and monic. +2.1. The prediction error method +In system identification, a parametric model set is used to +describe the true system S: +M(θ) : y(t) = G(q, θ)u(t) + H(q, θ)e(t). +(3) +In PEM, quadratic cost function of the one-step Prediction Error +(PE) is minimized: +ˆθN = arg min +θ +N +� +t=1 +� +y(t) − ˆypem(t; θ) +� +(4a) +ˆypem(t; θ) = +� +1 − H−1(q; θ) +� +y(t) + H−1(q; θ)G(q; θ)u(t). +(4b) +Notice that subsequently when talking about error criterion for +parameter estimation, all PE refers to one-step PE. If S ∈ M, +which means that the model structure M is flexible enough, +there exists θ0 such that G(q, θ0) = G0(q) and H(q, θ0) = H0(q). +Moreover, if M(θ) is globally identifiable at θ0 and the input +signal is persistently exciting, the PEM estimate is consistent: +ˆθN → θ0, w.p. 1 as N → ∞. +(5) +A more precise description of the conditions required for this +property can be found in Chapter 8 of [3]. The consistency +property implies that the estimate will approach the parameter +vector representing the true system, when large amount of data +are available. If e(t) in (1) is Guassian, the PEM estimate can +be further proved to have minimum variance [3, 4]. +2.2. Long short-term memory network +The below contents show the mathematical formulation of a +single-layer LSTM network. The cores of LSTM are four gates +and two states controlling the transfer of the information flow. +The four gates are defined of the form +∗(t) = σ (W∗uu(t) + W∗hh(t − 1) + b∗) +(6) +where ∗ can be i, f, g and o, corresponding to input, forget, cell +and output gates respectively. Suppose that the weighting ma- +trices and bias vectors above are of compact dimensions. Based +on these gates, the cell state c(t) and hidden state h(t) are up- +dated according to +c(t) = f(t) ∗ c(t − 1) + i(t) ∗ g(t) +(7a) +h(t) = o(t) ∗ tanh (c(t)) . +(7b) +Finally, the output of LSTM is +ˆylstm(t) = Wyhh(t) + by. +(8) +In above formulations, (6–7) constitute a LSTM layer, while +(8) is often referred as a linear fully connected layer. Introduce +also a vector ρ that contains all the parameters, i.e., W· and b· in +(6–8), which can be optimized according to +ˆρN = arg min +ρ +N +� +t=1 +(y(t) − ˆylstm(t; ρ))2 . +(9) +Notice that ˆylstm(t) is calculated according to (6–8). To obtain a +deep network structure, one can simply connect several LSTM +layer in series. +2 + +Identification +experiment +Choose +model sturcture +Choose +error criterion +Estimate +model +Model +validation +Priori knowledge, modeling purpose,... +PE? OE? +Flexible? +Compact? +Pass? +Yes: use +No: revise +Informative? +Economic? +Data +Estimation properties +Test what? +OE? K-step PE? +Figure 1: Typical identification procedure. +2.3. Cascade forward neural network +CFNN here refers to a specific FNN suggested in [14] for +nonlinear identification. It is characterized by that outputs of +all previous units are used as inputs in the next layer. It is al- +ready available in Matlab’s System Identification Toolbox, as +an idnlarx object. As has been mentioned in Section 1, lagged +input and output signals should be used to introduce dynamics. +The input of CFNN is the regressor +ϕ⊤(t) = �y(t − 1), · · · , y(t − n), u(t), · · · , u(t − n)� . +(10) +The output of a single layer CFNN is +ˆycfnn(t) = Wyuϕ(t) + by +���������������������� +Linear ++ Wyhσ (Whuϕ(t) + bh) +�������������������������������������������� +Nonlinear +. +(11) +Collect all parameters of CFNN into a vector ϑ, it can be opti- +mized according to +ˆϑN = arg min +ϑ +N +� +t=1 +(y(t) − ˆycfnn(t; ϑ))2 . +(12) +As shown in (11), CFNN can be divided into linear and non- +linear parts where the nonlinear part is due to the hidden layer. +Similar to LSTM, one can use multiple hidden layers to achieve +a deep structure. +3. Neural network based dynamic modeling from an iden- +tification perspective +Shown in Fig. 1 is a typical identification procedure [3, 4, +21] covering essential steps and key points of a general black- +box dynamic modeling. This section discusses the three models +introduced in Section 2 then points out problems that are often +neglected. The experiment design and parameter optimization +problems will not be discussed, only the steps with black dotted +line in Fig. 1 are focused. +3.1. Model structure +Among the three methods listed in Section 2, the structure of +PEM is exactly the same as the true system S. By introducing a +new state vector x⊤(t) = [c⊤(t), h⊤(t)], LSTM model (6–8) can +be summarized to a NonLinear State-Space (NLSS) form [1]: +x(t) = F (x(t − 1), u(t); ρ) +(13a) +ˆylstm(t; ρ) = H (x(t); ρ) . +(13b) +Notice that it is assumed that no time delay exists in input and +output in (13). According to [14, 10], CFNN belongs to the +Nonlinear ARX (NARX) model: +ˆycfnn(t; ϑ) = G (ϕ(t); ϑ) . +(14) +In (13–14) F , H, G are user-defined (nonlinear or linear) func- +tions. +As has been mentioned in Section 2, CFNN model contains +a linear part. When Wyh, Whu, by and bh all vanish and if +Wyu = [−a1 +0, · · · , −ana +0 , b1 +0, · · · , bnb +0 ], +(15) +CFNN will present exactly the same behavior as G0(q). That is +to say, CFNN covers the LTI model structure as its special case +and is theoretically capable of modeling such system, which +resembles Volterra model [22], block-oriented model [23], lin- +ear parameter-varying model [24], etc. In contrast, it is non- +trivial how LSTM network can be connected or reduced to a +common LTI model. In [19], after a careful design of the hy- +perparamters LSTM still gives worst performance among other +statistical models. The authors attribute this phenomenon to the +effects of gate functions in (6), disabling some of these gates +can improve the performance of LSTM. Simulation results in +[14] are similar. +The above discussions and the results in [14, 19] reveal that, +for process dynamic modeling, models that cannot be inter- +preted from process dynamics are undesirable even if they have +complex structures with strong approximation abilities. +3.2. Error criteria +Error criteria are the basics for optimization of model param- +eters, they are application-oriented and are closely related to the +properties of the estimated model. Three commonly used error +criteria of the linear model set (M) are, Prediction Error (PE), +Output Error (OE, also called simulation error), Equation Error +(EE): +εpe(t; θ) = H−1(q; θ) (y(t) − G(q; θ)u(t)) +(16a) +εoe(t; θ) = y(t) − G(q; θ)u(t) +(16b) +εee(t; θ) = A(q; θ)y(t) − B(q; θ)u(t) +(16c) +where A(q) and B(q) are defined similarly to A0(q) and B0(q) +in (2). Notice that EE equals PE of an ARX model [3] hence +only PE and OE will be considered. The difference between the +predicted output (4b) and the simulated output G(q; θ)u(t) is that +predicted output uses measured output signals while simulated +3 + +0 +5 +10 +15 +20 +25 +System order: n +100 +101 +102 +103 +104 +105 +Number of parameters +BJ +LSTM: nh=n +LSTM: nh=3n +CFNN: nh=n +CFNN: nh=3n +Figure 2: Parameter numbers of different models under different system orders. +output only uses input signals. Bearing this point in mind, the +extensions in LSTM and CFNN are straightforward: because +no measured outputs are used in ˆylstm, the LSTM criterion (9) is +an output error criterion; the CFNN criterion (12) is a prediction +error (equation error) criterion. +While the PEM model set M has a noise model that can +be parameterized independently to process model (e.g., Box- +Jenkins (BJ) model [3]), LSTM and CFNN are not equipped +with such ability to handle disturbance. In [20], the authors +argue that the noise effects can even be amplified in CFNN cri- +terion (12) and suggests the use of output error criterion. How- +ever, how to optimize CFNN model under an output error crite- +rion is an unsolved problem. +Remark 1. Sometimes the input of LSTM is chosen to be a +regressor containing measured outputs like ϕ(t) in (10), which +is called teacher forcing in Chapter 10.2.1 in [1]. In this case, +the LSTM criterion (9) becomes a prediction error criterion. +3.3. Estimation properties +Analyzing the (statistical) properties of the estimated model +is of particular interest and of vital importance in all model- +ing technique. The total model error can be divided into struc- +ture error caused by deficiencies in model structure and random +model error caused by stochastic disturbance. For linear identi- +fication, one often investigates the consistencies and variances +of parameters, step responses or frequency responses. The two +points concerning neural network will be respectively discussed +below. +Consistency +For nonlinear model structures, it makes less sense to con- +sider in the parameter space. Instead, the consistency in step +response can be defined analogously to (5): +Definition 1. Denote {U(t)} as a step signal of amplitude AU. +Then for some general system S and an estimated model +ˆ +MN +under structure M and with N data samples, as well as their +outputs subject to {U(t)}: {Y0(t)} and { ˆYN(t)}, if +ˆYN(t) → Y0(t), w.p. 1 as N → ∞, ∀t ∈ N+, AU � 0, +(17) +the estimate of M is consistent in step response to S . +In this regard, the estimated model +ˆ +MN can accurately de- +scribe the input-output relations of the true system S if suf- +ficient data samples are collected. For LTI system, PEM can +deliver consistent estimate in parameters, step response and fre- +quency response (under some conditions, see Section 2.1), and +for step response one only needs to consider the case AU = 1. +It is necessary to give some discussions on the differences be- +tween the above consistency concept and the well-known uni- +versal approximation theorem [1, 5, 6]. In Chapter 6.4.1 of[1], +this theorem is summarized as: +An FNN with a linear output layer and at least one hid- +den layer with the commonly used activation function (sig- +moid, ReLU, etc.) can approximate any Borel measurable func- +tion from one-dimensional space to another with any desired +amount of error. +The theorem states that even a single-layer FNN (which is +nearly the simplest network) is theoretically capable of approx- +imating any functions arbitrarily well, which seems to imply +that neural network is a powerful modeling tool. But note that +this is only an existence theorem that gives neither the guaran- +tee of consistent estimates, nor the guideline for how to obtain a +consistent estimate using training data. This theorem may have +made control community overly optimistic about the modeling +capability of neural networks for dynamic systems. +Variance +Consider first the asymptotic variance expression of the fre- +quency response for linear system [25]: +Var +� ˆGn +N(ejω) +� +≈ n +N +Φv(ω)λ0 +Φu(ω)λ0 − |Φue0(ω)| +(18) +where ˆGn +N(ejω) denotes frequency response function of a nth- +order estimate based on N data samples. The expression holds +exactly for ARX model when N → ∞, n → ∞. It reveals +that the model variance is proportional to number of param- +eters and is inversely proportional to sample size. Although +there is no such conclusion available for nonlinear systems and +neural network models, it is generally acknowledged that com- +plex models containing many parameters are more vulnerable +to stochastic disturbances. +Consider modeling a nth order SISO LTI system like S, sup- +pose that the correct order is used for PEM (a BJ model is used) +and CFNN, all biases in LSTM and CFNN are set to zero, and +use nh hidden units for both LSTM and CFNN. Then the pa- +rameter numbers of the three models are +BJ: nθ = 4n +(19a) +LSTM: nρ = 4 +� +n2 +h + nh +� ++ nh +(19b) +CFNN: nϑ = nh (2n + 1) + 2n +(19c) +Notice that only single-layer LSTM and CFNN with fully con- +nected output layers are considered. +Fig. 2 shows the parameter numbers of different models with +varying system orders and hidden unit settings. Among three +model structures, LSTM has the most parameters and is 1–4 +orders of magnitude higher than BJ model; CFNN ranges in the +4 + +middle. For dynamic process modeling, the training data must +be informative enough to contain the main process behavior, +which can be guaranteed by adding persistently exciting test +signals to the plant [21]. This procedure certainly introduces +some slight disturbance to normal operations and is the main +cost of black-box process modeling. Heuristically, according +to (18), to achieve a same level variance, LSTM requires 1–4 +orders of magnitude more data than BJ model, which means a +huge increase in modeling cost. This gives another explanation +about the poor model quality in [14, 19] apart from the model +structure issue. +3.4. Model validation +Model validation is the final step before a model can be put +into use. It should be application-oriented. In MPC, the model +is used to give multi-step predictions. If the step size of a multi- +step prediction tends to infinity, it becomes simulated output, +see Chapter 3.2 of [3] for details. Recently, it is proved that +output error serves as a useful tool for FDI as well [26]. Hence +it is important to check simulation error because it reflects the +real gap between the model and the plant. However in many +recently published papers, the model quality is only validated +through (one-step) prediction error. In [27], the author illus- +trated that two model with close prediction errors can have huge +differences in simulation errors. +Typically, a neural network model with complex structure +can easily deliver a loss function of very small value. When the +loss corresponds to prediction error and the model gives accu- +rate one-step prediction, one can tell nothing about the model +quality from the input to the output. If possible, check simu- +lation error; or at least test the multi-step prediction error for +validation. +3.5. Summary +LSTM has a very different structure from a LTI system hence +it may have difficulties to model such a system. In contrast, +CFNN has a structure which can be interpreted as a linear model +extension. However, as discussed above, there is no guarantee +for its model consistency and it may have high variance. Fur- +ther, a small prediction error delivered by CFNN may not imply +a good input-output model. +4. Simulation study +While LSTM has been comprehensively tested in [14, 19], +this section studies CFNN in a linear ARX system and a Ham- +merstein system. Step responses under different input ampli- +tudes will be calculated and normalized according to the input +amplitudes for comparison. Mean values are removed in the +training data so the bias vectors in CFNN will be disabled. +Simulations performed in this section are all performed using +Matlab and System Identification Toolbox. Specifically, nlarx +is used to estimate CFNN; BJ is used to estimate a BJ model; +nlhw is used to estimate a Hammerstein system. For above +estimates obtained in System Identification Toolbox, sim and +predict are used to calculate model simulation and k−step +prediction. +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Samples +-60 +-40 +-20 +0 +20 +40 +60 +Real output +Simulated output +0 +200 +400 +600 +800 +-2 +0 +2 +4200 +4400 +4600 +4800 +5000 +35 +40 +45 +50 +55 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +-60 +-40 +-20 +0 +20 +40 +60 +Real output +Predicted output +Comparsion with +one-step prediction +Comparsion with +simulation +0.2*GBN(20) +1*GBN(10) +1*GBN(20) +1*GBN(40) +5*GBN(20) +Figure 3: Model validation based on one-step prediction and simulation. +Table 1: Noise-free LTI system: REs(%) of residuals in different data sections. +Residual +0.2*GBN(20) +1*GBN(10) +1*GBN(20) +1*GBN(40) +5*GBN(20) +1-step PE +9.99E-07 +8.38E-08 +2.80E-08 +1.61E-08 +1.14E-09 +20-step OE +1.81E-02 +1.12E-03 +4.14E-04 +3.05E-04 +2.17E-04 +Simulation +5.73E+00 +1.20E-02 +3.56E-03 +3.56E-03 +5.12E-03 +4.1. Linear system +Consider the following ARX system: +y(t) = B0(q) +A0(q)u(t) + α +1 +A0(q)e0(t) +B0(q) = 0.0115 + 0.00639q−1 +A0(q) = 1 − 1.963q−1 + 0.965q−2 +(20) +where α is used to control the Noise-to-Signal Ratio (NSR), +defined as Var +� +v2 +0(t) +� +/Var +� +y2 +0(t) +� +, see (1) for definitions of y0(t) +and v0(t). +4.1.1. Noise-free system +The system without noise is considered. The training data +are generated with a Generalized Binary Noise (GBN) [28] as +input. Its average switching time is 20 samples and amplitude +is 1. The sample size N = 50000, the result below does not +change if N is further increased. Subsequently, a GBN signal +with amplitude A and average switching time T will be abbre- +viated as A ∗ GBN(T) for simplicity. The input in the valida- +tion data have 5 different types: 0.2 ∗ GBN(20), 1 ∗ GBN(10), +1 ∗ GBN(20), 1 ∗ GBN(40), 5 ∗ GBN(20). Each input lasts for +1000 samples. Among these inputs, the first and the fifth have +different amplitudes compared to the one in training data but +have the same spectral distribution; the second and the fourth +have different spectral distributions but have the same ampli- +tude; the third is entirely the same. A CFNN with one layer and +4 hidden units is trained. All activation functions are ReLUs, +which are suggested to use in [10, 14]. The model validation +based on 1-step prediction error and output error are shown in +5 + +0 +50 +100 +150 +200 +250 +Samples +2 +4 +6 +8 +10 +12 +14 +16 +Real +0.2 +0.5 +1 +2 +5 +180 +200 +220 +240 +260 +8.5 +9 +9.5 +10 +10.5 +(a) 1*GBN(20). +0 +50 +100 +150 +200 +250 +Samples +2 +4 +6 +8 +10 +12 +14 +16 +Real +0.2 +0.5 +1 +2 +5 +180 +200 +220 +240 +260 +9 +10 +11 +12 +13 +(b) 2*GBN(20). +0 +50 +100 +150 +200 +250 +Samples +0 +2 +4 +6 +8 +10 +12 +14 +16 +True +0.2 +0.5 +1 +2 +5 +180 +200 +220 +240 +260 +8.5 +9 +9.5 +10 +(c) Gaussian input. +0 +50 +100 +150 +200 +250 +Samples +2 +4 +6 +8 +10 +12 +14 +16 +Real +0.2 +0.5 +1 +2 +5 +180 +200 +220 +240 +260 +8.5 +9 +9.5 +(d) Uniform input. +Figure 4: Noise-free LTI system: step responses under different inputs. +0 +50 +100 +150 +200 +250 +Samples +0 +2 +4 +6 +8 +10 +12 +14 +BJ +True +Estimated +CFNN: 1 layer 4 hidden units +0 +50 +100 +150 +200 +250 +0 +2 +4 +6 +8 +10 +12 +14 +CFNN: 1 layer 6 hidden units +0 +50 +100 +150 +200 +250 +0 +2 +4 +6 +8 +10 +12 +14 +CFNN: 2 layer 4 hidden units +0 +50 +100 +150 +200 +250 +0 +2 +4 +6 +8 +10 +12 +14 +Figure 5: Noisy LTI system: comparison of step responses of BJ and different CFNN structures when NSR=5%. For CFNN, the best step response among input +amplitude [0.2, 0.5, 1, 2, 5] in each run is plotted. The edge of the red region is the envelope of 100 runs. +Table 2: Noise-free LTI system: FITs(%) of different step responses. The num- +bers in the first raw are input amplitudes. The red one denotes the best for each +input. +Input type +0.2 +0.5 +1 +2 +5 +1*GBN(20) +50.59 +85.95 +99.79 +93.23 +89.09 +2*GBN(20) +19.67 +60.94 +85.95 +99.79 +91.85 +Gaussian +83.81 +75.12 +72.26 +70.85 +70.01 +Uniform +80.31 +94.96 +87.66 +83.87 +81.60 +Table 3: Noisy LTI system: comparison of BJ and CFNN in mean FITs (%) of +step responses. CFNN has one layer with 4 hidden units. +NSR (%) BJ +CFNN (with different input amplitudes) +0.2 +0.5 +1 +2 +5 +Best +0 +100.00 +63.43 +89.35 +99.23 +94.42 +91.22 +99.26 +5 +95.67 +40.84 +77.30 +90.48 +85.16 +80.21 +93.50 +10 +93.64 +39.54 +73.85 +88.67 +83.48 +78.21 +92.50 +20 +92.42 +28.68 +66.80 +83.30 +76.84 +70.22 +89.08 +40 +86.61 +27.35 +62.78 +76.50 +68.80 +61.88 +84.62 +Fig. 3. The Relative Errors (RE) of one-step prediction, simu- +lation and additionally 20-step prediction are shown in Table 1. +RE is calculated according to Var +� +ε2(t) +� +/Var +� +y2(t) +� +where ε is +some residual and y the measured output. +In parameter optimization, CFNN uses 1-step PE, and in val- +idation data it remains small but OE becomes much larger. For +inputs having different characteristics with the training data, the +situation becomes even worse. The 20-step PE ranges in the +middle of 1-step PE and OE. The results confirm that a model +with small PE can have large OE. +To further analyze the estimated CFNN, the step responses +of the models are plotted in Fig. 4(a). The normalized step re- +sponse of a linear system is unique, but the estimated CFNN +gives different responses when step inputs with different ampli- +tudes are used. Therefore it cannot deliver consistent estimates +in step response (c.f. Definition 1). The FITs of these responses +to the true one are recorded in the second raw of Table 2. The +FIT between two sequence Z0 and ˆZ is +FIT = 1 − +���Z0 − ˆZ +��� +���Z0 − mean( ˆZ) +���. +(21) +The best result is achieved when input amplitude equals 1. This +is because GBN is a binary signal that only have two values. +For amplitudes different from the one in training data, the step +responses of CFNN deviate from the true one. +Three other types of inputs are also tested: 2*GBN(20), zero- +mean Gaussian white input with variance 0.332, uniform input +that ranges in (−1, 1). The results are shown in Fig. 4(c-d) and +Table 2. It is interesting to note that for 2*GBN(20) the best FIT +moves to amplitude 2; for Gaussian input the best FIT occurs +for the smallest amplitude 0.2 because Gaussian distribution has +a bell-shaped curve; for uniform input the gaps between the +best and the other are insignificant because the curve of uniform +distribution is flat in its range. +For each case, the MSE of training data have been optimized +to very small value (< 10−6) and obtained models differ, which +implies that CFNN that can give PE nearly to zero is non- +unique. The final estimated CFNN is strongly dependent on +the input characteristics, the initial conditions, etc. When en- +countering a ‘never-met’ (does not occur in training data) or +‘unfamiliar’ (does not occur frequently in training data) input, +CFNN gives wrong step response. +6 + +0 +100 +200 +300 +Samples +0 +100 +200 +300 +400 +500 +600 +700 +-2.5 +True +Polynomial Hammerstein +CFNN +0 +100 +200 +300 +0 +50 +100 +150 +200 +250 +-1.5 +0 +100 +200 +300 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +-0.5 +0 +100 +200 +300 +0 +10 +20 +30 +40 +50 +60 +0.5 +0 +100 +200 +300 +0 +50 +100 +150 +200 +250 +300 +350 +1.5 +0 +100 +200 +300 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +2.5 +Figure 6: Noise-free Hammerstein system: comparison of step responses under different input amplitudes. CFNN with 3 layers and 3 hidden units in each layer is +used. +Table 4: Noisy system: information about the mean estimated parameters of CFNN. +NSR (%) +∥Whu∥F +∥Wyh∥ +∥Wyu(3 : 4)∥ +Wyu(1) +Wyu(2) +Wyu(3) +Wyu(4) +0 +2.43 +7.33E-05 +9.48E-04 +1.9630 +-0.9650 +8.28E-04 +4.62E-04 +5 +2.45 +3.60E-04 +8.77E-04 +1.9630 +-0.9650 +7.62E-04 +4.33E-04 +10 +2.46 +6.16E-04 +8.03E-04 +1.9627 +-0.9648 +6.94E-04 +4.04E-04 +20 +2.49 +8.92E-04 +7.59E-04 +1.9631 +-0.9650 +6.56E-04 +3.81E-04 +40 +2.57 +1.51E-03 +7.32E-04 +1.9627 +-0.9647 +6.99E-04 +2.20E-04 +Table 5: Noisy LTI system: mean FITs of different CFNN structures. nl denotes +the layer number, nh denotes hidden units in each layer. The red one denotes +the best settings. +nl +nh +0.2 +0.5 +1 +2 +5 +Best +1 +1 +41.80 +76.04 +90.26 +85.40 +80.68 +93.81 +1 +2 +47.35 +77.84 +88.85 +84.24 +79.35 +92.31 +1 +4 +40.84 +77.30 +90.48 +85.16 +80.21 +93.50 +1 +6 +50.40 +78.37 +89.31 +86.95 +83.36 +92.80 +1 +10 +40.84 +70.97 +81.03 +78.90 +75.68 +83.47 +1 +4 +41.80 +76.04 +90.26 +85.40 +80.68 +93.81 +2 +4 +8.31 +21.12 +81.98 +-3.09 +-36.67 +82.24 +3 +4 +-12.34 +-4.88 +43.76 +-223.69 +-282.30 +63.32 +Table 6: Noisy Hammerstein system: mean FITs(%) of step responses. The +numbers in the first raw are input amplitudes. +Model +-2.5 +-1.5 +-0.5 +0.5 +1.5 +2.5 +Polynomial Hammerstein +94.61 +94.41 +52.81 +77.83 +96.12 +95.55 +CFNN (nl = 3, nh = 3) +-1.13 +88.79 +27.42 +45.03 +-3.52 +14.50 +CFNN (nl = 4, nh = 4) +-0.39 +87.01 +27.13 +56.05 +-3.54 +32.76 +4.1.2. Noisy system +In this part, the performance of CFNN will be compared to +PEM for noisy system. The true system has an ARX structure, +the parameter optimization of the same model has a closed- +form solution. However, a BJ model that can also give con- +sistent estimate is used. In this case, both CFNN and BJ re- +quire numerical optimization. For each NSR setting, 100 Monte +Carlo simulations are run. In each simulation, 10000 training +data are generated under the input is 1*GBN(20), then the step +response of CFNN is calculated under different input ampli- +tudes. Among these responses the one that gives the best FIT +will be recorded. +A single-layer CFNN with 4 hidden units +is first tested, the results are shown in Table 3. CFNN delivers +worse results than BJ for all NSRs, even for the best ones cho- +sen from the five candidates. Consistent with the discussions +in Section 4.1.1, the mean FITs under input amplitude 1 give +best result. Additionally, Table 4 presents information about +the mean estimated parameters of CFNN. Recall that Whu and +Wyh corresponds to the nonlinear part and Wyu corresponds to +the linear part. Concerning the linear part, Wyu(1 : 2) related +to A0(q) is consistent while Wyu(3 : 4) related to B0(q) is not, +∥Wyu(3 : 4)∥ decreases as NSR decreases; in the nonlinear part, +∥Whu∥F and ∥Wyh∥ increases as NSR increases. This reveals that +there is a competition between the weighting matrices of non- +linear and linear parts. When noise level increase, the nonlinear +part takes the advantage gradually and make the results devi- +ate further away from the true system. Notice that the consis- +tency of the parameters related to A0(q) does not hold generally. +When other types of inputs are used, such as Gaussian, uniform, +all parameters become inconsistent. +Different structure settings of CFNN are tested and the mean +FITs are shown in Table 5. One can see that increasing numbers +of hidden units or layers do not improve the situation. In fact, +the poorest result is obtained when three layers are used. The +step responses of 100 Monte Carlo runs of three selected cases +are plotted in Fig. 5. The more complex the structure of CFNN, +the higher the variance; the best setting also has higher variance +than BJ. +4.1.3. Summary and discussion +The simulation results give surprise finding: as a universal +approximator, CFNN cannot even give consistent estimate for +a simple LTI system that is contained in its model structure. +Although when all weighting matrices in its nonlinear part van- +ish, CFNN is simply a linear ARX model, there is no guarantee +that estimated model is consistent. The ‘flexible’ structure of +CFNN, enabled by the nonlinear part, i.e., the hidden layers, +becomes a nuisance factor for identification of LTI system. For +a nonlinear system that is more complex than a LTI system, +such neural network based models can perform poorer. +4.2. Hammerstein system +Consider the following Hammerstein system: +y(t) = B0(q) +A0(q) f (u(t)) + α +1 +A0(q)e0(t) +f(u) = 10u3 + 3.5u2 + u. +(22) +7 + +The linear part of this system is chosen entirely the same +as (20). 10000 training data are generated using a General- +ized Multiple-level Noise (GMN) signal (see Chapter 9.1.2 of +[21]) with average switching time 10s and amplitude ranging +in [−2, 2]. When using idnlhw function the orders of system +and polynomial are set to their correct values. Slightly different +from the LTI case, in the first layers the activation functions are +ReLU while in other layers they are hyperbolic tangent func- +tions. +Fig. 6 shows the result of noise-free case. +The step re- +sponses delivered by the polynomial Hammerstein model coin- +cides with the true system. CFNN only gives consistent results +for amplitude -1.5 and 1.5; for other amplitudes, the results are +very poor. The noisy system with 1% is also tested, shown +in Table 6. The polynomial Hammerstein has better FITs than +CFNN for all cases tested. The variances of step responses are +similar to Fig. 5 in which CFNN delivers large variances, and it +will not be shown here for brevity. +5. Concluding remarks +Many researchers in control community are optimistic about +the use of neural networks for dynamic system modeling, per- +haps due to their success in CV and NLP. In this work, three rep- +resentative models PEM, CFNN and LSTM are compared for +their ability in LTI system identification. As reported, LSTM is +unsuitable for dynamic system identification. CFNN has a rea- +sonable structure and can be reduced to a common LTI model. +However, no results exist to guarantee the model consistency. +Moreover, the large number of model parameters of LSTM and +CFNN will result in large model variance. In simulation studies +of the LTI system, CFNN fails to give consistent step responses +even in the noise-free case. In the noisy case, CFNN models +have larger model variances than the BJ model. When tested in +a Hammerstein system, CFNN gives poorer performance. In- +creasing hidden unit number and hidden layer number do not +improve model quality. +This study reveals that there is still a long way to go for neural +network based dynamic system identification/modeling. 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Tulleken, Generalized binary noise test-signal concept for improved +identification-experiment design, Automatica 26 (1) (1990) 37–49. +8 + diff --git a/_dFJT4oBgHgl3EQfqSzu/content/tmp_files/load_file.txt b/_dFJT4oBgHgl3EQfqSzu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a45bcf37a2d10dbf8d19c524b0ecd41b04588089 --- /dev/null +++ b/_dFJT4oBgHgl3EQfqSzu/content/tmp_files/load_file.txt @@ -0,0 +1,732 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf,len=731 +page_content='A critical look at deep neural network for dynamic system modeling Jinming Zhoua, Yucai Zhua,∗ aState Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China Abstract Neural network models become increasingly popular as dynamic modeling tools in the control community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' They have many appealing features including nonlinear structures, being able to approximate any functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' While most researchers hold optimistic attitudes towards such models, this paper questions the capability of (deep) neural networks for the modeling of dynamic systems using input-output data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For the identification of linear time-invariant (LTI) dynamic systems, two representative neural network models, Long Short-Term Memory (LSTM) and Cascade Foward Neural Network (CFNN) are compared to the standard Prediction Error Method (PEM) of system identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In the comparison, four essential aspects of system identification are considered, then several possible defects and neglected issues of neural network based modeling are pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Detailed simulation studies are performed to verify these defects: for the LTI system, both LSTM and CFNN fail to deliver consistent models even in noise-free cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' and they give worse results than PEM in noisy cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Keywords: Deep neural network, Model structure, Error criteria, Consistency, Model validation, System identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Introduction Process modeling is fundamental to most process applica- tions, from control, optimization to fault diagnosis and soft sen- sor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' An accurate model that well reflects the process behavior is essential for success of all these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Neural network and deep learning models have become increasingly popular in the control community, inspired by their tremendous success in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=', Computer Vision (CV), Natural Language Processing (NLP) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Unlike system identification theory that starts from the well-established theory for linear systems [3, 4], neu- ral networks have inherently nonlinear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Moreover, they are so-called universal approximators capable of approxi- mating any functions within any degree of accuracy [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Nu- merous papers have been published for the development of neu- ral network based methods for various applications in systems and control [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The input and output spaces of a neural network are very gen- eral: the input could be signals, images, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=', while the output could be numerical values or classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Among the many possibilities, this paper only focuses on the case where the neu- ral network is used to learn the dynamic relations between pro- cess inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Such model can be used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=', as the internal model in model predictive control (MPC), for residual generation in FDI, or as a soft sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' According to some state-of-the-art review papers [7, 8, 9, 10, 11], three representative network structures are Feedfoward Neural Network (FNN), Recurrent Neural Network (RNN), and Convolution Neural Network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' FNN is a classical struc- ture and its use in process industry can be dated back to 20 years ∗Corresponding author Email addresses: zhoujinming@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='cn (Jinming Zhou), zhuyucai@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='cn (Yucai Zhu) ago [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' It still receives great research interests nowadays in nonlinear modeling and MPC applications [10, 14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For dynamic modeling, lagged input and output signals should be fed into FNN to manually create dynamics[14], while RNN has naturally a dynamic structure which can be written in a non- linear state-space form [1], thus it is very popular in process control research where dynamic modeling is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' As re- ported [1], most successful applications of RNN use the LSTM structure, which can successfully handle the gradient vanish- ing/exploding problem of original RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' LSTM has already been extensively used in FDI [17], MPC [15], soft sensor [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' AutoEncoder (AE) is another popular RNN structure whose ap- plications are mainly in FDI and soft sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' CNN is most well- known for its success in computer vision, it can also be used in local dynamic modeling and frequency domain modeling [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Great efforts have been put into research on neural networks and deep learning methods with the hope that they can pro- mote developments of process control industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' However, un- like in the fields of CV and NLP, very few successful appli- cations based on neural networks are reported in process in- dustries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' What can the user really benefits from a neural net- work based modeling, compared to linear system identification and simple nonlinear models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Most papers investigating neu- ral networks tend to compare only the performance between several network structures and ignore this important question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Only recently, such comparison studies are carried out by sev- eral researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In [19], based on a 660MW boiler dataset, the model quality of LSTM is compared to those of simple (lin- ear) statistical models and the results show that LSTM gives the worst performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Based on the Silverbox dataset, in [14], LSTM gives worse result than FNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' in [20], the authors find that some network structures can act as noise amplifiers that deteriorate the model quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The two papers reveal a common Preprint January 30, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='11604v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='LG] 27 Jan 2023 phenomenon: a complex model structure mathematically capa- ble of learning arbitrary systems well can fail in practice, even if the real system is not complex at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' This paper further investigates the deficiencies of neural net- work based modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Besides LSTM that has been criticized in [19, 14], it will be shown that the CFNN suggested in [14, 10] also has problems that may hinder its use in modeling dynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Instead of considering modeling of nonlinear system as in [14, 10], this paper returns to the most fundamental prob- lem: modeling of LTI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' If a method cannot model such system well, it certainly cannot handle more complex nonlin- ear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' It will be shown that, although a CFNN contains both linear part and nonlinear part, it fails to identify an LTI system without noise with consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Increasing the number of hidden units or hidden layers do not improve the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' These findings will be discussed and explained through four es- sential aspects of system identification, model structure, error criterion, estimation properties and model validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Simula- tion studies will also be presented to support the declarations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The rest of the paper unfolds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Section 2 gives background knowledge about linear system identification, LSTM and CFNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Section 3 discusses and compares the three models from a system identification perspective, points out po- tential problems for neural network based modeling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Section 4 contains detailed simulation studies of a LTI system and a Hammerstein system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Section 5 gives conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Notations q denotes the forward time shifter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' σ(·) denotes activation function (vector) in neural networks, such as sigmoid function, Rectified Linear Unit (ReLU) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Details about these functions can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' ∗ is the Hadamard (element- wise) product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' N+ denotes positive natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Φz denotes power spectrum of signal {z(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Var[·] is the mathematical vari- ance operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' ‘With probability α’ is abbreviated as w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Euclidean norm and Frobenius norm are denoted as ∥ · ∥ and ∥ · ∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Background Consider a general Single-Input Single-Output (SISO) LTI system [3]: S : y(t) = G0(q)u(t) �������������� :=y0(t) + H0(q)e0(t) ���������������� :=v0(t) , (1) where u(t) and y(t) denote system input and output signals, v(t) denotes the disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' G0(q) denotes the system trans- fer function, G0(q) = B0(q) A0(q) = b0 0 + · · · + bnb 0 q−nb 1 + a1 0q−1 + · · · + ana 0 q−na .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (2) which is assumed stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For simplicity assume na = nb = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' e0(t) is a zero-mean white noise sequence with variance λ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' H0(q) is assumed to be stable and inversely stable and monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The prediction error method In system identification, a parametric model set is used to describe the true system S: M(θ) : y(t) = G(q, θ)u(t) + H(q, θ)e(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (3) In PEM, quadratic cost function of the one-step Prediction Error (PE) is minimized: ˆθN = arg min θ N � t=1 � y(t) − ˆypem(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ) � (4a) ˆypem(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ) = � 1 − H−1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ) � y(t) + H−1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ)G(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ)u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (4b) Notice that subsequently when talking about error criterion for parameter estimation, all PE refers to one-step PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' If S ∈ M, which means that the model structure M is flexible enough, there exists θ0 such that G(q, θ0) = G0(q) and H(q, θ0) = H0(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Moreover, if M(θ) is globally identifiable at θ0 and the input signal is persistently exciting, the PEM estimate is consistent: ˆθN → θ0, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 1 as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (5) A more precise description of the conditions required for this property can be found in Chapter 8 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The consistency property implies that the estimate will approach the parameter vector representing the true system, when large amount of data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' If e(t) in (1) is Guassian, the PEM estimate can be further proved to have minimum variance [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Long short-term memory network The below contents show the mathematical formulation of a single-layer LSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The cores of LSTM are four gates and two states controlling the transfer of the information flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The four gates are defined of the form ∗(t) = σ (W∗uu(t) + W∗hh(t − 1) + b∗) (6) where ∗ can be i, f, g and o, corresponding to input, forget, cell and output gates respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Suppose that the weighting ma- trices and bias vectors above are of compact dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Based on these gates, the cell state c(t) and hidden state h(t) are up- dated according to c(t) = f(t) ∗ c(t − 1) + i(t) ∗ g(t) (7a) h(t) = o(t) ∗ tanh (c(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (7b) Finally, the output of LSTM is ˆylstm(t) = Wyhh(t) + by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (8) In above formulations, (6–7) constitute a LSTM layer, while (8) is often referred as a linear fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Introduce also a vector ρ that contains all the parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=', W· and b· in (6–8), which can be optimized according to ˆρN = arg min ρ N � t=1 (y(t) − ˆylstm(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' ρ))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (9) Notice that ˆylstm(t) is calculated according to (6–8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' To obtain a deep network structure, one can simply connect several LSTM layer in series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 2 Identification experiment Choose model sturcture Choose error criterion Estimate model Model validation Priori knowledge, modeling purpose,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' PE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' OE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Flexible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Compact?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Pass?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Yes: use No: revise Informative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Economic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Data Estimation properties Test what?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' OE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' K-step PE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Figure 1: Typical identification procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Cascade forward neural network CFNN here refers to a specific FNN suggested in [14] for nonlinear identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' It is characterized by that outputs of all previous units are used as inputs in the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' It is al- ready available in Matlab’s System Identification Toolbox, as an idnlarx object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' As has been mentioned in Section 1, lagged input and output signals should be used to introduce dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The input of CFNN is the regressor ϕ⊤(t) = �y(t − 1), · · · , y(t − n), u(t), · · · , u(t − n)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (10) The output of a single layer CFNN is ˆycfnn(t) = Wyuϕ(t) + by ���������������������� Linear + Wyhσ (Whuϕ(t) + bh) �������������������������������������������� Nonlinear .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (11) Collect all parameters of CFNN into a vector ϑ, it can be opti- mized according to ˆϑN = arg min ϑ N � t=1 (y(t) − ˆycfnn(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' ϑ))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (12) As shown in (11), CFNN can be divided into linear and non- linear parts where the nonlinear part is due to the hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Similar to LSTM, one can use multiple hidden layers to achieve a deep structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Neural network based dynamic modeling from an iden- tification perspective Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 1 is a typical identification procedure [3, 4, 21] covering essential steps and key points of a general black- box dynamic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' This section discusses the three models introduced in Section 2 then points out problems that are often neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The experiment design and parameter optimization problems will not be discussed, only the steps with black dotted line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 1 are focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Model structure Among the three methods listed in Section 2, the structure of PEM is exactly the same as the true system S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' By introducing a new state vector x⊤(t) = [c⊤(t), h⊤(t)], LSTM model (6–8) can be summarized to a NonLinear State-Space (NLSS) form [1]: x(t) = F (x(t − 1), u(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' ρ) (13a) ˆylstm(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' ρ) = H (x(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (13b) Notice that it is assumed that no time delay exists in input and output in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' According to [14, 10], CFNN belongs to the Nonlinear ARX (NARX) model: ˆycfnn(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' ϑ) = G (ϕ(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' ϑ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (14) In (13–14) F , H, G are user-defined (nonlinear or linear) func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' As has been mentioned in Section 2, CFNN model contains a linear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' When Wyh, Whu, by and bh all vanish and if Wyu = [−a1 0, · · · , −ana 0 , b1 0, · · · , bnb 0 ], (15) CFNN will present exactly the same behavior as G0(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' That is to say, CFNN covers the LTI model structure as its special case and is theoretically capable of modeling such system, which resembles Volterra model [22], block-oriented model [23], lin- ear parameter-varying model [24], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In contrast, it is non- trivial how LSTM network can be connected or reduced to a common LTI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In [19], after a careful design of the hy- perparamters LSTM still gives worst performance among other statistical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The authors attribute this phenomenon to the effects of gate functions in (6), disabling some of these gates can improve the performance of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Simulation results in [14] are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The above discussions and the results in [14, 19] reveal that, for process dynamic modeling, models that cannot be inter- preted from process dynamics are undesirable even if they have complex structures with strong approximation abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Error criteria Error criteria are the basics for optimization of model param- eters, they are application-oriented and are closely related to the properties of the estimated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Three commonly used error criteria of the linear model set (M) are, Prediction Error (PE), Output Error (OE, also called simulation error), Equation Error (EE): εpe(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ) = H−1(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ) (y(t) − G(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ)u(t)) (16a) εoe(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ) = y(t) − G(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ)u(t) (16b) εee(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ) = A(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ)y(t) − B(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ)u(t) (16c) where A(q) and B(q) are defined similarly to A0(q) and B0(q) in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Notice that EE equals PE of an ARX model [3] hence only PE and OE will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The difference between the predicted output (4b) and the simulated output G(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' θ)u(t) is that predicted output uses measured output signals while simulated 3 0 5 10 15 20 25 System order: n 100 101 102 103 104 105 Number of parameters BJ LSTM: nh=n LSTM: nh=3n CFNN: nh=n CFNN: nh=3n Figure 2: Parameter numbers of different models under different system orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' output only uses input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Bearing this point in mind, the extensions in LSTM and CFNN are straightforward: because no measured outputs are used in ˆylstm, the LSTM criterion (9) is an output error criterion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' the CFNN criterion (12) is a prediction error (equation error) criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' While the PEM model set M has a noise model that can be parameterized independently to process model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=', Box- Jenkins (BJ) model [3]), LSTM and CFNN are not equipped with such ability to handle disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In [20], the authors argue that the noise effects can even be amplified in CFNN cri- terion (12) and suggests the use of output error criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' How- ever, how to optimize CFNN model under an output error crite- rion is an unsolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Sometimes the input of LSTM is chosen to be a regressor containing measured outputs like ϕ(t) in (10), which is called teacher forcing in Chapter 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1 in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In this case, the LSTM criterion (9) becomes a prediction error criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Estimation properties Analyzing the (statistical) properties of the estimated model is of particular interest and of vital importance in all model- ing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The total model error can be divided into struc- ture error caused by deficiencies in model structure and random model error caused by stochastic disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For linear identi- fication, one often investigates the consistencies and variances of parameters, step responses or frequency responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The two points concerning neural network will be respectively discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Consistency For nonlinear model structures, it makes less sense to con- sider in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Instead, the consistency in step response can be defined analogously to (5): Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Denote {U(t)} as a step signal of amplitude AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Then for some general system S and an estimated model ˆ MN under structure M and with N data samples, as well as their outputs subject to {U(t)}: {Y0(t)} and { ˆYN(t)}, if ˆYN(t) → Y0(t), w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 1 as N → ∞, ∀t ∈ N+, AU � 0, (17) the estimate of M is consistent in step response to S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In this regard, the estimated model ˆ MN can accurately de- scribe the input-output relations of the true system S if suf- ficient data samples are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For LTI system, PEM can deliver consistent estimate in parameters, step response and fre- quency response (under some conditions, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1), and for step response one only needs to consider the case AU = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' It is necessary to give some discussions on the differences be- tween the above consistency concept and the well-known uni- versal approximation theorem [1, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1 of[1], this theorem is summarized as: An FNN with a linear output layer and at least one hid- den layer with the commonly used activation function (sig- moid, ReLU, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=') can approximate any Borel measurable func- tion from one-dimensional space to another with any desired amount of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The theorem states that even a single-layer FNN (which is nearly the simplest network) is theoretically capable of approx- imating any functions arbitrarily well, which seems to imply that neural network is a powerful modeling tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' But note that this is only an existence theorem that gives neither the guaran- tee of consistent estimates, nor the guideline for how to obtain a consistent estimate using training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' This theorem may have made control community overly optimistic about the modeling capability of neural networks for dynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Variance Consider first the asymptotic variance expression of the fre- quency response for linear system [25]: Var � ˆGn N(ejω) � ≈ n N Φv(ω)λ0 Φu(ω)λ0 − |Φue0(ω)| (18) where ˆGn N(ejω) denotes frequency response function of a nth- order estimate based on N data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The expression holds exactly for ARX model when N → ∞, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' It reveals that the model variance is proportional to number of param- eters and is inversely proportional to sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Although there is no such conclusion available for nonlinear systems and neural network models, it is generally acknowledged that com- plex models containing many parameters are more vulnerable to stochastic disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Consider modeling a nth order SISO LTI system like S, sup- pose that the correct order is used for PEM (a BJ model is used) and CFNN, all biases in LSTM and CFNN are set to zero, and use nh hidden units for both LSTM and CFNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Then the pa- rameter numbers of the three models are BJ: nθ = 4n (19a) LSTM: nρ = 4 � n2 h + nh � + nh (19b) CFNN: nϑ = nh (2n + 1) + 2n (19c) Notice that only single-layer LSTM and CFNN with fully con- nected output layers are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 2 shows the parameter numbers of different models with varying system orders and hidden unit settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Among three model structures, LSTM has the most parameters and is 1–4 orders of magnitude higher than BJ model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' CFNN ranges in the 4 middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For dynamic process modeling, the training data must be informative enough to contain the main process behavior, which can be guaranteed by adding persistently exciting test signals to the plant [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' This procedure certainly introduces some slight disturbance to normal operations and is the main cost of black-box process modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Heuristically, according to (18), to achieve a same level variance, LSTM requires 1–4 orders of magnitude more data than BJ model, which means a huge increase in modeling cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' This gives another explanation about the poor model quality in [14, 19] apart from the model structure issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Model validation Model validation is the final step before a model can be put into use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' It should be application-oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In MPC, the model is used to give multi-step predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' If the step size of a multi- step prediction tends to infinity, it becomes simulated output, see Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 of [3] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Recently, it is proved that output error serves as a useful tool for FDI as well [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Hence it is important to check simulation error because it reflects the real gap between the model and the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' However in many recently published papers, the model quality is only validated through (one-step) prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In [27], the author illus- trated that two model with close prediction errors can have huge differences in simulation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Typically, a neural network model with complex structure can easily deliver a loss function of very small value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' When the loss corresponds to prediction error and the model gives accu- rate one-step prediction, one can tell nothing about the model quality from the input to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' If possible, check simu- lation error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' or at least test the multi-step prediction error for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Summary LSTM has a very different structure from a LTI system hence it may have difficulties to model such a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In contrast, CFNN has a structure which can be interpreted as a linear model extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' However, as discussed above, there is no guarantee for its model consistency and it may have high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Fur- ther, a small prediction error delivered by CFNN may not imply a good input-output model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Simulation study While LSTM has been comprehensively tested in [14, 19], this section studies CFNN in a linear ARX system and a Ham- merstein system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Step responses under different input ampli- tudes will be calculated and normalized according to the input amplitudes for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Mean values are removed in the training data so the bias vectors in CFNN will be disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Simulations performed in this section are all performed using Matlab and System Identification Toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Specifically, nlarx is used to estimate CFNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' BJ is used to estimate a BJ model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' nlhw is used to estimate a Hammerstein system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For above estimates obtained in System Identification Toolbox, sim and predict are used to calculate model simulation and k−step prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Samples 60 40 20 0 20 40 60 Real output Simulated output 0 200 400 600 800 2 0 2 4200 4400 4600 4800 5000 35 40 45 50 55 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 60 40 20 0 20 40 60 Real output Predicted output Comparsion with one-step prediction Comparsion with simulation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2*GBN(20) 1*GBN(10) 1*GBN(20) 1*GBN(40) 5*GBN(20) Figure 3: Model validation based on one-step prediction and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Table 1: Noise-free LTI system: REs(%) of residuals in different data sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Residual 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2*GBN(20) 1*GBN(10) 1*GBN(20) 1*GBN(40) 5*GBN(20) 1-step PE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='99E-07 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='38E-08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='80E-08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='61E-08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='14E-09 20-step OE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='81E-02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='12E-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='14E-04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='05E-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='17E-04 Simulation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='73E+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='20E-02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='56E-03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='56E-03 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='12E-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Linear system Consider the following ARX system: y(t) = B0(q) A0(q)u(t) + α 1 A0(q)e0(t) B0(q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='0115 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='00639q−1 A0(q) = 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='963q−1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='965q−2 (20) where α is used to control the Noise-to-Signal Ratio (NSR), defined as Var � v2 0(t) � /Var � y2 0(t) � , see (1) for definitions of y0(t) and v0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Noise-free system The system without noise is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The training data are generated with a Generalized Binary Noise (GBN) [28] as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Its average switching time is 20 samples and amplitude is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The sample size N = 50000, the result below does not change if N is further increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Subsequently, a GBN signal with amplitude A and average switching time T will be abbre- viated as A ∗ GBN(T) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The input in the valida- tion data have 5 different types: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 ∗ GBN(20), 1 ∗ GBN(10), 1 ∗ GBN(20), 1 ∗ GBN(40), 5 ∗ GBN(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Each input lasts for 1000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Among these inputs, the first and the fifth have different amplitudes compared to the one in training data but have the same spectral distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' the second and the fourth have different spectral distributions but have the same ampli- tude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' the third is entirely the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' A CFNN with one layer and 4 hidden units is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' All activation functions are ReLUs, which are suggested to use in [10, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The model validation based on 1-step prediction error and output error are shown in 5 0 50 100 150 200 250 Samples 2 4 6 8 10 12 14 16 Real 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 1 2 5 180 200 220 240 260 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 (a) 1*GBN(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 0 50 100 150 200 250 Samples 2 4 6 8 10 12 14 16 Real 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 1 2 5 180 200 220 240 260 9 10 11 12 13 (b) 2*GBN(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 0 50 100 150 200 250 Samples 0 2 4 6 8 10 12 14 16 True 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 1 2 5 180 200 220 240 260 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 10 (c) Gaussian input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 0 50 100 150 200 250 Samples 2 4 6 8 10 12 14 16 Real 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 1 2 5 180 200 220 240 260 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 (d) Uniform input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Figure 4: Noise-free LTI system: step responses under different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 0 50 100 150 200 250 Samples 0 2 4 6 8 10 12 14 BJ True Estimated CFNN: 1 layer 4 hidden units 0 50 100 150 200 250 0 2 4 6 8 10 12 14 CFNN: 1 layer 6 hidden units 0 50 100 150 200 250 0 2 4 6 8 10 12 14 CFNN: 2 layer 4 hidden units 0 50 100 150 200 250 0 2 4 6 8 10 12 14 Figure 5: Noisy LTI system: comparison of step responses of BJ and different CFNN structures when NSR=5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For CFNN, the best step response among input amplitude [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5, 1, 2, 5] in each run is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The edge of the red region is the envelope of 100 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Table 2: Noise-free LTI system: FITs(%) of different step responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The num- bers in the first raw are input amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The red one denotes the best for each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Input type 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 1 2 5 1*GBN(20) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='59 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='95 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='79 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='23 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='09 2*GBN(20) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='67 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='94 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='95 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='79 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='85 Gaussian 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='81 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='12 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='26 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='85 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='01 Uniform 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='31 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='96 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='66 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='87 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='60 Table 3: Noisy LTI system: comparison of BJ and CFNN in mean FITs (%) of step responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' CFNN has one layer with 4 hidden units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' NSR (%) BJ CFNN (with different input amplitudes) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 1 2 5 Best 0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='00 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='43 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='35 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='23 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='42 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='22 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='26 5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='67 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='84 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='30 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='48 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='16 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='21 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='50 10 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='64 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='54 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='85 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='67 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='48 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='21 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='50 20 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='42 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='68 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='80 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='30 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='84 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='22 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='08 40 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='61 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='35 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='78 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='50 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='80 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='88 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='62 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The Relative Errors (RE) of one-step prediction, simu- lation and additionally 20-step prediction are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' RE is calculated according to Var � ε2(t) � /Var � y2(t) � where ε is some residual and y the measured output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In parameter optimization, CFNN uses 1-step PE, and in val- idation data it remains small but OE becomes much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For inputs having different characteristics with the training data, the situation becomes even worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The 20-step PE ranges in the middle of 1-step PE and OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The results confirm that a model with small PE can have large OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' To further analyze the estimated CFNN, the step responses of the models are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The normalized step re- sponse of a linear system is unique, but the estimated CFNN gives different responses when step inputs with different ampli- tudes are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Therefore it cannot deliver consistent estimates in step response (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Definition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The FITs of these responses to the true one are recorded in the second raw of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The FIT between two sequence Z0 and ˆZ is FIT = 1 − ���Z0 − ˆZ ��� ���Z0 − mean( ˆZ) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (21) The best result is achieved when input amplitude equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' This is because GBN is a binary signal that only have two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For amplitudes different from the one in training data, the step responses of CFNN deviate from the true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Three other types of inputs are also tested: 2*GBN(20), zero- mean Gaussian white input with variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='332, uniform input that ranges in (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 4(c-d) and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' It is interesting to note that for 2*GBN(20) the best FIT moves to amplitude 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' for Gaussian input the best FIT occurs for the smallest amplitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 because Gaussian distribution has a bell-shaped curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' for uniform input the gaps between the best and the other are insignificant because the curve of uniform distribution is flat in its range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For each case, the MSE of training data have been optimized to very small value (< 10−6) and obtained models differ, which implies that CFNN that can give PE nearly to zero is non- unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The final estimated CFNN is strongly dependent on the input characteristics, the initial conditions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' When en- countering a ‘never-met’ (does not occur in training data) or ‘unfamiliar’ (does not occur frequently in training data) input, CFNN gives wrong step response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 6 0 100 200 300 Samples 0 100 200 300 400 500 600 700 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 True Polynomial Hammerstein CFNN 0 100 200 300 0 50 100 150 200 250 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 0 100 200 300 0 2 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 0 100 200 300 0 10 20 30 40 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 0 100 200 300 0 50 100 150 200 250 300 350 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 0 100 200 300 0 100 200 300 400 500 600 700 800 900 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 Figure 6: Noise-free Hammerstein system: comparison of step responses under different input amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' CFNN with 3 layers and 3 hidden units in each layer is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Table 4: Noisy system: information about the mean estimated parameters of CFNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' NSR (%) ∥Whu∥F ∥Wyh∥ ∥Wyu(3 : 4)∥ Wyu(1) Wyu(2) Wyu(3) Wyu(4) 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='43 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='33E-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='48E-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9630 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9650 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='28E-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='62E-04 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='60E-04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='77E-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9630 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9650 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='62E-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='33E-04 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='16E-04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='03E-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9648 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='94E-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='04E-04 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='49 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='92E-04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='59E-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9650 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='56E-04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='81E-04 40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='51E-03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='32E-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='9647 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='99E-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='20E-04 Table 5: Noisy LTI system: mean FITs of different CFNN structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' nl denotes the layer number, nh denotes hidden units in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The red one denotes the best settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' nl nh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 1 2 5 Best 1 1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='80 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='04 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='26 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='40 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='68 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='81 1 2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='35 77.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='81 2 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='31 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='12 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='09 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='67 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='24 3 4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='88 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='76 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='69 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='30 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='32 Table 6: Noisy Hammerstein system: mean FITs(%) of step responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The numbers in the first raw are input amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 Polynomial Hammerstein 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='61 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='41 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='81 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='83 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='12 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='55 CFNN (nl = 3, nh = 3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='13 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='79 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='42 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='52 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='50 CFNN (nl = 4, nh = 4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='39 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='01 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='13 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='54 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Noisy system In this part, the performance of CFNN will be compared to PEM for noisy system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The true system has an ARX structure, the parameter optimization of the same model has a closed- form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' However, a BJ model that can also give con- sistent estimate is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In this case, both CFNN and BJ re- quire numerical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For each NSR setting, 100 Monte Carlo simulations are run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In each simulation, 10000 training data are generated under the input is 1*GBN(20), then the step response of CFNN is calculated under different input ampli- tudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Among these responses the one that gives the best FIT will be recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' A single-layer CFNN with 4 hidden units is first tested, the results are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' CFNN delivers worse results than BJ for all NSRs, even for the best ones cho- sen from the five candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Consistent with the discussions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1, the mean FITs under input amplitude 1 give best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Additionally, Table 4 presents information about the mean estimated parameters of CFNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Recall that Whu and Wyh corresponds to the nonlinear part and Wyu corresponds to the linear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Concerning the linear part, Wyu(1 : 2) related to A0(q) is consistent while Wyu(3 : 4) related to B0(q) is not, ∥Wyu(3 : 4)∥ decreases as NSR decreases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' in the nonlinear part, ∥Whu∥F and ∥Wyh∥ increases as NSR increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' This reveals that there is a competition between the weighting matrices of non- linear and linear parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' When noise level increase, the nonlinear part takes the advantage gradually and make the results devi- ate further away from the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Notice that the consis- tency of the parameters related to A0(q) does not hold generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' When other types of inputs are used, such as Gaussian, uniform, all parameters become inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Different structure settings of CFNN are tested and the mean FITs are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' One can see that increasing numbers of hidden units or layers do not improve the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In fact, the poorest result is obtained when three layers are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The step responses of 100 Monte Carlo runs of three selected cases are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The more complex the structure of CFNN, the higher the variance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' the best setting also has higher variance than BJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Summary and discussion The simulation results give surprise finding: as a universal approximator, CFNN cannot even give consistent estimate for a simple LTI system that is contained in its model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Although when all weighting matrices in its nonlinear part van- ish, CFNN is simply a linear ARX model, there is no guarantee that estimated model is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The ‘flexible’ structure of CFNN, enabled by the nonlinear part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=', the hidden layers, becomes a nuisance factor for identification of LTI system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' For a nonlinear system that is more complex than a LTI system, such neural network based models can perform poorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Hammerstein system Consider the following Hammerstein system: y(t) = B0(q) A0(q) f (u(t)) + α 1 A0(q)e0(t) f(u) = 10u3 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5u2 + u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' (22) 7 The linear part of this system is chosen entirely the same as (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 10000 training data are generated using a General- ized Multiple-level Noise (GMN) signal (see Chapter 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='2 of [21]) with average switching time 10s and amplitude ranging in [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' When using idnlhw function the orders of system and polynomial are set to their correct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Slightly different from the LTI case, in the first layers the activation functions are ReLU while in other layers they are hyperbolic tangent func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 6 shows the result of noise-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The step re- sponses delivered by the polynomial Hammerstein model coin- cides with the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' CFNN only gives consistent results for amplitude -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' for other amplitudes, the results are very poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The noisy system with 1% is also tested, shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The polynomial Hammerstein has better FITs than CFNN for all cases tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' The variances of step responses are similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 5 in which CFNN delivers large variances, and it will not be shown here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Concluding remarks Many researchers in control community are optimistic about the use of neural networks for dynamic system modeling, per- haps due to their success in CV and NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In this work, three rep- resentative models PEM, CFNN and LSTM are compared for their ability in LTI system identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' As reported, LSTM is unsuitable for dynamic system identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' CFNN has a rea- sonable structure and can be reduced to a common LTI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' However, no results exist to guarantee the model consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Moreover, the large number of model parameters of LSTM and CFNN will result in large model variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In simulation studies of the LTI system, CFNN fails to give consistent step responses even in the noise-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In the noisy case, CFNN models have larger model variances than the BJ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' When tested in a Hammerstein system, CFNN gives poorer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' In- creasing hidden unit number and hidden layer number do not improve model quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' This study reveals that there is still a long way to go for neural network based dynamic system identification/modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' Fol- lowing remarks can be made based on the findings: 1) The success of neural network models in CV and NLP does not guarantee its success in dynamic system model- ing and control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 2) in the noise-free case, in numerical optimization for pa- rameter estimation a neural network model may not con- verge to a dynamic model that is contained in its model structure when the loss function tends to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 3) the theorem of universal approximator cannot guarantee model consistency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 4) if a neural network model is unsuitable for modeling an LTI system, it will have more difficulties to model a non- linear dynamic system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf'} +page_content=' 5) the performance of neural network based 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Askour1,2∗, A. Belhaj, +ID 3†, H. Belmahi ID 3‡, M. Benali ID 3§, H. El Moumni ID 4¶, Y. Sekhmani3‖∗∗ +1 Department of Mathematics, Sultan Moulay Slimane University, Faculty of Sciences and Technics +B´eni Mellal, BP 523, 23000, Morocco. +2 Department of Mathematics, Mohammed V University in Rabat, Faculty of Sciences +Rabat, P.O. Box 1014, Morocco. +3 D´epartement de Physique, Equipe des Sciences de la mati`ere et du rayonnement, ESMaR +Facult´e des Sciences, Universit´e Mohammed V de Rabat, Rabat, Morocco +4 LPTHE, D´epartement de Physique, Facult´e des Sciences, Universit´e Ibn Zohr, Agadir, Morocco +January 23, 2023 +Abstract +We study the deflection angle and the trajectory of the light rays around black +holes in M-theory scenarios. Using the Gauss-Bonnet theorem, we first compute and +examine the deflection angle of the light rays near four and seven dimensional AdS +black holes obtained from the M-theory compactifications on the real spheres on S7 +and S4, respectively. We discuss the effect of the M-theory brane number and the +rotating parameter on such an optical quantity. We then investigate the trajectories +of the light rays using the equation of motion associated with M2 and M5-branes. +∗askour2a@gmail.com +†a-belhaj@um5r.ac.ma +‡hajar belmahi@um5.ac.ma +§mohamed benali4@um5.ac.ma +¶h.elmoumni@uiz.ac.ma +‖yassine sekhmani@um5.ac.ma +∗∗Authors in alphabetical order. +1 +arXiv:2301.08321v1 [hep-th] 19 Jan 2023 + +Contents +1 +Introduction +2 +2 +Deflection angle formalism for d-dimensional AdS black holes +4 +3 +Light deviating behaviors of the black holes in M-theory compactification +5 +3.1 +Light ray deflection in the (11, 4, 3) model +. . . . . . . . . . . . . . . . . . . +7 +3.2 +Light deflection behaviors in the (11, 7, −3) model . . . . . . . . . . . . . . . +10 +4 +Light trajectories around black holes in M-theory +13 +4.1 +Trajectories of the light rays in the (11, 4, 3) model +. . . . . . . . . . . . . . +14 +4.2 +Light trajectories in the (11, 7, −3) model . . . . . . . . . . . . . . . . . . . . +16 +5 +Conclusion +18 +1 +Introduction +Black hole physics has received a remarkable interest encouraged by theoretical and observa- +tional findings [1,2]. These investigations have opened new gates, which could be exploited to +understand gravity models from nontrivial theories including supergravity in higher dimen- +sions [3]. Concerning the observational results, the detection of the gravitational waves has +been considered as a strong existence of the black hole objects [4]. These interesting observa- +tions are followed by the first image of the black hole shadows in M87∗ galaxy provided by the +Event Horizon Telescope (EHT) [5–7]. Motivated by such activities, the thermodynamic and +the optical properties of certain black holes have been extensively investigated using different +methods based on analytic and numerical computations [8–10]. Various theories have been +exploited including the compactification of the superstrings and M-theory on the spherical +internal compact manifolds. Inspired by the AdS/CFT correspondence, it has been shown +that the associated black holes exhibit interesting thermodynamic behaviors. Precisely, the +black holes on AdS geometries behave like Van der Wall fluid systems [11]. Interpreting the +cosmological constant as a pressure, many phase transitions have been examined showing cer- +tain critical and universal aspects [12,13]. These thermodynamical investigations have been +bridged with optical ones with the help of two essential concepts [14–16]. The first treated +concept is the deflection angle of the light rays near black holes [17,18]. Concretely, Gibbons +and Werner proposed a direct way to calculate this optical quantity by exploiting certain +results of the Gauss-Bonnet theorem using the optical metric analysis [19]. Based on such a +method, the weak field limits of the deflection angle of the light rays around various black +holes have been investigated [20–24]. Moreover, the strong field deflection angle has received +a large interest since it has been considered as a powerful tool to make contact with exper- +imental measurements [25, 26]. The second one concerns the shadow. In four dimensions, +2 + +this optical aspect has been approached in terms of one-dimensional real curves [27–29]. For +the non-rotating solutions, these curves are identified with perfect circles. These geometri- +cal configurations have been distorted by adding the rotating parameter [30–32]. In higher +dimensions, new geometrical configurations have been obtained where the shadows involve +cardioid shapes. Concretely, the shadow behaviors of the five-dimensional (5D) black holes +embedded in type IIB superstring/supergravity-inspired spacetimes have been dealt with +by considering solutions with and without rotations [33]. The geometrical properties have +been analyzed in terms of the D3-brane number and the rotation parameter. It has been +shown that the shadows shapes are distorted by such parameters. More precisely, the size of +the shadows decreases and gets deformed with the M-brane number. Similar behaviors have +been observed in four-dimensional black holes embedded in M-theory inspired models [34]. It +has been revealed that the M2-brane number can control the circular shadow sizes. The ge- +ometrical behaviors are distorted for rotating solutions exhibiting cardioid shapes in certain +moduli space regions. Possible connections with observations (from Event Horizon Telescope +or future devices) from a particular M-theory compactification have been proposed by de- +riving certain constraints on the M2-brane number in the light of the M87∗ observational +parameters. Generalizing the study to d-dimensional black holes surrounded by dark energy +(DE), embedded in D dimensional M-theory/superstring inspired models, the shadow ge- +ometries have been approached in terms of the M(d − 2)-brane number in the presence of +DE [35]. +In this work, we investigate the deflection angle and the trajectory of the light rays around +the black holes in M-theory spherical compactifications. Using the Gauss-Bonnet theorem, +we first compute and examine the deflection angle of the light rays by four and seven- +dimensional AdS black holes obtained from the M-theory compactification on the real spheres +S7 and S4, respectively. We inspect the effect of the M-theory brane number and the rotating +parameter on such an optical quantity. We then study the trajectories of the light rays using +the equation of motion corresponding to M2 and M5-branes. +This paper is organized as follows. In section 2, we generalize the deflection angle formal- +ism for d-dimensional dealing with non-rotating AdS black holes using the Gauss-Bonnet +theorem. In section 3, we investigate the deflection angle of four-dimensional non-rotating +and rotating AdS black holes by examining the M2-brane number effect in each case. We +extend the computation to seven-dimensional non-rotating black holes by providing a com- +parative study. +In section 4, we study the trajectory of the light rays around four and +seven-dimensional AdS black holes in the M-theory compactification on the real spheres S7 +and S4, respectively. The last section concerns concluding remarks. +3 + +2 +Deflection angle formalism for d-dimensional AdS +black holes +A close examination, in the recent theoretical results, shows many interesting optical proper- +ties concerning black holes. Certain investigations have been extensively encouraged by EHT +collaboration using different scenarios. In particular, the shadows and the light deflection of +the light rays have been examined providing interesting results in certain four dimensional +gravity theories. For such reasons, we give the crucial formalism used to compute the deflec- +tion angle of the light rays around d-dimensional AdS black holes. Such an optical quantity +has been extensively investigated using various methods and approximations [36, 37]. The +most powerful results have been derived from the Gauss-Bonnet theorem. The calculation of +the light deflection angle by d-dimensional non-rotating AdS black holes can be established +using the method developed in [38]. To start, we consider the ansatz metric +ds2 = −A(r)dt2 + B(r)dr2 + r2dΩ2 +d−2 +(2.1) +where dΩ2 +d−2 ≡ dθ2 + �d−3 +k=1 sin θ sin φk represents the metric of the (d − 2)-real dimensional +unite sphere. θ and φk are the spheric local coordinates. A(r) and B(r) are radial functions +specified letter one. The optical metric of the light rays γij can be determined from the null +condition, which gives +dt2 = γijdxidxj. +(2.2) +Such an optical metric can be expressed as +dt2 = B(r) +A(r)dr2 + +r2 +A(r)dΩ2 +d−2 +(2.3) +providing a (d − 1)dimensional Riemannian space. In this space, we can calculate the devi- +ation of the spatial curve by choosing an equatorial plane constrained by θ = π +2. This plane +is defined by the radial coordinate and one periodic coordinate of {φk}. The remaining ones +should be fixed. φ is considered as the only non fixed one. Before going ahead, certain con- +served quantities are needed including the energy E and the angular momentum L. These +conserved quantities are given by +E += +A(r) dt +dλ +(2.4) +L += +r2dφ +dλ +(2.5) +where λ is the affine parameter. They can be combined to define the impact parameter b of +the motion +b = L +E . +(2.6) +The computations provide +b = +r2 +A(r) +dφ +dλ. +(2.7) +4 + +It turns out that certain vectors are needed, being the unit tangential vector of the light ray +curve Ki and the radial vector Rj. They are defined as follows +Ki += +dxi +dλ = bA(r) +r2 +� +� dr +dφ, 0, . . . , 0 +� �� � +d-3 +, 1 +� +� , +(2.8) +Rj += +� +� +1 +√γrr +, 0, . . . , 0 +� �� � +d-2 +� +� . +(2.9) +Considering the plane coordinated by (r, φ), the angle between the light rays and the radial +direction can be extracted from the following relation +cos Ψ ≡ γijKiRj. +(2.10) +The orbit equation can be obtained from the unity of the tangential vector as +Fd(r) = +� dr +dφ +�2 += +r4 +b2A(r)B(r) − +r2 +B(r). +(2.11) +In the plane (r, φ), the calculations are reduced to the ones of the spherically symmetric +black hole in four dimensions. Considering the observer (R) and the source (S) at finite +distances, one can define the deflection angle of the light rays as follows +Θd = ΨR − ΨS + φRS +(2.12) +where φRS is the separation angle. Putting u = 1 +r in Eq (2.11), this optical angle can be +rewritten as +φRS += +� u0 +uS +1 +� +Fd(u) +du + +� u0 +uR +1 +� +Fd(u) +du +(2.13) +where uS and uR are the inverse of the source and the observer distance from the black +hole. It is denoted that u0 is the inverse of the closest approach r0 linked to the impact +parameter via the constraint F(u0) = 0. From the above formalism, we observe that A(r) +and B(r) functions encode all contribution effects including the dimension of the black hole +space-time. To visualize such effects, we investigate certain higher dimensional models. +3 +Light deviating behaviors of the black holes in M- +theory compactification +In this section, we study the deflection angle of the right rays around black holes in M-theory +living in 11 dimensions. At lower energy limits, this theory is described by 11-dimensional +supergravity with M2 and M5-branes [40]. Many black holes in such a theory have been +constructed [40–42]. Recently, the thermodynamics and the shadow optical properties have +5 + +been investigated by considering D−dimensional M-theory inspired models where the asso- +ciated d-dimensional AdS black holes have been embedded. It has been assumed that the +AdS models are supposed to be dual versions of (d − 1) dimensional CFT living on the as- +sociated boundaries, which could be modeled in terms of N coincides (d − 2) branes. It has +been suggested that these AdS black holes can be obtained from the compactification of the +D-dimensional theories on the real spheres. According to [43], the associated near horizon +geometries take the forms AdSd × Sd+k where the dimension D is constrained by +D = 2d + k, +(3.1) +with k is an integer indicating the dimension of the internal spherical spaces. In this way, the +higher dimensional theories are indexed by a triplet (D, d, k). For the non-rotating solutions, +the line element of the d-dimensional AdS black holes in such a theory is given by +ds2 += +−A(r)dt2 + dr2 +A(r) + r2d2Ωd−2, +(3.2) +where one has used the following metric function identification +B(r) = +1 +A(r). +(3.3) +An inspection reveals that A(r) is a relevant metric function encoding data on the involved +black hole parameters. In such (D, d, k) models, it takes the following form +A(r) = 1 − m +rd−3 + r2 +L2 AdS. +(3.4) +In this equation, m is a physical parameter related to the black hole mass via the relation +M = (d − 2)Ωd−2m +16πGd +. +(3.5) +where Gd and Ωd+k are given by +Gd = +ℓ2(d−1)+k +p +ℏΩd+KLd+K +AdS +, +Ωd+k = 2π(d+k+1)/2 +Γ( d+k+1 +2 +) . +(3.6) +It is denoted that LAdS is the radius of the AdS space related to the number of the M −(d−2) +branes as follows +L2(d−1)+k +AdS += 2−( +d(4−d)+3 +2 +) π7(k+2(d−5))−4 N +d−1 +2 ℓ 2(d−1)+k +p +. +(3.7) +where ℓp denotes the Planck length. To get a concrete solution derived from such M-theory +inspired models, the triplet (D, d, k) should be specified. +Phase transitions and shadow +behaviors of such models have been investigated in [34, 43, 44]. +After that, an universal +criticality of the thermodynamic geometry of the (D, d, k) models has been studied [35]. +Motivated by such activities, we examine the light behaviors around AdS black holes for +6 + +such (D, d, k) models.. We first deal with the corresponding deflection angle of the light +rays. For the sake of simplicities, we will limit our analysis to triplets associated with M- +theory. We expect that the calculations could be adopted to generic (D, d, k) models [45]. +However, the computations may need a deeper numerical computations. At the end of this +work, the light trajectories will be treated in section 4. +In the present section, however, we focus on the deflection angle of the light rays by +4-dimensional and 7-dimensional AdS black holes embedded in 11-dimensional M-theory +associated with the triplets (11, 4, 3) and (11, 7, −3), respectively. +3.1 +Light ray deflection in the (11, 4, 3) model +We start by the non-rotating solutions corresponding to the triplet (11, 4, 3) obtianed from +M-theory compactified on S7 with N coincide M2-branes. Using Eq( 3.5) and Eq( 3.7), we +get the metric function in terms of the M2-brane number which reads as +A(r) = 1 − 192 2 +1 +6π +2 +3ℓ2 +pM +N +7 +6r ++ +2 +1 +3r2 +π +2 +3ℓ2 +pN +1 +3 , +(3.8) +where one has used +LAdS = 2− 1 +6π +1 +3N +1 +3ℓp, +m = 192 × 21/6 π2/3 ℓ2 +p M +N 7/6 +. +(3.9) +The calculation will be expended to be in the leading order of m. The needed radial equation +in four dimensions is given by +F4(u) = 1 +b2 − +2 +1 +3 +π +2 +3ℓ2 +pN +1 +3 − u2 + 192 2 +1 +6π +2 +3ℓ2 +pM +N +7 +6 +u3. +(3.10) +Using the above equations, the angle φRS can be written as follows +φRS = (π − arcsin(buR) − arcsin(buS)) − +� +uR +� +1 − b2u2 +R ++ +uS +� +1 − b2u2 +S +� +b3 +(2π) +2 +3ℓ2 +pN +1 +3 ++ +� +2 − b2u2 +R +� +1 − b2u2 +R ++ +2 − b2u2 +S +� +1 − b2u2 +S +� +96 2 +1 +6π +2 +3ℓ2 +pM +bN +7 +6 ++ +� +3b2u2 +R − 2 +(1 − b2u2 +R) 3/2 + +3b2u2 +S − 2 +(1 − b2u2 +S) 3/2 +� 48 +√ +2bM +N 3/2 +. +(3.11) +According to Eq(2.10), the ΨR − ΨS term can be expressed as +ΨR − ΨS = (arcsin(buR) + arcsin(buS) − π) + +� +1 +uR +� +1 − b2u2 +R ++ +1 +uS +� +1 − b2u2 +S +� +b +(2π) +2 +3ℓ2 +pN +1 +3 +− +� +u2 +R +� +1 − b2u2 +R ++ +u2 +S +� +1 − b2u2 +S +� +96 2 +1 +6π +2 +3ℓ2 +pbM +N +7 +6 ++ +� +1 − 2b2u2 +R +(1 − b2u2 +R) 3/2 + +1 − 2b2u2 +S +(1 − b2u2 +S) 3/2 +� 48 +√ +2bM +N +3 +2 +. +(3.12) +7 + +Combining the above equations, we can obtain the deflection angle in terms of the M2-brane +number N and the black hole mass. this is found to be +Θ4 = +�� +1 − b2u2 +R +uR ++ +� +1 − b2u2 +S +uS +� +b +(2π) +2 +3ℓ2 +pN +1 +3 + +�� +1 − b2u2 +R + +� +1 − b2u2 +S +� 192π +2 +32 +1 +6ℓ2 +pM +bN +7 +6 +− +� +1 +� +1 − b2u2 +R ++ +1 +� +1 − b2u2 +S +� +48 +√ +2bM +N +3 +2 +. +(3.13) +It has been observed that the above expression diverges by taking the limits buS → 0 and +buR → 0. This is due to the fact that the spacetime is not asymptotically flat. Hence, the +finite deflection angle of the light rays by AdS black holes from M-theory takes the form +Θ4 ∼ +b +(2π) +2 +3ℓ2 +pN +1 +3 +� 1 +uR ++ 1 +uS +� ++ 384 2 +1 +6π +2 +3ℓ2 +pM +bN +7 +6 +− 96 +√ +2bM +N +3 +2 +. +(3.14) +In Fig.(1), we illustrate the M2-brane effect on such a deflection angle. In the left panel of +this figure, we present the variation of the deflection angle in terms of the impact parameter +by taking different values of N. Examining such AdS black holes, we show that the deflection +angle of the light rays decreases for small values of the impact parameter then it becomes +an increasing function. It has been observed from the left and the right panels of Fig.(1) +that the M2-brane number decreases the deflection angle. In the right panel, we consider +two values of the impact parameter. Plotting the deflection angle in terms of the M2-brane +number, the two curves meet a particular point, where the deflection angle of the AdS space +changes the behavior from a decreasing to an increasing function of the impact parameter. +A close examination shows that the behavior of the deflection angle of the AdS-Shwarzchild +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +100 +150 +200 +250 +b +θ4 +N +30 +32 +34 +36 +38 +40 +b=0.1 +b=0.6 +0 +50 +100 +150 +200 +50 +100 +150 +200 +250 +N +θ4 +Figure 1: Right panel: Variation of the deflection angle of 4-dimensional black holes in M-theory in terms of +the impact parameter for different values of N. Lift panel: Variation of the deflection angle of 4-dimensional +black holes in M-theory in terms of the brane number for b = 0.1 and b = 0.6 +black holes for M = 1 and L = 3 is similar to the present one for N ≃ 147. +It seems possible to extend this analysis of the deflection angle behaviors in M-theory +by introducing the rotating parameter a. According [46], indeed, the metric line element +8 + +becomes +ds2 = −∆r +W +� +dt − a +Ξ sin2 θdφ +�2 ++ W +�dr2 +∆r ++ dθ2 +∆θ +� ++ ∆θ sin2 θ +W +� +adt − r2 + a2 +Ξ +dφ +�2 +. (3.15) +The involved terms are given by +∆r = r2 − mr + a2 + r2 +L2(r2 + a2), +∆θ = 1 − a2 +L2 cos2 θ, +(3.16) +Ξ = 1 − a2 +L2, +W = r2 + a2 cos2 θ +(3.17) +where a is the rotating parameter. In this way, the computation will be expanded to the +first order of m and a. To compute the Ψ and φRS angles for such a rotating black hole +in four dimensions, we follow the method developed in [36], since we have the same metric +form. Considering the equatorial plane, we can elaborate the orbit equation in terms of the +M2-brane number N and the rotating parameter a. In this way, it is found to be +F4(a, u) = 1 +b2−u2− +3√ +2 +π2/3 3√ +Nℓ2 +p ++192 +6√ +2π2/3Mu3ℓ2 +p +N 7/6 ++ +2 +3√ +2a +π2/3b3 3√ +Nu2ℓ2 +p +−384 +6√ +2π2/3aMuℓ2 +p +b3N 7/6 ++768 +√ +2aM +b3N 3/2u . +(3.18) +The longitudinal angle can be expressed as follows +φRS += (π − arcsin (buR) − arcsin (buS)) − +� +uR +� +1 − b2u2 +R ++ +uS +� +1 − b2u2 +S +� +b3 +(2π)2/3 3√ +Nℓ2 +p +− +� +1 − 2b2u2 +R +uR +� +1 − b2u2 +R ++ +1 − 2b2u2 +S +uS +� +1 − b2u2 +S +� +3√ +2a +π2/3 3√ +Nℓ2 +p ++ +� +2 − b2u2 +R +� +1 − b2u2 +R ++ +2 − b2u2 +S +� +1 − b2u2 +S +� +96 +6√ +2π2/3Mℓ2 +p +bN 7/6 +− +� +1 +� +1 − b2u2 +R ++ +1 +� +1 − b2u2 +S +� +192 +6√ +2π2/3aMℓ2 +p +b2N 7/6 +− +� +2 − 3b2u2 +R +(1 − b2u2 +R) 3/2 + +2 − 3b2u2 +S +(1 − b2u2 +S) 3/2 +� 48 +√ +2bM +N 3/2 +. +Using the previous computations, we get the Ψ angle +ΨRS += +(arcsin (buR) + arcsin (buS) − π) + +� +1 +uR +� +1 − b2u2 +R ++ +1 +uS +� +1 − b2u2 +S +� +b +(2π)2/3 3√ +Nℓ2 +p +− +� +u2 +R +� +1 − b2u2 +R ++ +u2 +S +� +1 − b2u2 +S +� +96 +6√ +2π2/3bMℓ2 +p +N 7/6 ++ +� +1 − 2b2u2 +R +(1 − b2u2 +R) 3/2 + +1 − 2b2u2 +S +(1 − b2u2 +S) 3/2 +� 48 +√ +2bM +N 3/2 +. +Combining the obtained expressions, we can obtain the deflection angle of four dimensional +rotating AdS black holes from M-theory in terms of the involved parameters. The compu- +9 + +tations give +Θ4(a) += +� +1 − b2u2 +R +uR +� +1 − b2u2 +R ++ +1 − b2u2 +S +uS +� +1 − b2u2 +S +� +b +(2π)2/3 3√ +Nℓ2 +p +− +� +1 − 2b2u2 +R +uR +� +1 − b2u2 +R ++ +1 − 2b2u2 +S +uS +� +1 − b2u2 +S +� +3√ +2a +π2/3 3√ +Nℓ2 +p +− +� +1 +� +1 − b2u2 +R ++ +1 +� +1 − b2u2 +S +� +84 +√ +2bM +N 3/2 ++ +�� +1 − b2u2 +R + +� +1 − b2u2 +S +� 192 +6√ +2π2/3Mℓ2 +p +bN 7/6 +− +� +1 +� +1 − b2u2 +R ++ +1 +� +1 − b2u2 +S +� +192 +6√ +2π2/3aMℓ2 +p +b2N 7/6 +. +(3.19) +Taking a finite distance limit by sending buR and buS to 0, we get the reduced expression of +the deflection angle +Θ4(a) +∼ +−96 +√ +2bM +N 3/2 ++ +� 1 +uR ++ 1 +uS +� +b +(2π)2/3 3√ +Nℓ2 +p ++ 384 +6√ +2π2/3Mℓ2 +p +bN 7/6 +− +� 1 +uR ++ 1 +uS +� +3√ +2a +π2/3 3√ +Nℓ2 +p +− +384 +6√ +2π2/3aMℓ2 +p +b2N 7/6 +. +(3.20) +Putting a = 0, we recover the deflection angle of four dimensional non-rotating black hole +in M-theory given in Eq.(3.14). +To examine the effect of the M2-brane number for the rotating black holes, we vary the +involved parameters. Taking two values of the rotating parameter a, we plot in the top of +the left and the right panels of Fig.(2) the variation of the deflection angle in terms of the +impact parameter b by varying the M2-brane number from 20 to 30. The M2-brane number +parameter still decreases the deflection angle of the light rays. In the left one, where we have +a smaller value of a, the deflection angle of the light rays decreases for small values of b, and +then it becomes an increasing function of the impact parameter b. In the right one, where the +contribution of the rotating parameter is relevant, the deflection angle is only an increasing +function of b without any minimum values. In the bottom of Fig.(2), we take a small (left +side) and a large value (right side) of the M2-brane number and vary the rotating parameter +a from 0.1 to 0.9. This shows that the rotating parameter a still decreases the deflection +angle as expected. For generic values of the M2-brane number, the deflection angle decreases +by increasing the rotating parameter a. Near b = 2, the deflection angle behaviors depend on +the M2-brane number. For generic values of a, the deflection angle increases by decreasing +N. Similar behaviors have been obtained in the previous results. The only difference around +b = 2 is the linear behavior for large values of the M2-brane number. It has been remarked +that these optical behaviors could be related to the AdS spacetime backgrounds [20,21,24]. +3.2 +Light deflection behaviors in the (11, 7, −3) model +Here, we consider the 7-dimensional AdS black holes by considering the triplet (11, 7, −3). +This model can be obtained from the compactification of M-theory on four dimensional real +10 + +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +30 +35 +40 +45 +50 +55 +60 +65 +b +θ +M=1, a=0.1 +N +20 +22 +24 +26 +28 +30 +2 +3 +4 +5 +6 +0 +20 +40 +60 +80 +b +θ +M=1, a=0.8 +N +20 +22 +24 +26 +28 +30 +2 +4 +6 +8 +10 +0 +50 +100 +150 +200 +b +θ +M=1, N=10 +a +0.2 +0.4 +0.6 +0.8 +2 +4 +6 +8 +10 +0 +20 +40 +60 +80 +100 +120 +b +θ +M=1, N=100 +a +0.2 +0.4 +0.6 +0.8 +Figure 2: Variation of the deflection angle in terms of the impact parameter by varying a and N. +sphere S4 in the presence of M5-branes. The corresponding metric function in terms of the +M5-brane number is expressed as follows +A(r) = 1 + +r2 +4π2/3N 2/3ℓ2 +p +− 6π5/3Mℓ3 +p +5N 4/3r4 . +(3.21) +Using the orbital equation Eq(2.11), we obtain +F7(u) = 1 +b2 − +1 +4π2/3N 2/3ℓ2 +p +− u2 + 6π5/3Mu6ℓ3 +p +5N 4/3 +. +(3.22) +By the help of this finding, we get the φRS term +φRS = (π − arcsin(buR) − arcsin(buS)) − +� +uR +� +1 − b2u2 +R ++ +uS +� +1 − b2u2 +S +� +b3 +8π2/3N 2/3ℓ2 +p ++ +� +−2b5u5 +R − 5b3u3 +R + 15buR +� +1 − b2u2 +R ++ −2b5u5 +S − 5b3u3 +S + 15buS +� +1 − b2u2 +S +� +3π5/3Mℓ3 +p +40b4N 4/3 +− +�buR (3b4u4 +R − 20b2u2 +R + 15) +(1 − b2u2 +R) 3/2 ++ buS (3b4u4 +S − 20b2u2 +S + 15) +(1 − b2u2 +S) 3/2 +� 3πMℓp +80b2N 2 ++ (π − arcsin(buR) − arcsin(buS)) +� +9π5/3Mℓ3 +p +8b4N 4/3 − 9πMℓp +16b2N 2 +� +. +(3.23) +11 + +For the Ψ part of these black holes, we find the following expression +ΨR − ΨS = (arcsin(buR) + arcsin(buS) − π) + +� +1 +uR +� +1 − b2u2 +R ++ +1 +uS +� +1 − b2u2 +S +� +b +8π2/3N 2/3ℓ2 +p +− +� +u5 +R +� +1 − b2u2 +R ++ +u5 +S +� +1 − b2u2 +S +� +3π5/3bMℓ3 +p +5N 4/3 +− +�u3 +R (2b2u2 +R − 1) +(1 − b2u2 +R) 3/2 + u3 +S (2b2u2 +S − 1) +(1 − b2u2 +S) 3/2 +� 3πbMℓp +40N 2 . +(3.24) +Combining Eq(3.23) and Eq(3.24), we obtain the deflection angle expression +Θ7 = +�� +1 − b2u2 +R +uR ++ +� +1 − b2u2 +S +uS +� +b +8π2/3N 2/3ℓ2 +p +− +� +buR (2b4u4 +R + b2u2 +R − 3) +� +1 − b2u2 +R ++ buS (2b4u4 +S + b2u2 +S − 3) +� +1 − b2u2 +S +� +3π5/3Mℓ3 +p +40b4N 4/3 ++ +� +buR (7b2u2 +R − 15) +� +1 − b2u2 +R ++ buS (7b2u2 +S − 15) +� +1 − b2u2 +S +� +3πMℓp +80b2N 2 ++ (π − arcsin(buR) − arcsin(buS)) +� +9π5/3Mℓ3 +p +8b4N 4/3 − 9πMℓp +16b2N 2 +� +. +(3.25) +Taking the finite distance limits by sending buS and buR to zero, this deflection angle could +be approximated by the following form +Θ7 ∼ 9π8/3Mℓ3 +p +8b4N 4/3 − 9π2Mℓp +16b2N 2 + +b +8π2/3N 2/3ℓ2 +p +� 1 +uR ++ 1 +uS +� +. +(3.26) +0.5 +1.0 +1.5 +2.0 +10 +15 +20 +25 +30 +35 +b +θ7 +N +30 +32 +34 +36 +38 +40 +b=0.5 +b=1 +0 +20 +40 +60 +80 +100 +10 +20 +30 +40 +N +θ7 +Figure 3: Right panel: Variation of the deflection angle of non rotating 7-dimensional black holes in M- +theory in terms of the impact parameter for different values of N. Left panel: Variation of the deflection +angle of 7-dimensional black holes in M-theory in terms of the brane number for b = 0.5 and b = 1. +12 + +To examine the associated behaviors, we plot in the left panel of Fig.(3) the variation of +the deflection angle of the light rays by a 7-dimensional black hole in M-theory in terms of +the impact parameter by varying the M5-brane number. Similar to the four-dimensional +case, the deflection angle of the light rays decreases for small values of the impact parameter +to a critical value and then it becomes an increasing function. An examination of the right +panel of Fig.(3) reveals that when we increase the number of M2-branes the deflection angle +decreases. The critical transition behavior of the deflection angle from an increasing function +to a decreasing one is illustrated in the right panel of Fig.(3) by the intersection point of the +two curves. +To compare the effect of the dimension on the variation of the deflection angle, we plot +in the Fig.(4) the variation of the deflection angle in terms of the impact parameter for +(11, 4, 3) and (11, 7, −3) models by fixing N. For small values of the impact parameter, the +θ4 +θ7 +0.5 +1.0 +1.5 +2.0 +0 +200 +400 +600 +800 +1000 +b +θ +Figure 4: The variation of the deflection angle in terms of the impact parameter of four and seven dimen- +sional non rotating black holes by taking N = 10. +four-dimensional black hole bends the light rays larger than the seven-dimensional black +hole. However, this behavior is inverted for large values of b. +Having discussed the behaviors of the deflection angle of the light rays casted by AdS black +holes in M-theory in the presence of M2 and M5-branes, we move to investigate the involved +trajectories. This has been motivated from the fact that the light trajectories of the black +holes rely on the orbit equation exploited in the deflection angle computations. The study +of the light trajectories around the black holes for M2 and M5-brane models could confirm +the behavior of the deflection angle obtained in the previous subsections. +4 +Light trajectories around black holes in M-theory +In this section, we study the trajectory of the light rays by the black holes in M-theory +scenarios. Concretely, we study the light trajectories near the AdS black holes of (D, d, +k) models by varying the M(d − 2)-brane number. It is denoted that the light trajectories +around black holes can be generally approached via the numerical computations adopted to +the Eq.(3.10) and Eq.(3.22). Concretely, we can solve φ with respect to u in order to depict +13 + +the behaviors of the light rays around the involved black holes. To establish such trajectories, +we need to identify the regions corresponding to the light ray trajectory possibilities. These +regions can be determined by the help of the effective potential which is expressed as follows +V eff +d +(r) = − +� dr +dλ +�2 +. +(4.1) +This can be rewritten as +V eff +d +(r) = −Fd(r) +�bA(r) +r2 +�2 +. +(4.2) +For simplicity reasons, we restrict ourselves to the special models embedded in 11-dimensional +supergravity limits of M-theory. +4.1 +Trajectories of the light rays in the (11, 4, 3) model +We start by considering the (11, 4, 3) model developed in [34, 35]. Evincing the rotating +parameter, the four dimensional effective potential, in the presence of the M2-branes, takes +the following form +V eff +4 +(r) = L2 +r2 +� +1 − 192 2 +1 +6π +2 +3ℓ2 +pM +N +7 +6r ++ +2 +1 +3r2 +π +2 +3ℓ2 +pN +1 +3 +� +− E2. +(4.3) +This effective potential will be illustrated as a function of the radial coordinate r for different +values of the M2-brane number N by taking ℓp = 1 and M = 1. According to [34], the +maximum value of the shadow radius corresponds to N = 80. In Fig.(5), we plot such a +potential for two M2-brane number values being N = 100 and N = 80. +5 +10 +15 +20 +0.140 +0.145 +0.150 +0.155 +0.160 +5 +10 +15 +20 +0.13 +0.14 +0.15 +0.16 +0.17 +N = 80 +N = 100 +Region 3 +b > bsp +Region 2 +bsp = 2.5364 +Region 1 +b < bsp +Region 3 +b > bsp +Region 2 +bsp = 2.5101 +Region 1 +b < bsp +rsp = 3.218 +rsp = 4.175 +Veff +4 (r) +Veff +4 (r) +r +r +Figure 5: +The effective potential behaviors of 4-dimensional AdS black holes embedded in 11-dimensional +M-theory by varying taking to values of the brane number. +This potential increases and reaches a maximum at the photon sphere associated with bsp +which represents the impact parameter of the spinning light rays around the black holes. +14 + +This verifies the following constraint +V eff(rsp) = 1 +b2 +sp +. +(4.4) +It has been found that the two values of the M2-brane number N = 80, 100 provide two +impact parameter values bsp = 2.5364 and bsp = 2.5101 corresponding to the photon sphere +radius rsp = 4.175 and rsp = 3.218, respectively as shown in Fig.(5). Indeed, the associated +impact parameter and the photon sphere radius decrease by increasing the M2-brane number. +It has been remarked that the impact parameter value bsp provides trajectories of the light +rays in three different regions. These regions are denoted by region 1 , region 2 and region +3 corresponding to b < bsp, b = bsp and b > bsp, respectively. +In the first region 1 , the light ray falls into the black hole due to the values of the impact +parameter lower to bsp. In the third region 3 , however, the light ray near the black hole can +be reflected back. In the second region 2 , however, the light ray comes into the photon sphere +making an infinite number of turns around the black hole due to the a non vanishing angular +velocity. The associated orbit is circular and unstable. To illustrate these regions, we plot in +Fig.(6) the trajectories of the light rays in the polar coordinates (r, φ) for different values of +the M2-brane number N. To analyze the effect of the M2-brane on the light ray trajectories, +we vary the impact parameter b by using the step between two values of the impact parameter +as 1/20 for all light rays. A close examination reveals that the horizon and the sphere photon +20 +15 +10 +5 +0 +5 +10 +15 +20 +x +10 +5 +0 +5 +10 +y +20 +15 +10 +5 +0 +5 +10 +15 +20 +x +10 +5 +0 +5 +10 +y +N = 80 +N = 100 +Figure 6: The trajectories of the light ray for different values of M2-brane number. The black and the +dashed red circle is the horizon and the photon sphere of the M2-brane, respectively. +radius decrease by increasing the M2-brane number. However, the variation of the impact +parameter bsp is small by varying the M2-brane number, N. It has been observed that the +region 1 , 2 and 3 are the same for different values of the M2-brane number. Taking +two values of N, we observe that the distance between two light rays increases for an impact +parameter value closer and bigger to bsp for all regions. Indeed, this distance decreases by +increasing N. This distinction comes from the values of the angular velocity. More analysis +shows that the reflected of the light ray is more intense by decreasing the M2-brane number. +Comparing this result with many works concerning the trivial solution [47–49], we observe +15 + +a different behavior. First, bsp is almost the same by varying the N. However, rsp decreases +by increasing the M2-brane number. This is completely different than the previous results. +Second, for small values of the impact parameter, we remark that the light ray falls into the +black hole by keeping the parallel trajectories with the r-axis. However, for values close to +bsp the light ray falls into the black hole without keeping the parallel trajectory with the +r-axis. Indeed, the parallel trajectory is replaced by a critical angle between the light ray +and the r-axis. Fixing the M2-brane number, this angle increases for values near to bsp +(or bigger). Varying N, it increases by decreasing the M2-brane number. This shows that +such a M2-brane number can be considered as a relevant quantity modifying the light ray +behaviors near a black hole in M-theory compactifications. This distinction comes from the +geometry of the black holes in M-theory with brane backgrounds. +4.2 +Light trajectories in the (11, 7, −3) model +Here, we deal with the trajectory of the light rays near black holes in the presence of M5- +branes in M-theory compactifications on the four-dimensional sphere S4. For the (11, 7, −3) +model, such light behaviors can be determined with the help of the effective potential +V eff +7 +(r) = L2 +r2 +� +1 + +r2 +4π2/3N 2/3ℓ2 +p +− 6π5/3Mℓ3 +p +5N 4/3r4 +� +− E2 +(4.5) +In Fig.(7), we illustrate the associated effective potential as a function of the radial coor- +dinate r for two values of the M5-brane number being N = 1 and N = 80. Using the +2 +4 +6 +8 +10 +0.08 +0.10 +0.12 +0.14 +0.16 +1 +2 +3 +4 +5 +6 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +N = 1 +N = 80 +Region 3 +b > bsp +Region 2 +bsp = 2.7180 +Region 1 +b < bsp +Region 3 +b > bsp +Region 2 +bsp = 0.6308 +Region 1 +b < bsp +rsp = 0.515 +rsp = 2.219 +Veff +7 (r) +Veff +7 (r) +r +r +Figure 7: +The effective potential behaviors of 7-dimensional AdS black holes embedded in 11-dimensional +M-theory by varying taking to values of the brane number. +constraint given by Eq.(4.4), we can get the values of the photon sphere and the critical +impact parameter of the photon sphere associated with the maximal value of the effective +potential V eff +7 +(r). +It follows that the two values of the M5-brane number N = 1 and N = 80 provide +two impact parameter values bsp = 2.7180 and bsp = 0.6308 associated with the values of +16 + +the photon sphere rsp = 2.219 and rsp = 0.515, respectively. The corresponding impact +parameter and the photon sphere radius decrease by increasing the M5-brane number. For +the fixed value of N = 80, however, we observe that bsp and rsp in the M5-brane model is +small compared to the M2-brane model. This distinction affects the light ray trajectories +around the black holes in the M5-brane model. +Placing the observer in the equatorial +hyperplane ensured by θ1 = θ2 = π +2, we plot in Fig. (8) the trajectories of the light rays in +the polar coordinates (r,φ) for different values of the M5-brane number. +8 +6 +4 +2 +0 +2 +4 +6 +8 +x +8 +6 +4 +2 +0 +2 +4 +6 +8 +y +3 +2 +1 +0 +1 +2 +3 +x +3 +2 +1 +0 +1 +2 +3 +y +N = 1 +N = 80 +Figure 8: The trajectories of the light ray for different values of M5-brane number. The black and the +dashed red circle is the horizon and the photon sphere of the M5-brane, respectively. +Due to small values of bsp, we vary the impact parameter b by using the step between +two values of the impact parameter as 1/40 for all light rays. It has been suggested that +the M5-brane number and the extra dimension can be considered as relevant parameters +modifying the light ray behaviors near a black hole in M-theory compactifications. Such +modifications come from the black hole geometry in the presence of the M5-branes in M- +theory compacatifications. The obtained behaviors match perfectly with the previous works +[34,35]. +It has been remarked that the previous behaviors of the M2-brane model associated with +the horizon and the photon sphere radius have been conserved in the M5-brane model. +Concretely, one can show that the reflected light ray becomes more intense by deceasing the +M5-brane number. It has been remarked that these behaviors are contrary to the ones of the +M2-brane model. This is due to the extra dimension contributions of the M5-brane model +appearing in the associated metric function. Moreover, bsp and rsp vary by taking different +values of the M5-brane number. For all values of the impact parameter, we observe that if +the light ray falls into the black hole or refracted it keeps a parallel trajectory with respect +17 + +to the r-axis. This behavior is completely different than the previous results related to the +M2-brane model. Moreover, the values of the angular velocity in the M5-brane model is +different than the one in the M2-brane model. A close examination shows that the results +concerning the light trajectories behaviors confirm the results associated with the deflection +angle behavior. It has been remarked that the deflection angle increases by decreasing the +M-theory brane number N. Moreover, the deflection of the light ray is more intense by +decreasing the M-brane number. Both results inter-match perfectly. Finally, the behaviors +of the optical quantities including the light trajectories and the deflection angle of black +holes for M-theory brane models are interesting and similar. +5 +Conclusion +In this work, we have investigated the deflection angle and the trajectory of the light rays +casted by black holes in M-theory scenarios. Using the Gauss-Bonnet theorem, we have com- +puted and examined the deflection angle of the light rays around four and seven dimensional +AdS black holes derived from the M-theory compacatification on the real spheres S7 and +S4, respectively. First, we have generalized the deflection angle formalism for d-dimensional +AdS black hole solutions using the Gauss-Bonnet theorem. Then, we have studied the de- +flection angle of four dimensional rotating and non-rotating AdS black holes by examining +the M2-brane number effect for both cases. Concretely, we have shown that the deflection +angle of light rays decreases for small value of the impact parameter then it becomes an +increasing function. It has been observed that the M-theory brane number decreases the de- +flection angle. Taking two values of the impact parameter and varying N, we have observed +that the two curves meets a particular point, where the deflection angle of the AdS space +has changed the behavior from a decreasing to an increasing function in terms the impact +parameter. For four dimensional rotating model, we have revealed that the behaviors of the +deflection angle by varying the M2-brane number is similar to the non-rotating case. For +the small value of a, however, the deflection angle of the light rays decreases for small values +of b and then it becomes an increasing function of the impact parameter b. In general, the +rotating parameter is a relevant quantity decreasing the deflection angle. We have shown +that, around b = 2, the deflection angle behaviors depend on the M2-brane number. For +generic values of a, the deflection angle increases by decreasing N. Similar behaviors have +been obtained in the previous results. The only difference around b = 2 is the linear behavior +for large values of the M2-brane number. +Then, we have extended the calculations to seven dimensional non-rotating black holes by +providing a comparative study. For small values of the impact parameter, we have shown +that the four-dimensional black hole bends the light rays larger than the seven-dimensional +one. However, this behavior is inverted for large values of the impact parameter b. +Finally, we have discussed the trajectories of the lights rays around four and seven- +dimensional AdS black hole in M-theory. We have shown that the regions +1 , +2 +and +18 + +3 are the same for different values of the M2-brane number. Taking two values of N, we +have observed that the distance between two light rays increases for an impact parameter +value closer and bigger to bsp value for all regions. This distance decreases by increasing +N. Concretely, this distinction is originated from the values of the angular velocity. We +have remarked that the reflected of the light ray is more intense by decreasing the M2-brane +number. Comparing the present results with many investigations associated with the trivial +solution, we have observed a different behavior. For the M5-brane model, we have remarked +a different behavior with respect to the M2-brane model. This difference could come from +the extra dimension being a relevant parameter appearing in the metric function. +This work comes up with certain open questions. A possible project concerns generic models +associated with (D, d, k) M-theory inspired models proposed in [35]. In particular, the effect +of the M(d−2)-branes in such models could be examined by performing non trivial numerical +computations. Another issue is to approach certain M-theory compactifications using the +orbifold of spheres providing possible ways to implement G2-manifolds in such black hole +activities. We hope address these questions in future works. +Acknowledgments +The present paper is dedicated to the memory of Pr. Ahmed Intissar. This work is partially +supported by the ICTP through AF. +References +[1] S. W. Hawking, Black hole explosions, Nature, 248 (1974) 30. +[2] J. L. Synge Relativity: The General Theory. North Holland, Amsterdam, 1960). +[3] R. Emparan, and H. S. 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D 100, (2019) 024018, arXiv: +1906.00873. +22 + diff --git a/a9E_T4oBgHgl3EQfzByO/content/tmp_files/load_file.txt b/a9E_T4oBgHgl3EQfzByO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..67cfce01d6722bb09fcbcd7231c34c5253df6d2a --- /dev/null +++ b/a9E_T4oBgHgl3EQfzByO/content/tmp_files/load_file.txt @@ -0,0 +1,1008 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf,len=1007 +page_content='Light Behaviors around Black Holes in M-theory N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Askour1,2∗, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Belhaj, ID 3†, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Belmahi ID 3‡, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Benali ID 3§, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' El Moumni ID 4¶, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Sekhmani3‖∗∗ 1 Department of Mathematics, Sultan Moulay Slimane University, Faculty of Sciences and Technics B´eni Mellal, BP 523, 23000, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 2 Department of Mathematics, Mohammed V University in Rabat, Faculty of Sciences Rabat, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Box 1014, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 3 D´epartement de Physique, Equipe des Sciences de la mati`ere et du rayonnement, ESMaR Facult´e des Sciences, Universit´e Mohammed V de Rabat, Rabat, Morocco 4 LPTHE, D´epartement de Physique, Facult´e des Sciences, Universit´e Ibn Zohr, Agadir, Morocco January 23, 2023 Abstract We study the deflection angle and the trajectory of the light rays around black holes in M-theory scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Using the Gauss-Bonnet theorem, we first compute and examine the deflection angle of the light rays near four and seven dimensional AdS black holes obtained from the M-theory compactifications on the real spheres on S7 and S4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We discuss the effect of the M-theory brane number and the rotating parameter on such an optical quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We then investigate the trajectories of the light rays using the equation of motion associated with M2 and M5-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' ∗askour2a@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='com †a-belhaj@um5r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ma ‡hajar belmahi@um5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ma §mohamed benali4@um5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ma ¶h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='elmoumni@uiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ma ‖yassine sekhmani@um5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='ma ∗∗Authors in alphabetical order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='08321v1 [hep-th] 19 Jan 2023 Contents 1 Introduction 2 2 Deflection angle formalism for d-dimensional AdS black holes 4 3 Light deviating behaviors of the black holes in M-theory compactification 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 Light ray deflection in the (11, 4, 3) model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 Light deflection behaviors in the (11, 7, −3) model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 10 4 Light trajectories around black holes in M-theory 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 Trajectories of the light rays in the (11, 4, 3) model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 Light trajectories in the (11, 7, −3) model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 16 5 Conclusion 18 1 Introduction Black hole physics has received a remarkable interest encouraged by theoretical and observa- tional findings [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' These investigations have opened new gates, which could be exploited to understand gravity models from nontrivial theories including supergravity in higher dimen- sions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Concerning the observational results, the detection of the gravitational waves has been considered as a strong existence of the black hole objects [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' These interesting observa- tions are followed by the first image of the black hole shadows in M87∗ galaxy provided by the Event Horizon Telescope (EHT) [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Motivated by such activities, the thermodynamic and the optical properties of certain black holes have been extensively investigated using different methods based on analytic and numerical computations [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Various theories have been exploited including the compactification of the superstrings and M-theory on the spherical internal compact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Inspired by the AdS/CFT correspondence, it has been shown that the associated black holes exhibit interesting thermodynamic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Precisely, the black holes on AdS geometries behave like Van der Wall fluid systems [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Interpreting the cosmological constant as a pressure, many phase transitions have been examined showing cer- tain critical and universal aspects [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' These thermodynamical investigations have been bridged with optical ones with the help of two essential concepts [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The first treated concept is the deflection angle of the light rays near black holes [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Concretely, Gibbons and Werner proposed a direct way to calculate this optical quantity by exploiting certain results of the Gauss-Bonnet theorem using the optical metric analysis [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Based on such a method, the weak field limits of the deflection angle of the light rays around various black holes have been investigated [20–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Moreover, the strong field deflection angle has received a large interest since it has been considered as a powerful tool to make contact with exper- imental measurements [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The second one concerns the shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In four dimensions, 2 this optical aspect has been approached in terms of one-dimensional real curves [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For the non-rotating solutions, these curves are identified with perfect circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' These geometri- cal configurations have been distorted by adding the rotating parameter [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In higher dimensions, new geometrical configurations have been obtained where the shadows involve cardioid shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Concretely, the shadow behaviors of the five-dimensional (5D) black holes embedded in type IIB superstring/supergravity-inspired spacetimes have been dealt with by considering solutions with and without rotations [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The geometrical properties have been analyzed in terms of the D3-brane number and the rotation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been shown that the shadows shapes are distorted by such parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' More precisely, the size of the shadows decreases and gets deformed with the M-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Similar behaviors have been observed in four-dimensional black holes embedded in M-theory inspired models [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been revealed that the M2-brane number can control the circular shadow sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The ge- ometrical behaviors are distorted for rotating solutions exhibiting cardioid shapes in certain moduli space regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Possible connections with observations (from Event Horizon Telescope or future devices) from a particular M-theory compactification have been proposed by de- riving certain constraints on the M2-brane number in the light of the M87∗ observational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Generalizing the study to d-dimensional black holes surrounded by dark energy (DE), embedded in D dimensional M-theory/superstring inspired models, the shadow ge- ometries have been approached in terms of the M(d − 2)-brane number in the presence of DE [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In this work, we investigate the deflection angle and the trajectory of the light rays around the black holes in M-theory spherical compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Using the Gauss-Bonnet theorem, we first compute and examine the deflection angle of the light rays by four and seven- dimensional AdS black holes obtained from the M-theory compactification on the real spheres S7 and S4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We inspect the effect of the M-theory brane number and the rotating parameter on such an optical quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We then study the trajectories of the light rays using the equation of motion corresponding to M2 and M5-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In section 2, we generalize the deflection angle formal- ism for d-dimensional dealing with non-rotating AdS black holes using the Gauss-Bonnet theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In section 3, we investigate the deflection angle of four-dimensional non-rotating and rotating AdS black holes by examining the M2-brane number effect in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We extend the computation to seven-dimensional non-rotating black holes by providing a com- parative study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In section 4, we study the trajectory of the light rays around four and seven-dimensional AdS black holes in the M-theory compactification on the real spheres S7 and S4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The last section concerns concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 3 2 Deflection angle formalism for d-dimensional AdS black holes A close examination, in the recent theoretical results, shows many interesting optical proper- ties concerning black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Certain investigations have been extensively encouraged by EHT collaboration using different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In particular, the shadows and the light deflection of the light rays have been examined providing interesting results in certain four dimensional gravity theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For such reasons, we give the crucial formalism used to compute the deflec- tion angle of the light rays around d-dimensional AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Such an optical quantity has been extensively investigated using various methods and approximations [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The most powerful results have been derived from the Gauss-Bonnet theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The calculation of the light deflection angle by d-dimensional non-rotating AdS black holes can be established using the method developed in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' To start, we consider the ansatz metric ds2 = −A(r)dt2 + B(r)dr2 + r2dΩ2 d−2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1) where dΩ2 d−2 ≡ dθ2 + �d−3 k=1 sin θ sin φk represents the metric of the (d − 2)-real dimensional unite sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' θ and φk are the spheric local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' A(r) and B(r) are radial functions specified letter one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The optical metric of the light rays γij can be determined from the null condition, which gives dt2 = γijdxidxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2) Such an optical metric can be expressed as dt2 = B(r) A(r)dr2 + r2 A(r)dΩ2 d−2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='3) providing a (d − 1)dimensional Riemannian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In this space, we can calculate the devi- ation of the spatial curve by choosing an equatorial plane constrained by θ = π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This plane is defined by the radial coordinate and one periodic coordinate of {φk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The remaining ones should be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' φ is considered as the only non fixed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Before going ahead, certain con- served quantities are needed including the energy E and the angular momentum L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' These conserved quantities are given by E = A(r) dt dλ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='4) L = r2dφ dλ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5) where λ is the affine parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' They can be combined to define the impact parameter b of the motion b = L E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6) The computations provide b = r2 A(r) dφ dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='7) 4 It turns out that certain vectors are needed, being the unit tangential vector of the light ray curve Ki and the radial vector Rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' They are defined as follows Ki = dxi dλ = bA(r) r2 � � dr dφ, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' , 0 � �� � d-3 , 1 � � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='8) Rj = � � 1 √γrr , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' , 0 � �� � d-2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='9) Considering the plane coordinated by (r, φ), the angle between the light rays and the radial direction can be extracted from the following relation cos Ψ ≡ γijKiRj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='10) The orbit equation can be obtained from the unity of the tangential vector as Fd(r) = � dr dφ �2 = r4 b2A(r)B(r) − r2 B(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='11) In the plane (r, φ), the calculations are reduced to the ones of the spherically symmetric black hole in four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Considering the observer (R) and the source (S) at finite distances, one can define the deflection angle of the light rays as follows Θd = ΨR − ΨS + φRS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='12) where φRS is the separation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Putting u = 1 r in Eq (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='11), this optical angle can be rewritten as φRS = � u0 uS 1 � Fd(u) du + � u0 uR 1 � Fd(u) du (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='13) where uS and uR are the inverse of the source and the observer distance from the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It is denoted that u0 is the inverse of the closest approach r0 linked to the impact parameter via the constraint F(u0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' From the above formalism, we observe that A(r) and B(r) functions encode all contribution effects including the dimension of the black hole space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' To visualize such effects, we investigate certain higher dimensional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 3 Light deviating behaviors of the black holes in M- theory compactification In this section, we study the deflection angle of the right rays around black holes in M-theory living in 11 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' At lower energy limits, this theory is described by 11-dimensional supergravity with M2 and M5-branes [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Many black holes in such a theory have been constructed [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Recently, the thermodynamics and the shadow optical properties have 5 been investigated by considering D−dimensional M-theory inspired models where the asso- ciated d-dimensional AdS black holes have been embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been assumed that the AdS models are supposed to be dual versions of (d − 1) dimensional CFT living on the as- sociated boundaries, which could be modeled in terms of N coincides (d − 2) branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been suggested that these AdS black holes can be obtained from the compactification of the D-dimensional theories on the real spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' According to [43], the associated near horizon geometries take the forms AdSd × Sd+k where the dimension D is constrained by D = 2d + k, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1) with k is an integer indicating the dimension of the internal spherical spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In this way, the higher dimensional theories are indexed by a triplet (D, d, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For the non-rotating solutions, the line element of the d-dimensional AdS black holes in such a theory is given by ds2 = −A(r)dt2 + dr2 A(r) + r2d2Ωd−2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2) where one has used the following metric function identification B(r) = 1 A(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='3) An inspection reveals that A(r) is a relevant metric function encoding data on the involved black hole parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In such (D, d, k) models, it takes the following form A(r) = 1 − m rd−3 + r2 L2 AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='4) In this equation, m is a physical parameter related to the black hole mass via the relation M = (d − 2)Ωd−2m 16πGd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5) where Gd and Ωd+k are given by Gd = ℓ2(d−1)+k p ℏΩd+KLd+K AdS , Ωd+k = 2π(d+k+1)/2 Γ( d+k+1 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6) It is denoted that LAdS is the radius of the AdS space related to the number of the M −(d−2) branes as follows L2(d−1)+k AdS = 2−( d(4−d)+3 2 ) π7(k+2(d−5))−4 N d−1 2 ℓ 2(d−1)+k p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='7) where ℓp denotes the Planck length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' To get a concrete solution derived from such M-theory inspired models, the triplet (D, d, k) should be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Phase transitions and shadow behaviors of such models have been investigated in [34, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' After that, an universal criticality of the thermodynamic geometry of the (D, d, k) models has been studied [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Motivated by such activities, we examine the light behaviors around AdS black holes for 6 such (D, d, k) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='. We first deal with the corresponding deflection angle of the light rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For the sake of simplicities, we will limit our analysis to triplets associated with M- theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We expect that the calculations could be adopted to generic (D, d, k) models [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' However, the computations may need a deeper numerical computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' At the end of this work, the light trajectories will be treated in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In the present section, however, we focus on the deflection angle of the light rays by 4-dimensional and 7-dimensional AdS black holes embedded in 11-dimensional M-theory associated with the triplets (11, 4, 3) and (11, 7, −3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 Light ray deflection in the (11, 4, 3) model We start by the non-rotating solutions corresponding to the triplet (11, 4, 3) obtianed from M-theory compactified on S7 with N coincide M2-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Using Eq( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5) and Eq( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='7), we get the metric function in terms of the M2-brane number which reads as A(r) = 1 − 192 2 1 6π 2 3ℓ2 pM N 7 6r + 2 1 3r2 π 2 3ℓ2 pN 1 3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='8) where one has used LAdS = 2− 1 6π 1 3N 1 3ℓp, m = 192 × 21/6 π2/3 ℓ2 p M N 7/6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='9) The calculation will be expended to be in the leading order of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The needed radial equation in four dimensions is given by F4(u) = 1 b2 − 2 1 3 π 2 3ℓ2 pN 1 3 − u2 + 192 2 1 6π 2 3ℓ2 pM N 7 6 u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='10) Using the above equations, the angle φRS can be written as follows φRS = (π − arcsin(buR) − arcsin(buS)) − � uR � 1 − b2u2 R + uS � 1 − b2u2 S � b3 (2π) 2 3ℓ2 pN 1 3 + � 2 − b2u2 R � 1 − b2u2 R + 2 − b2u2 S � 1 − b2u2 S � 96 2 1 6π 2 3ℓ2 pM bN 7 6 + � 3b2u2 R − 2 (1 − b2u2 R) 3/2 + 3b2u2 S − 2 (1 − b2u2 S) 3/2 � 48 √ 2bM N 3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='11) According to Eq(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='10), the ΨR − ΨS term can be expressed as ΨR − ΨS = (arcsin(buR) + arcsin(buS) − π) + � 1 uR � 1 − b2u2 R + 1 uS � 1 − b2u2 S � b (2π) 2 3ℓ2 pN 1 3 − � u2 R � 1 − b2u2 R + u2 S � 1 − b2u2 S � 96 2 1 6π 2 3ℓ2 pbM N 7 6 + � 1 − 2b2u2 R (1 − b2u2 R) 3/2 + 1 − 2b2u2 S (1 − b2u2 S) 3/2 � 48 √ 2bM N 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='12) 7 Combining the above equations, we can obtain the deflection angle in terms of the M2-brane number N and the black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' this is found to be Θ4 = �� 1 − b2u2 R uR + � 1 − b2u2 S uS � b (2π) 2 3ℓ2 pN 1 3 + �� 1 − b2u2 R + � 1 − b2u2 S � 192π 2 32 1 6ℓ2 pM bN 7 6 − � 1 � 1 − b2u2 R + 1 � 1 − b2u2 S � 48 √ 2bM N 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='13) It has been observed that the above expression diverges by taking the limits buS → 0 and buR → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This is due to the fact that the spacetime is not asymptotically flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Hence, the finite deflection angle of the light rays by AdS black holes from M-theory takes the form Θ4 ∼ b (2π) 2 3ℓ2 pN 1 3 � 1 uR + 1 uS � + 384 2 1 6π 2 3ℓ2 pM bN 7 6 − 96 √ 2bM N 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='14) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (1), we illustrate the M2-brane effect on such a deflection angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In the left panel of this figure, we present the variation of the deflection angle in terms of the impact parameter by taking different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Examining such AdS black holes, we show that the deflection angle of the light rays decreases for small values of the impact parameter then it becomes an increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been observed from the left and the right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (1) that the M2-brane number decreases the deflection angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In the right panel, we consider two values of the impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Plotting the deflection angle in terms of the M2-brane number, the two curves meet a particular point, where the deflection angle of the AdS space changes the behavior from a decreasing to an increasing function of the impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' A close examination shows that the behavior of the deflection angle of the AdS-Shwarzchild 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 100 150 200 250 b θ4 N 30 32 34 36 38 40 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6 0 50 100 150 200 50 100 150 200 250 N θ4 Figure 1: Right panel: Variation of the deflection angle of 4-dimensional black holes in M-theory in terms of the impact parameter for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Lift panel: Variation of the deflection angle of 4-dimensional black holes in M-theory in terms of the brane number for b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 and b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6 black holes for M = 1 and L = 3 is similar to the present one for N ≃ 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It seems possible to extend this analysis of the deflection angle behaviors in M-theory by introducing the rotating parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' According [46], indeed, the metric line element 8 becomes ds2 = −∆r W � dt − a Ξ sin2 θdφ �2 + W �dr2 ∆r + dθ2 ∆θ � + ∆θ sin2 θ W � adt − r2 + a2 Ξ dφ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='15) The involved terms are given by ∆r = r2 − mr + a2 + r2 L2(r2 + a2), ∆θ = 1 − a2 L2 cos2 θ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='16) Ξ = 1 − a2 L2, W = r2 + a2 cos2 θ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='17) where a is the rotating parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In this way, the computation will be expanded to the first order of m and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' To compute the Ψ and φRS angles for such a rotating black hole in four dimensions, we follow the method developed in [36], since we have the same metric form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Considering the equatorial plane, we can elaborate the orbit equation in terms of the M2-brane number N and the rotating parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In this way, it is found to be F4(a, u) = 1 b2−u2− 3√ 2 π2/3 3√ Nℓ2 p +192 6√ 2π2/3Mu3ℓ2 p N 7/6 + 2 3√ 2a π2/3b3 3√ Nu2ℓ2 p −384 6√ 2π2/3aMuℓ2 p b3N 7/6 +768 √ 2aM b3N 3/2u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='The longitudinal angle can be expressed as follows ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='φRS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='= (π − arcsin (buR) − arcsin (buS)) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='uR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 − b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='uS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 − 2b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='uR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 − b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 − 2b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='uS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 − b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='3√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='π2/3 3√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='Nℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='p ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 − b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 − b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2π2/3Mℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='bN 7/6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 − b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2π2/3aMℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='b2N 7/6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 − 3b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='(1 − b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='R) 3/2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 − 3b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='(1 − b2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='S) 3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='� 48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2bM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='N 3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Using the previous computations, we get the Ψ angle ΨRS = (arcsin (buR) + arcsin (buS) − π) + � 1 uR � 1 − b2u2 R + 1 uS � 1 − b2u2 S � b (2π)2/3 3√ Nℓ2 p − � u2 R � 1 − b2u2 R + u2 S � 1 − b2u2 S � 96 6√ 2π2/3bMℓ2 p N 7/6 + � 1 − 2b2u2 R (1 − b2u2 R) 3/2 + 1 − 2b2u2 S (1 − b2u2 S) 3/2 � 48 √ 2bM N 3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Combining the obtained expressions, we can obtain the deflection angle of four dimensional rotating AdS black holes from M-theory in terms of the involved parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The compu- 9 tations give Θ4(a) = � 1 − b2u2 R uR � 1 − b2u2 R + 1 − b2u2 S uS � 1 − b2u2 S � b (2π)2/3 3√ Nℓ2 p − � 1 − 2b2u2 R uR � 1 − b2u2 R + 1 − 2b2u2 S uS � 1 − b2u2 S � 3√ 2a π2/3 3√ Nℓ2 p − � 1 � 1 − b2u2 R + 1 � 1 − b2u2 S � 84 √ 2bM N 3/2 + �� 1 − b2u2 R + � 1 − b2u2 S � 192 6√ 2π2/3Mℓ2 p bN 7/6 − � 1 � 1 − b2u2 R + 1 � 1 − b2u2 S � 192 6√ 2π2/3aMℓ2 p b2N 7/6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='19) Taking a finite distance limit by sending buR and buS to 0, we get the reduced expression of the deflection angle Θ4(a) ∼ −96 √ 2bM N 3/2 + � 1 uR + 1 uS � b (2π)2/3 3√ Nℓ2 p + 384 6√ 2π2/3Mℓ2 p bN 7/6 − � 1 uR + 1 uS � 3√ 2a π2/3 3√ Nℓ2 p − 384 6√ 2π2/3aMℓ2 p b2N 7/6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='20) Putting a = 0, we recover the deflection angle of four dimensional non-rotating black hole in M-theory given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' To examine the effect of the M2-brane number for the rotating black holes, we vary the involved parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Taking two values of the rotating parameter a, we plot in the top of the left and the right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (2) the variation of the deflection angle in terms of the impact parameter b by varying the M2-brane number from 20 to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The M2-brane number parameter still decreases the deflection angle of the light rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In the left one, where we have a smaller value of a, the deflection angle of the light rays decreases for small values of b, and then it becomes an increasing function of the impact parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In the right one, where the contribution of the rotating parameter is relevant, the deflection angle is only an increasing function of b without any minimum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In the bottom of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (2), we take a small (left side) and a large value (right side) of the M2-brane number and vary the rotating parameter a from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This shows that the rotating parameter a still decreases the deflection angle as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For generic values of the M2-brane number, the deflection angle decreases by increasing the rotating parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Near b = 2, the deflection angle behaviors depend on the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For generic values of a, the deflection angle increases by decreasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Similar behaviors have been obtained in the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The only difference around b = 2 is the linear behavior for large values of the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been remarked that these optical behaviors could be related to the AdS spacetime backgrounds [20,21,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 Light deflection behaviors in the (11, 7, −3) model Here, we consider the 7-dimensional AdS black holes by considering the triplet (11, 7, −3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This model can be obtained from the compactification of M-theory on four dimensional real 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 30 35 40 45 50 55 60 65 b θ M=1, a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 N 20 22 24 26 28 30 2 3 4 5 6 0 20 40 60 80 b θ M=1, a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='8 N 20 22 24 26 28 30 2 4 6 8 10 0 50 100 150 200 b θ M=1, N=10 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='8 2 4 6 8 10 0 20 40 60 80 100 120 b θ M=1, N=100 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='8 Figure 2: Variation of the deflection angle in terms of the impact parameter by varying a and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' sphere S4 in the presence of M5-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The corresponding metric function in terms of the M5-brane number is expressed as follows A(r) = 1 + r2 4π2/3N 2/3ℓ2 p − 6π5/3Mℓ3 p 5N 4/3r4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='21) Using the orbital equation Eq(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='11), we obtain F7(u) = 1 b2 − 1 4π2/3N 2/3ℓ2 p − u2 + 6π5/3Mu6ℓ3 p 5N 4/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='22) By the help of this finding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' we get the φRS term φRS = (π − arcsin(buR) − arcsin(buS)) − � uR � 1 − b2u2 R + uS � 1 − b2u2 S � b3 8π2/3N 2/3ℓ2 p + � −2b5u5 R − 5b3u3 R + 15buR � 1 − b2u2 R + −2b5u5 S − 5b3u3 S + 15buS � 1 − b2u2 S � 3π5/3Mℓ3 p 40b4N 4/3 − �buR (3b4u4 R − 20b2u2 R + 15) (1 − b2u2 R) 3/2 + buS (3b4u4 S − 20b2u2 S + 15) (1 − b2u2 S) 3/2 � 3πMℓp 80b2N 2 + (π − arcsin(buR) − arcsin(buS)) � 9π5/3Mℓ3 p 8b4N 4/3 − 9πMℓp 16b2N 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='23) 11 For the Ψ part of these black holes, we find the following expression ΨR − ΨS = (arcsin(buR) + arcsin(buS) − π) + � 1 uR � 1 − b2u2 R + 1 uS � 1 − b2u2 S � b 8π2/3N 2/3ℓ2 p − � u5 R � 1 − b2u2 R + u5 S � 1 − b2u2 S � 3π5/3bMℓ3 p 5N 4/3 − �u3 R (2b2u2 R − 1) (1 − b2u2 R) 3/2 + u3 S (2b2u2 S − 1) (1 − b2u2 S) 3/2 � 3πbMℓp 40N 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='24) Combining Eq(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='23) and Eq(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='24), we obtain the deflection angle expression Θ7 = �� 1 − b2u2 R uR + � 1 − b2u2 S uS � b 8π2/3N 2/3ℓ2 p − � buR (2b4u4 R + b2u2 R − 3) � 1 − b2u2 R + buS (2b4u4 S + b2u2 S − 3) � 1 − b2u2 S � 3π5/3Mℓ3 p 40b4N 4/3 + � buR (7b2u2 R − 15) � 1 − b2u2 R + buS (7b2u2 S − 15) � 1 − b2u2 S � 3πMℓp 80b2N 2 + (π − arcsin(buR) − arcsin(buS)) � 9π5/3Mℓ3 p 8b4N 4/3 − 9πMℓp 16b2N 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='25) Taking the finite distance limits by sending buS and buR to zero, this deflection angle could be approximated by the following form Θ7 ∼ 9π8/3Mℓ3 p 8b4N 4/3 − 9π2Mℓp 16b2N 2 + b 8π2/3N 2/3ℓ2 p � 1 uR + 1 uS � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='26) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 10 15 20 25 30 35 b θ7 N 30 32 34 36 38 40 b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 b=1 0 20 40 60 80 100 10 20 30 40 N θ7 Figure 3: Right panel: Variation of the deflection angle of non rotating 7-dimensional black holes in M- theory in terms of the impact parameter for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Left panel: Variation of the deflection angle of 7-dimensional black holes in M-theory in terms of the brane number for b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 12 To examine the associated behaviors, we plot in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3) the variation of the deflection angle of the light rays by a 7-dimensional black hole in M-theory in terms of the impact parameter by varying the M5-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Similar to the four-dimensional case, the deflection angle of the light rays decreases for small values of the impact parameter to a critical value and then it becomes an increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' An examination of the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3) reveals that when we increase the number of M2-branes the deflection angle decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The critical transition behavior of the deflection angle from an increasing function to a decreasing one is illustrated in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3) by the intersection point of the two curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' To compare the effect of the dimension on the variation of the deflection angle, we plot in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (4) the variation of the deflection angle in terms of the impact parameter for (11, 4, 3) and (11, 7, −3) models by fixing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For small values of the impact parameter, the θ4 θ7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 0 200 400 600 800 1000 b θ Figure 4: The variation of the deflection angle in terms of the impact parameter of four and seven dimen- sional non rotating black holes by taking N = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' four-dimensional black hole bends the light rays larger than the seven-dimensional black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' However, this behavior is inverted for large values of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Having discussed the behaviors of the deflection angle of the light rays casted by AdS black holes in M-theory in the presence of M2 and M5-branes, we move to investigate the involved trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This has been motivated from the fact that the light trajectories of the black holes rely on the orbit equation exploited in the deflection angle computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The study of the light trajectories around the black holes for M2 and M5-brane models could confirm the behavior of the deflection angle obtained in the previous subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 4 Light trajectories around black holes in M-theory In this section, we study the trajectory of the light rays by the black holes in M-theory scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Concretely, we study the light trajectories near the AdS black holes of (D, d, k) models by varying the M(d − 2)-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It is denoted that the light trajectories around black holes can be generally approached via the numerical computations adopted to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Concretely, we can solve φ with respect to u in order to depict 13 the behaviors of the light rays around the involved black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' To establish such trajectories, we need to identify the regions corresponding to the light ray trajectory possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' These regions can be determined by the help of the effective potential which is expressed as follows V eff d (r) = − � dr dλ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1) This can be rewritten as V eff d (r) = −Fd(r) �bA(r) r2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2) For simplicity reasons, we restrict ourselves to the special models embedded in 11-dimensional supergravity limits of M-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='1 Trajectories of the light rays in the (11, 4, 3) model We start by considering the (11, 4, 3) model developed in [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Evincing the rotating parameter, the four dimensional effective potential, in the presence of the M2-branes, takes the following form V eff 4 (r) = L2 r2 � 1 − 192 2 1 6π 2 3ℓ2 pM N 7 6r + 2 1 3r2 π 2 3ℓ2 pN 1 3 � − E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='3) This effective potential will be illustrated as a function of the radial coordinate r for different values of the M2-brane number N by taking ℓp = 1 and M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' According to [34], the maximum value of the shadow radius corresponds to N = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (5), we plot such a potential for two M2-brane number values being N = 100 and N = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='160 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='17 N = 80 N = 100 Region 3 b > bsp Region 2 bsp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5364 Region 1 b < bsp Region 3 b > bsp Region 2 bsp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5101 Region 1 b < bsp rsp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='218 rsp = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='175 Veff 4 (r) Veff 4 (r) r r Figure 5: The effective potential behaviors of 4-dimensional AdS black holes embedded in 11-dimensional M-theory by varying taking to values of the brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This potential increases and reaches a maximum at the photon sphere associated with bsp which represents the impact parameter of the spinning light rays around the black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 14 This verifies the following constraint V eff(rsp) = 1 b2 sp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='4) It has been found that the two values of the M2-brane number N = 80, 100 provide two impact parameter values bsp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5364 and bsp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5101 corresponding to the photon sphere radius rsp = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='175 and rsp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='218, respectively as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Indeed, the associated impact parameter and the photon sphere radius decrease by increasing the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been remarked that the impact parameter value bsp provides trajectories of the light rays in three different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' These regions are denoted by region 1 , region 2 and region 3 corresponding to b < bsp, b = bsp and b > bsp, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In the first region 1 , the light ray falls into the black hole due to the values of the impact parameter lower to bsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In the third region 3 , however, the light ray near the black hole can be reflected back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In the second region 2 , however, the light ray comes into the photon sphere making an infinite number of turns around the black hole due to the a non vanishing angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The associated orbit is circular and unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' To illustrate these regions, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (6) the trajectories of the light rays in the polar coordinates (r, φ) for different values of the M2-brane number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' To analyze the effect of the M2-brane on the light ray trajectories, we vary the impact parameter b by using the step between two values of the impact parameter as 1/20 for all light rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' A close examination reveals that the horizon and the sphere photon 20 15 10 5 0 5 10 15 20 x 10 5 0 5 10 y 20 15 10 5 0 5 10 15 20 x 10 5 0 5 10 y N = 80 N = 100 Figure 6: The trajectories of the light ray for different values of M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The black and the dashed red circle is the horizon and the photon sphere of the M2-brane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' radius decrease by increasing the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' However, the variation of the impact parameter bsp is small by varying the M2-brane number, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been observed that the region 1 , 2 and 3 are the same for different values of the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Taking two values of N, we observe that the distance between two light rays increases for an impact parameter value closer and bigger to bsp for all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Indeed, this distance decreases by increasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This distinction comes from the values of the angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' More analysis shows that the reflected of the light ray is more intense by decreasing the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Comparing this result with many works concerning the trivial solution [47–49], we observe 15 a different behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' First, bsp is almost the same by varying the N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' However, rsp decreases by increasing the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This is completely different than the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Second, for small values of the impact parameter, we remark that the light ray falls into the black hole by keeping the parallel trajectories with the r-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' However, for values close to bsp the light ray falls into the black hole without keeping the parallel trajectory with the r-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Indeed, the parallel trajectory is replaced by a critical angle between the light ray and the r-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Fixing the M2-brane number, this angle increases for values near to bsp (or bigger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Varying N, it increases by decreasing the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This shows that such a M2-brane number can be considered as a relevant quantity modifying the light ray behaviors near a black hole in M-theory compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This distinction comes from the geometry of the black holes in M-theory with brane backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='2 Light trajectories in the (11, 7, −3) model Here, we deal with the trajectory of the light rays near black holes in the presence of M5- branes in M-theory compactifications on the four-dimensional sphere S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For the (11, 7, −3) model, such light behaviors can be determined with the help of the effective potential V eff 7 (r) = L2 r2 � 1 + r2 4π2/3N 2/3ℓ2 p − 6π5/3Mℓ3 p 5N 4/3r4 � − E2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (7), we illustrate the associated effective potential as a function of the radial coor- dinate r for two values of the M5-brane number being N = 1 and N = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Using the 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='16 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='5 N = 1 N = 80 Region 3 b > bsp Region 2 bsp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='7180 Region 1 b < bsp Region 3 b > bsp Region 2 bsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6308 Region 1 b < bsp rsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='515 rsp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='219 Veff 7 (r) Veff 7 (r) r r Figure 7: The effective potential behaviors of 7-dimensional AdS black holes embedded in 11-dimensional M-theory by varying taking to values of the brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' constraint given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='4), we can get the values of the photon sphere and the critical impact parameter of the photon sphere associated with the maximal value of the effective potential V eff 7 (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It follows that the two values of the M5-brane number N = 1 and N = 80 provide two impact parameter values bsp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='7180 and bsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='6308 associated with the values of 16 the photon sphere rsp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='219 and rsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content='515, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The corresponding impact parameter and the photon sphere radius decrease by increasing the M5-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For the fixed value of N = 80, however, we observe that bsp and rsp in the M5-brane model is small compared to the M2-brane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This distinction affects the light ray trajectories around the black holes in the M5-brane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Placing the observer in the equatorial hyperplane ensured by θ1 = θ2 = π 2, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' (8) the trajectories of the light rays in the polar coordinates (r,φ) for different values of the M5-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 8 6 4 2 0 2 4 6 8 x 8 6 4 2 0 2 4 6 8 y 3 2 1 0 1 2 3 x 3 2 1 0 1 2 3 y N = 1 N = 80 Figure 8: The trajectories of the light ray for different values of M5-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The black and the dashed red circle is the horizon and the photon sphere of the M5-brane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Due to small values of bsp, we vary the impact parameter b by using the step between two values of the impact parameter as 1/40 for all light rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been suggested that the M5-brane number and the extra dimension can be considered as relevant parameters modifying the light ray behaviors near a black hole in M-theory compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Such modifications come from the black hole geometry in the presence of the M5-branes in M- theory compacatifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The obtained behaviors match perfectly with the previous works [34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been remarked that the previous behaviors of the M2-brane model associated with the horizon and the photon sphere radius have been conserved in the M5-brane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Concretely, one can show that the reflected light ray becomes more intense by deceasing the M5-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been remarked that these behaviors are contrary to the ones of the M2-brane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This is due to the extra dimension contributions of the M5-brane model appearing in the associated metric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Moreover, bsp and rsp vary by taking different values of the M5-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For all values of the impact parameter, we observe that if the light ray falls into the black hole or refracted it keeps a parallel trajectory with respect 17 to the r-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This behavior is completely different than the previous results related to the M2-brane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Moreover, the values of the angular velocity in the M5-brane model is different than the one in the M2-brane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' A close examination shows that the results concerning the light trajectories behaviors confirm the results associated with the deflection angle behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been remarked that the deflection angle increases by decreasing the M-theory brane number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Moreover, the deflection of the light ray is more intense by decreasing the M-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Both results inter-match perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Finally, the behaviors of the optical quantities including the light trajectories and the deflection angle of black holes for M-theory brane models are interesting and similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' 5 Conclusion In this work, we have investigated the deflection angle and the trajectory of the light rays casted by black holes in M-theory scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Using the Gauss-Bonnet theorem, we have com- puted and examined the deflection angle of the light rays around four and seven dimensional AdS black holes derived from the M-theory compacatification on the real spheres S7 and S4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' First, we have generalized the deflection angle formalism for d-dimensional AdS black hole solutions using the Gauss-Bonnet theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Then, we have studied the de- flection angle of four dimensional rotating and non-rotating AdS black holes by examining the M2-brane number effect for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Concretely, we have shown that the deflection angle of light rays decreases for small value of the impact parameter then it becomes an increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' It has been observed that the M-theory brane number decreases the de- flection angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Taking two values of the impact parameter and varying N, we have observed that the two curves meets a particular point, where the deflection angle of the AdS space has changed the behavior from a decreasing to an increasing function in terms the impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For four dimensional rotating model, we have revealed that the behaviors of the deflection angle by varying the M2-brane number is similar to the non-rotating case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For the small value of a, however, the deflection angle of the light rays decreases for small values of b and then it becomes an increasing function of the impact parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In general, the rotating parameter is a relevant quantity decreasing the deflection angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We have shown that, around b = 2, the deflection angle behaviors depend on the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For generic values of a, the deflection angle increases by decreasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Similar behaviors have been obtained in the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' The only difference around b = 2 is the linear behavior for large values of the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Then, we have extended the calculations to seven dimensional non-rotating black holes by providing a comparative study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For small values of the impact parameter, we have shown that the four-dimensional black hole bends the light rays larger than the seven-dimensional one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' However, this behavior is inverted for large values of the impact parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Finally, we have discussed the trajectories of the lights rays around four and seven- dimensional AdS black hole in M-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We have shown that the regions 1 , 2 and 18 3 are the same for different values of the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Taking two values of N, we have observed that the distance between two light rays increases for an impact parameter value closer and bigger to bsp value for all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This distance decreases by increasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Concretely, this distinction is originated from the values of the angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We have remarked that the reflected of the light ray is more intense by decreasing the M2-brane number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Comparing the present results with many investigations associated with the trivial solution, we have observed a different behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' For the M5-brane model, we have remarked a different behavior with respect to the M2-brane model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This difference could come from the extra dimension being a relevant parameter appearing in the metric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' This work comes up with certain open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' A possible project concerns generic models associated with (D, d, k) M-theory inspired models proposed in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' In particular, the effect of the M(d−2)-branes in such models could be examined by performing non trivial numerical computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Another issue is to approach certain M-theory compactifications using the orbifold of spheres providing possible ways to implement G2-manifolds in such black hole activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' We hope address these questions in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf'} +page_content=' Acknowledgments The present paper is dedicated to the memory of Pr.' metadata={'source': 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sha256:b35ca5b25994b93d15ad26034a25872be73787d4e33e108203399e824a6eba63 +size 5111853 diff --git a/c9AyT4oBgHgl3EQfXfd-/content/tmp_files/2301.00184v1.pdf.txt b/c9AyT4oBgHgl3EQfXfd-/content/tmp_files/2301.00184v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2760923b49bd27336287b5a802b68914ca54f4ef --- /dev/null +++ b/c9AyT4oBgHgl3EQfXfd-/content/tmp_files/2301.00184v1.pdf.txt @@ -0,0 +1,1563 @@ +Cap4Video: What Can Auxiliary Captions Do for Text-Video Retrieval? +Wenhao Wu1,2* +Haipeng Luo3∗ +Bo Fang3 +Jingdong Wang1 +Wanli Ouyang2,4 +1Baidu Inc. +2The University of Sydney +3University of Chinese Academy of Sciences +4Shanghai AI Laboratory +whwu.ucas@gmail.com +Abstract +Most existing text-video retrieval methods focus on +cross-modal matching between the visual content of offline +videos and textual query sentences. However, in real sce- +narios, online videos are frequently accompanied by rele- +vant text information such as titles, tags, and even subti- +tles, which can be utilized to match textual queries. This in- +spires us to generate associated captions from offline videos +to help with existing text-video retrieval methods. To do +so, we propose to use the zero-shot video captioner with +knowledge of pre-trained web-scale models (e.g., CLIP and +GPT-2) to generate captions for offline videos without any +training. Given the captions, one question naturally arises: +what can auxiliary captions do for text-video retrieval? +In this paper, we present a novel framework Cap4Video, +which makes use of captions from three aspects: i) Input +data: The video and captions can form new video-caption +pairs as data augmentation for training. ii) Feature inter- +action: We perform feature interaction between video and +caption to yield enhanced video representations. iii) Output +score: The Query-Caption matching branch can be com- +plementary to the original Query-Video matching branch +for text-video retrieval. We conduct thorough ablation stud- +ies to demonstrate the effectiveness of our method. Without +any post-processing, our Cap4Video achieves state-of-the- +art performance on MSR-VTT (51.4%), VATEX (66.6%), +MSVD (51.8%), and DiDeMo (52.0%). +1. Introduction +Text-Video retrieval is a fundamental task for video-and- +language understanding. +With the rapid development of +image-language pre-training (e.g., CLIP [29], ALIGN [13], +CoCa [41], Florence [42], etc.), more recent research has +concentrated on expanding the pre-trained image-language +models (e.g., CLIP [29]) to the text-video retrieval task. +The research path has evolved from the most direct global +*Equal contribution. +Query +Caption +Video +Query-Video +Matching +New training pairs +Query-Caption +Matching +Cross-modal interaction +Video +Query +Query-Video Matching +Video +Caption +CLIP +GPT-2 +Captioner +(a) +(b) +(c) +Figure 1. +(a) Existing learning framework for text-video re- +trieval. (b) Zero-shot video captioning by guiding a large-scale +pre-trained language model (e.g. GPT-2 [30]) with CLIP [29]. (c) +Our Cap4Video framework employs the generated captions from +three aspects: input data augmentation, intermediate feature inter- +action, output score fusion. +matching (i.e., video-sentence alignment [9, 23]) to fine- +grained matching (e.g., frame-word alignment [34], video- +word alignment [11], multi-hierarchical alignment [7, 27], +etc.). +These works demonstrate remarkable performance +and significantly outperform previous models. The key rea- +sons for the improvement are as follows: First, CLIP offers +powerful visual and textual representations that are roughly +pre-aligned in the semantic embedding space, hence re- +ducing the challenge of cross-modal learning on video-text +matching. Second, these works can finetune the pre-trained +vision and text encoders using the sparsely sampled frames +in an end-to-end manner. All of these methods aim to learn +1 +arXiv:2301.00184v1 [cs.CV] 31 Dec 2022 + +cross-modal alignment between the visual representation of +offline videos and the textual representation of the corre- +sponding query, as depicted in Figure 1(a). +However, in a real scenario, online videos are usually +accompanied by related content. For example, we may sim- +ply obtain the video’s title or tag from the video website. +In addition to the visual information in the video, the asso- +ciated textual information can also be used to describe the +video content to some extent and to match the query (i.e., +the common text-to-text retrieval). This leads to a ques- +tion: How to generate associated text descriptions for of- +fline videos? A possible solution is to crawl the video title +from the video website. However, this method relies on an- +notation, and there is also the risk that the video website +has become invalid. Another automated solution is to gen- +erate captions utilizing zero-shot video caption models. We +turn our attention to knowledge-rich pre-trained models to +handle such challenging open-set scenarios. We find that +the recent study ZeroCap [32] provides a good practice to +use frozen CLIP [29] and GPT-2 [30] for zero-shot image +captioning. So we extend it to the video domain for video +captioning without any further training or tuning stages. +Given the auxiliary captions, one question naturally +arises: +How to make full use of these generated captions +to improve the performance of text-video retrieval? In this +paper, as shown in Figure 1(c), we present a Cap4Video +learning framework which utilizes captions from three as- +pects: (i) Input Data: The simplest way is to augment the +training data with the generated captions. Specifically, the +given video and its generated caption are a matched pair, +so they can be regarded as additional positive sample pairs +other than the query-video pair for training. (ii) Feature +Interaction: we can perform cross-modal interaction be- +tween the video and captions to improve video representa- +tion. Specifically, we may take advantage of the information +complementarity between videos and captions to reduce re- +dundant features from videos and learn more discrimina- +tive video representations. (iii) Output score: The gener- +ated caption can also represent the video’s content, so we +can employ query-caption matching to complement stan- +dard query-video matching for the text-video retrieval task. +We can further utilize the two-stream architecture to reduce +the bias of the model and produce more robust results. +To demonstrate the effectiveness of the aforementioned +explorations, we conduct extensive experiments on four +well-known video datasets. We hope our new paradigm will +stimulate more investigation into text-video retrieval task. +We summarize the contributions as follows: +• We investigate a novel problem: how to use captions +generated by web-scale language models to help with +the text-video retrieval task. Our motivation is to lever- +age the vast knowledge of web-scale pre-trained lan- +guage models to automatically generate extra text in- +formation for offline videos rather than labor-intensive +annotations, to benefit text-video retrieval. +• We present a Cap4Video learning framework that +makes full use of the generated captions from three as- +pects (i.e., input data, feature interaction, output score) +and can bring further performance improvement to +the existing query-video matching mechanisms (i.e., +global matching, fine-grained matching). +• Extensive experiments on four text-video retrieval +datasets demonstrates the effectiveness of our method. +Our Cap4Video achieves state-of-the-art performance +on MSR-VTT [39] (51.4%), VATEX [36] (66.6%), +MSVD [38] (51.8%), and DiDeMo [1] (52.0%). +2. Methodology +2.1. Preliminaries: Text-Video Matching +The main goal of text-video matching is to develop a +function s(Qi, Vj) to determine how similar the video Vj +is to a sentence Qi. In the text-to-video retrieval, the ob- +jective is to rank all the videos given the query sentence ac- +cording to their similarity score. To transfer the image-text +pre-training knowledge into video-text learning, we follow +recent works [7, 9, 23] to apply CLIP [29] for initialization +to improve text-video retrieval. Figure 2(b) depicts two typ- +ical text-video matching mechanisms (i.e., global matching, +and fine-grained matching) that serve as our baseline. +Global Matching is widely used in cross-model contrastive +learning [14,23,29]. In text-video contrastive learning, we +train the visual encoder for a given video that samples F +frames and the query encoder for a given query sentence +that contains W words. The visual encoder then outputs +F frame embeddings, while the query encoder returns two +types of embeddings: W word embeddings and the [CLS] +embedding as the global representation. These frame em- +beddings are then integrated with average pooling to yield +the global video embedding. Finally, we can get the simi- +larity between this global video embedding and the global +query embedding for matching. +Fine-grained Matching. +The global matching encodes +each modality independently to get the global features be- +fore calculating the similarity between them. The solution +is simple and intuitive, but it ignores the fine-grained align- +ment between the two modalities (e.g., frame-word align- +ment). To model the token-wise word-patch alignment for +image-text learning, FILIP [40] and ColBERT [15] use a +Max-Mean pipeline. Specifically, they get the token-wise +maximum similarity between the patch and word tokens, +then use the average token-wise maximum similarity of to- +kens in the image (resp. text) as the similarity of an image +to a text (resp. a text to an image). DRL [34] expands the +token-wise alignment into text-video retrieval and further +2 + +Video shows a girl singing +in front of the audience. +a girls performing a song on +the stage for competition +Video +Encoder +Query +Encoder +Caption +Encoder +Video-Caption +Interaction +QC Matching +QV Matching +(b) Query-Video Matching: Two typical mechanisms +SQV +Frame embeddings +① Global embedding matching +Average +Pooling +F × D +1 × D +1 × D +Query [CLS] embedding +F × D +W × D +Word embeddings +② Fine-grained embedding matching +Frame embeddings +Max +W×1 +SQ2V +Max +F×1 +SV2Q +SQV +F×W +(c) Query-Caption Matching +Captions embeddings +Caption +Aggregation +C × D +1 × D +1 × D +Query [CLS] embedding +SQV +SQC +SQVC +SQC +(a) Our framework +Captioner +V1 V2 +Vn +Q1 +Q2 +Qn +… +… +Similarity matrix +Extra query augmentation for training +Contrastive training +··· +FC +FC +W×1 +F×1 +Figure 2. An overview of our Cap4Video for text-video retrieval. In Cap4Video, we first generate captions using a zero-shot video +captioner which combines CLIP [29] with GPT-2 [30], leveraging knowledge from both frozen web-scale models. Then we utilize the +pre-extracted caption information from three aspects: i) Input data: The video and captions can form new pairs as data augmentation for +training. ii) Feature interaction: We perform feature interaction between video and caption to capture intra- and inter-modality context +to yield enhanced video representations. iii) Output score: The Query-Caption matching branch can be complementary to the original +Query-Video matching branch for text-video retrieval. +proposes to use the attention mechanism to learn a weighted +pooling instead of mean pooling. Thus, we use fine-grained +matching to guide the contrastive objective and as our en- +hanced baseline. +2.2. Preprocessing: Caption Generation +To get the auxiliary caption for the given video, we uti- +lize the knowledge from the pre-trained language model to +generate rich and diverse captions. To compare captions +from different sources, we also choose a common case: the +video’s title as an auxiliary caption from the video URL. +Zero-shot Video Captioner. Considering the paradigm’s +scalability, we aim to generate captions directly from down- +stream video data without extra training, a process known +as zero-shot video captioning. +Following ZeroCap [32], +we use the GPT-2 [30] to infer the next word from an ini- +tial prompt, e.g., “Video shows”. A calibrated CLIP [29] +loss drives the model to generate sentences that describe the +video to incorporate video-related knowledge into the auto- +regression process. See Supplementary for more details. +2.3. Data Augmentation with Auxiliary Captions +Given the generated captions, the most natural applica- +tion of the captions is to augment training data. For exam- +ple, given a dataset consisting of N videos and correspond- +ing query sentences, each video of this dataset and its gen- +erated caption are a matched pair, so they can be regarded as +additional positive sample pairs other than the query-video +pair for training. In this way, if we choose one caption per +video, then we can increase at least N pairs as additional +data augmentation during the training phase. +The video captioner can generate multiple captions (e.g., +20) for each offline video. We think that some of these cap- +tions could contain noise, meaning that they may not be en- +tirely relevant to the video content. If all of them are used +as training data directly, this could have a negative effect on +training. Therefore, we design a filtering mechanism that +evaluates the semantic similarity between each caption and +the ground-truth query of the video (using a pre-trained text +encoder). The caption with the highest similarity is then +chosen for data augmentation. Note that we use the ground- +truth query for caption filtering only in the training phase. +2.4. Video-Caption Cross-Modal Interaction +We further perform cross-modal interactions between the +video and generated caption to improve video representa- +tion. Our motivation is that we may take advantage of the +information complementarity between videos and captions +3 + +Concat +MLP +Transformer +Encoder +× L +Transformer +Encoder +Q +K V +Transformer +Encoder +× L +(a) Sum +(b) MLP +(c) Cross Transformer +(d) Co-attention Transformer +Caption Embedding +Frame Embedding +Figure 3. Illustration of four Video-Caption interaction strategies. +The enhanced frame embeddings will be followed by a mean pool- +ing for global matching or will remain for fine-grained matching. +to reduce redundant features from videos and learn more +discriminative video representations. To make this more +concrete, we now discuss several ways of interaction be- +tween two modalities as depicted in Figure 3. +In order not to change the original pre-trained CLIP en- +coders as much as possible for better transferring, we only +perform interaction between the final caption and frame em- +beddings. To obtain the compact representation from all +generated captions, we simply utilize average pooling for +the output of the caption encoder as follows: +ec = 1 +C +C +� +i=1 +ci, +(1) +where ci ∈ RD represents the [CLS] embedding of i-th +caption and C is the number of generated captions. Then +we feed the frame embeddings ev = {v1, v2, · · · , vF } and +global caption embedding ec into the interaction module. +Sum. It is an intuitive solution to compute the sum of +each frame embedding and the global caption embedding to +yield enhanced frame embedding: +Sum(vi, ec) = vi + ec +for i = 1, · · · , F, +(2) +where F is the number of frames. +MLP. We concatenate each frame embedding with the +global caption embedding as a whole, then use the learnable +Multi-layer Perceptron (MLP) to model weighted combina- +tions of the two embeddings: +MLP(vi, ec) = fθ([vi, ec]) +for i = 1, · · · , F, +(3) +where [·, ·] is the denotes concatenation operation, fθ is the +MLP with parameter θ. +Cross Transformer. We also consider adopt the self- +attention [33] for cross-modal interactions. As shown in +Figure 3(c), the Cross Transformer operates on sequences +of embeddings {v1, · · · , vF , ec}. These embeddings are +passed through L “encoder-style” transformers blocks to +produce final representations: +Cross(ev, ec) = fψ({ev, ec}), +(4) +where {} means that ev, ec form a sequence, and fψ is the +transformer encoders with parameter ψ. +Co-attention Transformer. Another typical informa- +tion exchange mechanism is co-attention [22], which passes +the keys and values from one modality as input to the other +modality’s multi-headed attention block of a standard trans- +former encoder block. Here we introduce one co-attentional +transformer layer to enable information exchange between +modalities, followed by L standard transformer layers to +model temporal information: +CoAttn(ev, ec) = fφ2(fφ1(ev, ec)), +(5) +where fφ1 is the co-attentional transformer with parameter +φ1 and fφ2 is the transformer encoders with parameter φ2. +Next, the frame embeddings generated by the Video- +Caption interaction module will be then averaged for global +matching or will be kept for fine-grained matching. +2.5. Complementary Query-Caption Matching +In addition to the aforementioned uses of caption for data +augmentation and video feature enhancement, the generated +caption itself can represent the content of the video, making +it possible to use it directly for retrieval (i.e., text-text re- +trieval). Specifically, each of the C captions generated by +the video is then passed through the caption encoder to ob- +tain its [CLS] text embedding. As shown in Figure 2(c), +these caption embeddings are then aggregated to form a +global representation. Then the cosine similarity between +this global caption embedding and the global query embed- +ding is calculated to complement the query-video matching. +Denotation. Given a batch of B triples {evi, eti, eci}B +i=1, +where evi, eti, eci denote the i-th video, query, and caption +embedding, respectively. Note that the term “embedding” +used here is more general for convenience, the exact mean- +ing of embedding will vary depending on the situation. For +example, in query-video global matching, evi and eti rep- +resent the averaged video feature and global [CLS] text fea- +ture, respectively. In query-video fine-grain matching, evi +and eti represent a sequence of frame embeddings and a se- +quence of word embeddings, respectively. In query-caption +matching, eci represents a sequence of caption embeddings +and eti represents a global [CLS] text feature, respectively. +Learning objectives. For the Query-Caption branch, we +want the caption embedding ec and the query embedding +4 + +et to be close while they are related and far apart when they +are not during training phase. We follow the common prac- +tice [23,34] to consider the bidrectional learning objective. +We employ symmetric cross-entropy loss to maximize the +similarity between matched Query-Caption pairs and mini- +mize the similarity for other pairs: +LC2Q = − 1 +B +B +� +i +log +exp(sqc(eti, eci)/τ) +�B +j exp(sqc(eti, ecj)/τ) +, +LQ2C = − 1 +B +B +� +i +log +exp(sqc(eti, eci)/τ) +�B +j exp(sqc(etj, eci)/τ) +, +LQC = 1 +2(LC2Q + LQ2C), +(6) +where sqc(·, ·) represents the query-caption matching simi- +larity function shown in Figure 2(c), and τ refers to the tem- +perature hyper-parameter for scaling. Similarly, the con- +trastive loss for Query-Video branch is formulated as: +LV 2Q = − 1 +B +B +� +i +log +exp(sqv(eti, evi)/τ) +�B +j exp(sqv(eti, evj)/τ) +, +LQ2V = − 1 +B +B +� +i +log +exp(sqv(eti, evi)/τ) +�B +j exp(sqv(etj, evi)/τ) +, +LQV = 1 +2(LV 2Q + LQ2V ), +(7) +where sqv(·, ·) represents the query-video matching (e.g., +global matching, fine-grained matching) similarity function +shown in Figure 2(b). The total loss L is the sum of Query- +Video loss LQV and Query-Caption loss LQC: +L = LQV + LQC. +(8) +3. Experiments: Text-Video Retrieval +3.1. Setups +Datasets. We conduct experiment on four popular bench- +marks for video-to-text retrieval and text-to-video retrieval +tasks. MSR-VTT [39] contains a total of 10K video clips, +each having 20 captions. Following the data splits from +[8,23,26], we train models with associated captions on the +Training-9K set and report results on the test 1K-A +set. DiDeMo [1] has 10K videos paired with 40K descrip- +tions. Following previous works [2, 17, 23], we concate- +nate all descriptions of one video to a single query, acting +as a video-paragraph retrieval task. VATEX [36] collects +∼35K videos, each with multiple annotations. There are +∼26K videos for training, 1,500 videos for validation and +1,500 videos for testing. MSVD [38] contains 1,970 videos +with 80K captions, with ∼40 captions on average per video. +There are 1,200, 100, and 670 videos in the train, validation, +and test sets, respectively. +Evaluation Metrics. For brevity, we abbreviate Recall at +K to R@K (K = 1, 5, 10) upon all datasets, which com- +putes the percentage of correct videos among the top K re- +trieved videos given textual queries (Text→Video, and vice +versa). MdR, Median Rank, computes the median of the +ground-truth in the retrieval ranking list. MnR, Mean Rank, +computes the mean rank of the correct results in the retrieval +ranking list. Note that for MdR and MnR, the lower score +means the better (indicated as ↓). +Implementation Details. In all experiments, we use the +visual encoder of CLIP [29] as our video encoder and use +the textual encoder of CLIP as both the caption and query +encoders. To reduce conflict between the two branches, we +train the query-video branch first, then the query-caption +branch. The caption length is 32 and the video length is +12 for all datasets except DiDeMo (64 max words and 64 +frames). The network is optimized by Adam [16] with a +batch size of 128 and epoch 5. The initial learning rate is 1e- +7 for the clip parameters and 1e-4 for the non-clip param- +eters, respectively. Following previous works [11, 21, 23], +we train model for 5 epochs with Adam [16] optimizer and +adopt a warmup [12] strategy. All learning rates follow the +cosine learning rate schedule with a linear warmup. The +number C of generated captions per video is set to 30. The +number L of transformer layers is set to 4 for VATEX and +MSR-VTT, and 1 for Didemo and MSVD. +3.2. Comparison with State-of-the-arts +Here we compare our Cap4Video with recent state-of- +the-art methods on the four benchmarks, MSR-VTT [39], +MSVD [38], VATEX [36] and DiDeMo [1]. Table 1 lists the +comparisons on DiDeMo. We can see that our Cap4Video +significantly surpasses CLIP4Clip [23] by 9.2% R@1 and +exceeds DRL [34] by 3.0%, which proves the effectiveness +of our method. +Method +R@1 +R@5 +R@10 +MdR MnR +CE [19] +15.6 +40.9 +- +8.2 +- +ClipBERT [17] +21.1 +47.3 +61.1 +6.3 +- +Frozen [2] +31.0 +59.8 +72.4 +3.0 +- +TMVM [18] +36.5 +64.9 +75.4 +3.0 +- +CLIP4Clip [23] +42.8 +68.5 +79.2 +2.0 +18.9 +TS2-Net [21] +41.8 +71.6 +82.0 +2.0 +14.8 +HunYuan [27] +45.0 +75.6 +83.4 +2.0 +12.0 +DRL [34] +49.0 +76.5 +84.5 +2.0 +- +Cap4Video (Ours) +52.0 +79.4 +87.5 +1 +10.5 +Table 1. Results of text-to-video retrieval on the DiDeMo [1]. +Comparisons with recent state-of-the-art models on +MSR-VTT are provided in Table 2. Our approach signif- +icantly outperforms previous works and achieves new state- +of-the-art performance with both ViT-B/32 and ViT-B/16 +5 + +Method +Date +Text → Video +Video → Text +R@1 +R@5 +R@10 +MdR↓ +MnR↓ +R@1 +R@5 +R@10 +MdR↓ +MnR↓ +ClipBERT [17] +CVPR’20 +22.0 +46.8 +59.9 +6.0 +- +- +- +- +- +MMT [8] +ECCV’20 +26.6 +57.1 +69.6 +4.0 +- +27.0 +57.5 +69.7 +3.7 +21.3 +SupportSet [28] +ICLR’21 +30.1 +58.5 +69.3 +3.0 +- +28.5 +58.6 +71.6 +3.0 +- +Frozen [2] +ICCV’21 +32.5 +61.5 +71.2 +3.0 +- +- +- +- +- +- +BridgeFormer [10] +CVPR’22 +37.6 +64.8 +75.1 +- +- +- +- +- +- +- +TMVM [18] +NeurIPS’22 +36.2 +64.2 +75.7 +3.0 +- +34.8 +63.8 +73.7 +3.0 +- +CLIP-ViT-B/32 +CLIP4Clip [23] +ArXiv’21 +44.5 +71.4 +81.6 +2.0 +15.3 +42.7 +70.9 +80.6 +2.0 +11.6 +CenterCLIP [43] +SIGIR’22 +44.2 +71.6 +82.1 +2.0 +15.1 +42.8 +71.7 +82.2 +2.0 +10.9 +CAMoE [6] +ArXiv’21 +44.6 +72.6 +81.8 +2.0 +13.3 +45.1 +72.4 +83.1 +2.0 +10.0 +CLIP2Video [7] +ArXiv’21 +45.6 +72.6 +81.7 +2.0 +14.6 +43.5 +72.3 +82.1 +2.0 +10.2 +X-Pool [11] +CVPR’22 +46.9 +72.8 +82.2 +2.0 +14.3 +- +- +- +- +- +QB-Norm [3] +CVPR’22 +47.2 +73.0 +83.0 +2.0 +- +- +- +- +- +- +TS2-Net [21] +ECCV’22 +47.0 +74.5 +83.8 +2.0 +13.0 +45.3 +74.1 +83.7 +2.0 +9.2 +DRL [34] +ArXiv’22 +47.4 +74.6 +83.8 +2.0 +- +45.3 +73.9 +83.3 +2.0 +- +Cap4Video (Ours) +49.3 +74.3 +83.8 +2.0 +12.0 +47.1 +73.7 +84.3 +2.0 +8.7 +CLIP-ViT-B/16 +CLIP2TV [9] +ArXiv’21 +48.3 +74.6 +82.8 +2.0 +14.9 +46.5 +75.4 +84.9 +2.0 +10.2 +CenterCLIP [43] +SIGIR’22 +48.4 +73.8 +82.0 +2.0 +13.8 +47.7 +75.0 +83.3 +2.0 +10.2 +TS2-Net [21] +ECCV’22 +49.4 +75.6 +85.3 +2.0 +13.5 +46.6 +75.9 +84.9 +2.0 +8.9 +DRL [34] +ArXiv’22 +50.2 +76.5 +84.7 +1.0 +- +48.9 +76.3 +85.4 +2.0 +- +Cap4Video (Ours) +51.4 +75.7 +83.9 +1.0 +12.4 +49.0 +75.2 +85.0 +2.0 +8 +Table 2. Retrieval results on the validation set of MSR-VTT 1K [39]. Here we report results without any post-processing operations (e.g., +DSL [6] or QB-Norm [3]) during inference. +Method +R@1 R@5 R@10 MdR +MnR +CE [19] +19.8 +49.0 +63.8 +6.0 +- +SUPPORT [28] +28.4 +60.0 +72.9 +4.0 +- +CLIP [29] +37.0 +64.1 +73.8 +3.0 +- +Frozen [2] +33.7 +64.7 +76.3 +3.0 +- +TMVM [18] +36.7 +67.4 +81.3 +2.5 +- +CLIP4Clip [23] +45.2 +75.5 +84.3 +2.0 +10.3 +X-Pool [11] +47.2 +77.4 +86.0 +2.0 +9.3 +Cap4Video (Ours) +51.8 +80.8 +88.3 +1 +8.3 +Table 3. Results of text-to-video retrieval on the MSVD [38]. +backbones. +For example, on text-to-video retrieval, we +achieve +4.8% higher R@1 than CLIP4Clip with the same +ViT-B/32. Also, our Cap4Video outperforms the brand-new +method TS2-Net [21] by 2.3% and 2.0% with ViT-B/32 +and ViT-B/16, respectively. +Table 3 and Table 4 show results for the MSVD dataset +and VATEX dataset, respectively. We use the ViT-B/16 as +our backbone. For MSVD, our model achieves remarkable +performance 51.8% R@1 and a performance improvement +of 6.6%, 4.6% on text-to-video retrieval when compared +to CLIP-based models CLIP4Clip [23] and X-Pool [24], re- +Method +R@1 +R@5 +R@10 +MdR MnR +HGR [5] +35.1 +73.5 +83.5 +2.0 +- +CLIP [29] +39.7 +72.3 +82.2 +2.0 +12.8 +SUPPORT [28] +44.9 +82.1 +89.7 +1.0 +- +CLIP4Clip [23] +55.9 +89.2 +95.0 +1.0 +3.9 +Clip2Video [7] +57.3 +90.0 +95.5 +1.0 +3.6 +QB-Norm [3] +58.8 +88.3 +93.8 +1.0 +- +TS2-Net [21] +59.1 +90.0 +95.2 +1.0 +3.5 +Cap4Video (Ours) +66.6 +93.1 +97.0 +1 +2.7 +Table 4. Results of text-to-video retrieval on the VATEX [36]. +spectively. For VATEX, our approach also outperforms the +recent state-of-the-art methods and achieves +7.5% R@1 +improvement over TS2-Net [21] for text-to-video retrieval. +Recently, a few methods post-process the similarity gener- +ated by the model to significantly improve performance. It +should be noted that in all of the Tables, the results are re- +ported without any unfair post-processing processes, such +as DSL [6] and QB-Norm [3]. +Overall, +the consistent state-of-the-art performance +across four benchmarks demonstrates the effectiveness of +our Cap4Video. +6 + +Method +Global embedding matching +Fine-grained embedding matching +R@1 +R@5 +R@10 +MdR↓ +MnR↓ +R@1 +R@5 +R@10 +MdR↓ +MnR↓ +Baseline +42.8 +70.4 +79.0 +2 +16.6 +45.7 +73.7 +82.6 +2 +13.1 ++Different Sources of Caption as Data Augmentation +Video Title from Source URL +43.8 +71.1 +80.9 +2 +15.1 +44.3 +72.7 +83.5 +2 +13.1 +Zero-shot Video Captioning +44.2 +70.7 +81.5 +2 +16.2 +46.3 +72.5 +81.7 +2 +12.9 ++Different Number of Captions for Data Augmentation +Top-1 +44.2 +70.7 +81.5 +2 +16.2 +46.3 +72.5 +81.7 +2 +12.9 +Top-3 +43.3 +71.7 +81.6 +2 +15.0 +45.5 +73.8 +82.4 +2 +12.7 +Top-5 +43.4 +70.6 +80.4 +2 +16.2 +45.6 +72.7 +82.7 +2 +12.9 ++Video-Caption Feature Interaction +Video Only +44.2 +70.7 +81.5 +2 +16.2 +46.3 +72.5 +81.7 +2 +12.9 +Sum +43.8 +71.5 +80.3 +2 +16.1 +47.2 +73.3 +82.8 +2 +13.1 +Concat-MLP +37.5 +66.1 +78.4 +3 +15.7 +40.0 +68.7 +79.9 +2 +12.7 +Cross Transformer +44.6 +71.6 +80.3 +2 +14.6 +47.9 +75.4 +83.0 +2 +11.5 +Co-attention Transformer +45.3 +71.2 +80.9 +2 +15.0 +48.5 +74.0 +82.5 +2 +12.7 ++Query-Caption Matching Score +Query-Video Only +45.3 +71.2 +80.9 +2 +15.0 +48.5 +74.0 +82.5 +2 +12.7 +Query-Caption Only +30.7 +55.2 +67.5 +4 +26.4 +30.7 +55.2 +67.5 +4 +26.4 +Query-Video + Query-Caption +45.6 +71.7 +81.2 +2 +14.8 +49.3 +74.2 +83.4 +2 +12.1 +Table 5. Component-wise evaluation of our framework on the MSR-VTT 1K validation set. With the ViT-B/32 backbone, we report the +text-to-video retrieval results for two representative Query-Video matching mechanisms: global matching and fine-grained matching. The +consistent improvement on two typical matching mechanisms demonstrates the generalization ability and effectiveness of our method. +3.3. Ablation Study +In this section, we provide detailed ablation studies to +clarify the effects of each part of our design. Results are +obtained using the ViT-B/32 backbone. +Auxiliary caption as data augmentation. We begin by in- +vestigating the impact of captions on data augmentation for +training. In a real-world scenario, we believe that the orig- +inal video title would naturally be an additional auxiliary +caption. As a result, we use the annotation of the dataset to +manually extract the title from the video’s original webpage, +ignoring the expired source link, and compare it to the cap- +tion generated by the GPT-2 model. The results of different +sources of the caption are shown in Table 5, from which we +can see that using the captions generated by the web-scale +model as data augmentation for training can directly bring +additional R@1 improvements (+1.4%, +0.6%) under both +matching mechanisms. For global matching, using video +titles can also bring a 1% improvement. +Then we explore how many generated captions to use +as augmentation. We use the caption filtering mechanism +mentioned in Sec. 2.3 to rank the relevancy of captions and +ground-truth query, then select different numbers of cap- +tions for training. Results show that one caption is enough. +Video-Caption feature interaction. As stated in Sec. 2.4, +we design four ways for Video-Caption feature interaction. +From Table 5, we summarize the following observations: 1) +The most basic feature interaction approach, Sum, can en- +hance fine-grained matching by 0.9% R@1, but there is no +discernible gain in global matching. 2) MLP is difficult to +optimize and perform poorly in both matching settings. We +conjecture that, while the MLP provides a nonlinear metric +space, its operation in a black-box environment may cause +degradation. 3) Cross Transformer brings +0.4%, +1.6% +improvements in two matching settings, respectively, which +may be attributed to the self-attention mechanism that can +capture the inter-modal relationship between video and cap- +tion. 4) Furthermore, Co-attention Transformer can boost +the performance by +1.1% and +2.2% for these two match- +ing mechanisms. +In summary, the experimental results +show that proper interaction between video and the gen- +erated caption can produce better video representation for +improving Query-Video matching. +Query-Caption matching. +We further investigate the +Query-Caption matching branch for text-video retrieval. +We use the mean pooling to aggregate caption embed- +dings to yield a global embedding. As shown in Table 5, +the single Query-Caption matching branch can achieve +30.7% R@1 on text-to-video retrieval, outperforming sev- +eral previous query-video matching methods such as Clip- +BERT [17](22.0%) and MMT [17](26.6%). This encour- +ages us to combine its score with Query-Video match- +ing branch, further improving the performance (+0.8%). +7 + +Query7765：a person is discussing a car. +Rank ++ Caption +video of a car camera +recording the driver’s voice. +Video +Rank +2 +1 +3 +6 +4 +5 +Query9616：person is recording the brown horse which is having fun. +Rank ++ Caption +Video +Rank +2 +1 +3 +2 +1 +4 +video showing the car in +a parking spot. +video of SUV in the video +below shows a salesman +talking to an audience. +video of the horse jumping over +a fence at Ranch in Nevada +was captured on camera. +video showing animation of +a horse’s simulation, which +simulates the game. +video showing a horse +simulation video game in which +you could see your avatar +being animated by the camera. +Figure 4. The text-video results on the MSR-VTT 1K-A test set. +The left is the ranking results of query-video matching model, and +the right is the ranking results of Cap4Video which involve gener- +ated caption to enhance retrieval. Please zoom in for best view. +The results demonstrate that the Query-Caption matching +branch can complement the regular Query-Video matching +branch for enhanced text-video retrieval. +Overall, Our Cap4Video employs the generated captions +from three aspects: input data augmentation, intermedi- +ate feature interaction, and output score fusion, resulting +in consistent improvements (+2.8% and +3.6%) for both +matching mechanisms. +3.4. Visualization +We present two examples of videos retrieved by our +method and the model without auxiliary captions. As shown +in Figure 4, with the help of the caption, our Cap4Video +successfully retrieves the ground-truth video, whereas the +video-only model returns several videos that are somewhat +relevant to the query but are not precise. See more qualita- +tive results in Supplementary. +4. Related Works +Zero-shot Image Captioning. +In NLP, OpenAI pre- +sented the transformer-based [33] GPT models [4,30] which +are trained on large-scale text corpora and can then gener- +ate text given a prompt. In computer vision, CLIP [29] has +emerged as a successful vision-language alignment model +by training on 400M noisy web-collected image-text pairs. +On tasks such as image classification and image-text re- +trieval, the learned joint model shows impressive zero-shot +performance. However, the research on transferring web- +scale models to zero-shot image captioning is still limited. +ZeroCap [32] first proposes to employ CLIP together with +the GPT-2 language model to generate a textual description +of the input image, benefiting from the knowledge in both +web-scale models. ZeroCap is truly zero-shot, where the +optimization is performed “ex post facto” in the activation +space without re-training or fine-tuning the model parame- +ters. More recently, MAGIC [31] also employs CLIP scores +to shift GPT-2 logits towards image correspondence. De- +spite this, it is necessary to fine-tune the GPT-2 on the text +corpus of MS-COCO captions. In this paper, we expand the +zero-shot capability of ZeroCap [32] to the video domain to +generate auxiliary captions without additional training. +Text-Video Retrieval. +Text-Video Retrieval aims to +find relevant video content based on natural language de- +scriptions. Early studies [5, 8, 20, 35, 37] focus on knowl- +edge transfer from “expert” models and capture intra-modal +and cross-modal interaction based on pre-extracted features. +However, the performance of these methods is limited since +they cannot perform end-to-end optimization. +Recently, +more methods involve end-to-end training for text-video re- +trieval. One typical line [2, 25, 26] is to first do large-scale +text-video pre-training, then transfer the model to the down- +stream text-video retrieval tasks. +Meanwhile, mismatch- +ing noise in text-video retrieval datasets may exist. An- +other training-efficient line is to directly expand the pre- +trained image-text model to the text-video retrieval task. +CLIPBERT [17] enables affordable pioneering end-to-end +training with a sparse sampling strategy. After that, more +recent works [3, 7, 9, 11, 21, 23, 43] focus on transferring +knowledge from publicly available CLIP models that have +been pre-trained on 400M image-text pairs. CLIP4Clip [23] +firstly provides a strong baseline with CLIP’s initialization. +The research path has evolved from the most direct global +matching (i.e., video-sentence alignment [9, 23]) to fine- +grained matching (e.g., frame-word alignment [34], video- +word alignment [11], multi-hierarchical alignment [7, 27], +etc.). Unlike these previous efforts on query-video match- +ing, we propose to generate auxiliary captions from offline +videos to improve text-video retrieval. Thus our method is +compatible with both global and fine-grained matching. +5. Conclusion +We present a new framework Cap4Video which use cap- +tions generated by web-scale language models to benefit the +text-video matching in three aspects: 1) Input data augmen- +tation for training. 2) Intermediate video-caption feature +interaction for compact video representations. 3) Output +score fusion for enhance text-video retrieval. 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T2vlad: global- +local sequence alignment for text-video retrieval. +In Pro- +ceedings of the IEEE/CVF Conference on Computer Vision +and Pattern Recognition, pages 5079–5088, 2021. 8 +[38] Zuxuan Wu, Ting Yao, Yanwei Fu, and Yu-Gang Jiang. Deep +learning for video classification and captioning. In Frontiers +of multimedia research, pages 3–29. 2017. 2, 5, 6 +[39] Jun Xu, Tao Mei, Ting Yao, and Yong Rui. Msr-vtt: A large +video description dataset for bridging video and language. In +CVPR, pages 5288–5296, 2016. 2, 5, 6 +[40] Lewei Yao, Runhui Huang, Lu Hou, Guansong Lu, Minzhe +Niu, Hang Xu, Xiaodan Liang, Zhenguo Li, Xin Jiang, and +Chunjing Xu. Filip: Fine-grained interactive language-image +pre-training. In ICLR, 2021. 2 +[41] Jiahui Yu, Zirui Wang, Vijay Vasudevan, Legg Yeung, Mo- +jtaba Seyedhosseini, and Yonghui Wu. Coca: Contrastive +captioners are image-text foundation models. arXiv preprint +arXiv:2205.01917, 2022. 1 +[42] Lu Yuan, Dongdong Chen, Yi-Ling Chen, Noel Codella, +Xiyang Dai, Jianfeng Gao, Houdong Hu, Xuedong Huang, +Boxin Li, +Chunyuan Li, +et al. +Florence: +A new +foundation model for computer vision. +arXiv preprint +arXiv:2111.11432, 2021. 1 +[43] Shuai Zhao, Linchao Zhu, Xiaohan Wang, and Yi Yang. Cen- +terclip: Token clustering for efficient text-video retrieval. +arXiv preprint arXiv:2205.00823, 2022. 6, 8 +10 + diff --git a/c9AyT4oBgHgl3EQfXfd-/content/tmp_files/load_file.txt b/c9AyT4oBgHgl3EQfXfd-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8337aac51f304c78fb5b6cea347f69a43c823cc4 --- /dev/null +++ b/c9AyT4oBgHgl3EQfXfd-/content/tmp_files/load_file.txt @@ -0,0 +1,952 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf,len=951 +page_content='Cap4Video: What Can Auxiliary Captions Do for Text-Video Retrieval?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Wenhao Wu1,2* Haipeng Luo3∗ Bo Fang3 Jingdong Wang1 Wanli Ouyang2,4 1Baidu Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2The University of Sydney 3University of Chinese Academy of Sciences 4Shanghai AI Laboratory whwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='ucas@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='com Abstract Most existing text-video retrieval methods focus on cross-modal matching between the visual content of offline videos and textual query sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' However, in real sce- narios, online videos are frequently accompanied by rele- vant text information such as titles, tags, and even subti- tles, which can be utilized to match textual queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' This in- spires us to generate associated captions from offline videos to help with existing text-video retrieval methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' To do so, we propose to use the zero-shot video captioner with knowledge of pre-trained web-scale models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', CLIP and GPT-2) to generate captions for offline videos without any training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Given the captions, one question naturally arises: what can auxiliary captions do for text-video retrieval?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In this paper, we present a novel framework Cap4Video, which makes use of captions from three aspects: i) Input data: The video and captions can form new video-caption pairs as data augmentation for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' ii) Feature inter- action: We perform feature interaction between video and caption to yield enhanced video representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' iii) Output score: The Query-Caption matching branch can be com- plementary to the original Query-Video matching branch for text-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We conduct thorough ablation stud- ies to demonstrate the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Without any post-processing, our Cap4Video achieves state-of-the- art performance on MSR-VTT (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4%), VATEX (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6%), MSVD (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8%), and DiDeMo (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Introduction Text-Video retrieval is a fundamental task for video-and- language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' With the rapid development of image-language pre-training (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', CLIP [29], ALIGN [13], CoCa [41], Florence [42], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' ), more recent research has concentrated on expanding the pre-trained image-language models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', CLIP [29]) to the text-video retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The research path has evolved from the most direct global Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Query Caption Video Query-Video Matching New training pairs Query-Caption Matching Cross-modal interaction Video Query Query-Video Matching Video Caption CLIP GPT-2 Captioner (a) (b) (c) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' (a) Existing learning framework for text-video re- trieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' (b) Zero-shot video captioning by guiding a large-scale pre-trained language model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' GPT-2 [30]) with CLIP [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' (c) Our Cap4Video framework employs the generated captions from three aspects: input data augmentation, intermediate feature inter- action, output score fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' matching (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', video-sentence alignment [9, 23]) to fine- grained matching (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', frame-word alignment [34], video- word alignment [11], multi-hierarchical alignment [7, 27], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' These works demonstrate remarkable performance and significantly outperform previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The key rea- sons for the improvement are as follows: First, CLIP offers powerful visual and textual representations that are roughly pre-aligned in the semantic embedding space, hence re- ducing the challenge of cross-modal learning on video-text matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Second, these works can finetune the pre-trained vision and text encoders using the sparsely sampled frames in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' All of these methods aim to learn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='00184v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='CV] 31 Dec 2022 cross-modal alignment between the visual representation of offline videos and the textual representation of the corre- sponding query, as depicted in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' However, in a real scenario, online videos are usually accompanied by related content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' For example, we may sim- ply obtain the video’s title or tag from the video website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In addition to the visual information in the video, the asso- ciated textual information can also be used to describe the video content to some extent and to match the query (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', the common text-to-text retrieval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' This leads to a ques- tion: How to generate associated text descriptions for of- fline videos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' A possible solution is to crawl the video title from the video website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' However, this method relies on an- notation, and there is also the risk that the video website has become invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Another automated solution is to gen- erate captions utilizing zero-shot video caption models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We turn our attention to knowledge-rich pre-trained models to handle such challenging open-set scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We find that the recent study ZeroCap [32] provides a good practice to use frozen CLIP [29] and GPT-2 [30] for zero-shot image captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' So we extend it to the video domain for video captioning without any further training or tuning stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Given the auxiliary captions, one question naturally arises: How to make full use of these generated captions to improve the performance of text-video retrieval?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In this paper, as shown in Figure 1(c), we present a Cap4Video learning framework which utilizes captions from three as- pects: (i) Input Data: The simplest way is to augment the training data with the generated captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Specifically, the given video and its generated caption are a matched pair, so they can be regarded as additional positive sample pairs other than the query-video pair for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' (ii) Feature Interaction: we can perform cross-modal interaction be- tween the video and captions to improve video representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Specifically, we may take advantage of the information complementarity between videos and captions to reduce re- dundant features from videos and learn more discrimina- tive video representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' (iii) Output score: The gener- ated caption can also represent the video’s content, so we can employ query-caption matching to complement stan- dard query-video matching for the text-video retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We can further utilize the two-stream architecture to reduce the bias of the model and produce more robust results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' To demonstrate the effectiveness of the aforementioned explorations, we conduct extensive experiments on four well-known video datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We hope our new paradigm will stimulate more investigation into text-video retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We summarize the contributions as follows: We investigate a novel problem: how to use captions generated by web-scale language models to help with the text-video retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Our motivation is to lever- age the vast knowledge of web-scale pre-trained lan- guage models to automatically generate extra text in- formation for offline videos rather than labor-intensive annotations, to benefit text-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We present a Cap4Video learning framework that makes full use of the generated captions from three as- pects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', input data, feature interaction, output score) and can bring further performance improvement to the existing query-video matching mechanisms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', global matching, fine-grained matching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Extensive experiments on four text-video retrieval datasets demonstrates the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Our Cap4Video achieves state-of-the-art performance on MSR-VTT [39] (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4%), VATEX [36] (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6%), MSVD [38] (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8%), and DiDeMo [1] (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Preliminaries: Text-Video Matching The main goal of text-video matching is to develop a function s(Qi, Vj) to determine how similar the video Vj is to a sentence Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In the text-to-video retrieval, the ob- jective is to rank all the videos given the query sentence ac- cording to their similarity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' To transfer the image-text pre-training knowledge into video-text learning, we follow recent works [7, 9, 23] to apply CLIP [29] for initialization to improve text-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Figure 2(b) depicts two typ- ical text-video matching mechanisms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', global matching, and fine-grained matching) that serve as our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Global Matching is widely used in cross-model contrastive learning [14,23,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In text-video contrastive learning, we train the visual encoder for a given video that samples F frames and the query encoder for a given query sentence that contains W words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The visual encoder then outputs F frame embeddings, while the query encoder returns two types of embeddings: W word embeddings and the [CLS] embedding as the global representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' These frame em- beddings are then integrated with average pooling to yield the global video embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Finally, we can get the simi- larity between this global video embedding and the global query embedding for matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Fine-grained Matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The global matching encodes each modality independently to get the global features be- fore calculating the similarity between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The solution is simple and intuitive, but it ignores the fine-grained align- ment between the two modalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', frame-word align- ment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' To model the token-wise word-patch alignment for image-text learning, FILIP [40] and ColBERT [15] use a Max-Mean pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Specifically, they get the token-wise maximum similarity between the patch and word tokens, then use the average token-wise maximum similarity of to- kens in the image (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' text) as the similarity of an image to a text (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' a text to an image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' DRL [34] expands the token-wise alignment into text-video retrieval and further 2 Video shows a girl singing in front of the audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='a girls performing a song on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='the stage for competition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Video ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Query ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Caption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Video-Caption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Interaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='QC Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='QV Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='(b) Query-Video Matching: Two typical mechanisms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='SQV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Frame embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='① Global embedding matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='F × D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 × D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 × D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Query [CLS] embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='F × D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='W × D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Word embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='② Fine-grained embedding matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Frame embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='W×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='SQ2V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='F×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='SV2Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='SQV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='F×W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='(c) Query-Caption Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Captions embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Caption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='C × D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 × D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 × D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Query [CLS] embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='SQV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='SQC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='SQVC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='SQC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='(a) Our framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Captioner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='V1 V2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Vn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Q1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Qn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Similarity matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Extra query augmentation for training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Contrastive training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='··· ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='FC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='FC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='W×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='F×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' An overview of our Cap4Video for text-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In Cap4Video, we first generate captions using a zero-shot video captioner which combines CLIP [29] with GPT-2 [30], leveraging knowledge from both frozen web-scale models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Then we utilize the pre-extracted caption information from three aspects: i) Input data: The video and captions can form new pairs as data augmentation for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' ii) Feature interaction: We perform feature interaction between video and caption to capture intra- and inter-modality context to yield enhanced video representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' iii) Output score: The Query-Caption matching branch can be complementary to the original Query-Video matching branch for text-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' proposes to use the attention mechanism to learn a weighted pooling instead of mean pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Thus, we use fine-grained matching to guide the contrastive objective and as our en- hanced baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Preprocessing: Caption Generation To get the auxiliary caption for the given video, we uti- lize the knowledge from the pre-trained language model to generate rich and diverse captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' To compare captions from different sources, we also choose a common case: the video’s title as an auxiliary caption from the video URL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Zero-shot Video Captioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Considering the paradigm’s scalability, we aim to generate captions directly from down- stream video data without extra training, a process known as zero-shot video captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Following ZeroCap [32], we use the GPT-2 [30] to infer the next word from an ini- tial prompt, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', “Video shows”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' A calibrated CLIP [29] loss drives the model to generate sentences that describe the video to incorporate video-related knowledge into the auto- regression process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' See Supplementary for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Data Augmentation with Auxiliary Captions Given the generated captions, the most natural applica- tion of the captions is to augment training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' For exam- ple, given a dataset consisting of N videos and correspond- ing query sentences, each video of this dataset and its gen- erated caption are a matched pair, so they can be regarded as additional positive sample pairs other than the query-video pair for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In this way, if we choose one caption per video, then we can increase at least N pairs as additional data augmentation during the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The video captioner can generate multiple captions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', 20) for each offline video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We think that some of these cap- tions could contain noise, meaning that they may not be en- tirely relevant to the video content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' If all of them are used as training data directly, this could have a negative effect on training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Therefore, we design a filtering mechanism that evaluates the semantic similarity between each caption and the ground-truth query of the video (using a pre-trained text encoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The caption with the highest similarity is then chosen for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Note that we use the ground- truth query for caption filtering only in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Video-Caption Cross-Modal Interaction We further perform cross-modal interactions between the video and generated caption to improve video representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Our motivation is that we may take advantage of the information complementarity between videos and captions 3 Concat MLP Transformer Encoder × L Transformer Encoder Q K V Transformer Encoder × L (a) Sum (b) MLP (c) Cross Transformer (d) Co-attention Transformer Caption Embedding Frame Embedding Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Illustration of four Video-Caption interaction strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The enhanced frame embeddings will be followed by a mean pool- ing for global matching or will remain for fine-grained matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' to reduce redundant features from videos and learn more discriminative video representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' To make this more concrete, we now discuss several ways of interaction be- tween two modalities as depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In order not to change the original pre-trained CLIP en- coders as much as possible for better transferring, we only perform interaction between the final caption and frame em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' To obtain the compact representation from all generated captions, we simply utilize average pooling for the output of the caption encoder as follows: ec = 1 C C � i=1 ci, (1) where ci ∈ RD represents the [CLS] embedding of i-th caption and C is the number of generated captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Then we feed the frame embeddings ev = {v1, v2, · · · , vF } and global caption embedding ec into the interaction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' It is an intuitive solution to compute the sum of each frame embedding and the global caption embedding to yield enhanced frame embedding: Sum(vi, ec) = vi + ec for i = 1, · · · , F, (2) where F is the number of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We concatenate each frame embedding with the global caption embedding as a whole, then use the learnable Multi-layer Perceptron (MLP) to model weighted combina- tions of the two embeddings: MLP(vi, ec) = fθ([vi, ec]) for i = 1, · · · , F, (3) where [·, ·] is the denotes concatenation operation, fθ is the MLP with parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Cross Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We also consider adopt the self- attention [33] for cross-modal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' As shown in Figure 3(c), the Cross Transformer operates on sequences of embeddings {v1, · · · , vF , ec}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' These embeddings are passed through L “encoder-style” transformers blocks to produce final representations: Cross(ev, ec) = fψ({ev, ec}), (4) where {} means that ev, ec form a sequence, and fψ is the transformer encoders with parameter ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Co-attention Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Another typical informa- tion exchange mechanism is co-attention [22], which passes the keys and values from one modality as input to the other modality’s multi-headed attention block of a standard trans- former encoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Here we introduce one co-attentional transformer layer to enable information exchange between modalities, followed by L standard transformer layers to model temporal information: CoAttn(ev, ec) = fφ2(fφ1(ev, ec)), (5) where fφ1 is the co-attentional transformer with parameter φ1 and fφ2 is the transformer encoders with parameter φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Next, the frame embeddings generated by the Video- Caption interaction module will be then averaged for global matching or will be kept for fine-grained matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Complementary Query-Caption Matching In addition to the aforementioned uses of caption for data augmentation and video feature enhancement, the generated caption itself can represent the content of the video, making it possible to use it directly for retrieval (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', text-text re- trieval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Specifically, each of the C captions generated by the video is then passed through the caption encoder to ob- tain its [CLS] text embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' As shown in Figure 2(c), these caption embeddings are then aggregated to form a global representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Then the cosine similarity between this global caption embedding and the global query embed- ding is calculated to complement the query-video matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Denotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Given a batch of B triples {evi, eti, eci}B i=1, where evi, eti, eci denote the i-th video, query, and caption embedding, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Note that the term “embedding” used here is more general for convenience, the exact mean- ing of embedding will vary depending on the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' For example, in query-video global matching, evi and eti rep- resent the averaged video feature and global [CLS] text fea- ture, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In query-video fine-grain matching, evi and eti represent a sequence of frame embeddings and a se- quence of word embeddings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In query-caption matching, eci represents a sequence of caption embeddings and eti represents a global [CLS] text feature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Learning objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' For the Query-Caption branch, we want the caption embedding ec and the query embedding 4 et to be close while they are related and far apart when they are not during training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We follow the common prac- tice [23,34] to consider the bidrectional learning objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We employ symmetric cross-entropy loss to maximize the similarity between matched Query-Caption pairs and mini- mize the similarity for other pairs: LC2Q = − 1 B B � i log exp(sqc(eti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' eci)/τ) �B j exp(sqc(eti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' ecj)/τ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' LQ2C = − 1 B B � i log exp(sqc(eti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' eci)/τ) �B j exp(sqc(etj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' eci)/τ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' LQC = 1 2(LC2Q + LQ2C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' (6) where sqc(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' ·) represents the query-caption matching simi- larity function shown in Figure 2(c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' and τ refers to the tem- perature hyper-parameter for scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Similarly, the con- trastive loss for Query-Video branch is formulated as: LV 2Q = − 1 B B � i log exp(sqv(eti, evi)/τ) �B j exp(sqv(eti, evj)/τ) , LQ2V = − 1 B B � i log exp(sqv(eti, evi)/τ) �B j exp(sqv(etj, evi)/τ) , LQV = 1 2(LV 2Q + LQ2V ), (7) where sqv(·, ·) represents the query-video matching (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', global matching, fine-grained matching) similarity function shown in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The total loss L is the sum of Query- Video loss LQV and Query-Caption loss LQC: L = LQV + LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' (8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Experiments: Text-Video Retrieval 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Setups Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We conduct experiment on four popular bench- marks for video-to-text retrieval and text-to-video retrieval tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' MSR-VTT [39] contains a total of 10K video clips, each having 20 captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Following the data splits from [8,23,26], we train models with associated captions on the Training-9K set and report results on the test 1K-A set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' DiDeMo [1] has 10K videos paired with 40K descrip- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Following previous works [2, 17, 23], we concate- nate all descriptions of one video to a single query, acting as a video-paragraph retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' VATEX [36] collects ∼35K videos, each with multiple annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' There are ∼26K videos for training, 1,500 videos for validation and 1,500 videos for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' MSVD [38] contains 1,970 videos with 80K captions, with ∼40 captions on average per video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' There are 1,200, 100, and 670 videos in the train, validation, and test sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' For brevity, we abbreviate Recall at K to R@K (K = 1, 5, 10) upon all datasets, which com- putes the percentage of correct videos among the top K re- trieved videos given textual queries (Text→Video, and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' MdR, Median Rank, computes the median of the ground-truth in the retrieval ranking list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' MnR, Mean Rank, computes the mean rank of the correct results in the retrieval ranking list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Note that for MdR and MnR, the lower score means the better (indicated as ↓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In all experiments, we use the visual encoder of CLIP [29] as our video encoder and use the textual encoder of CLIP as both the caption and query encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' To reduce conflict between the two branches, we train the query-video branch first, then the query-caption branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The caption length is 32 and the video length is 12 for all datasets except DiDeMo (64 max words and 64 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The network is optimized by Adam [16] with a batch size of 128 and epoch 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The initial learning rate is 1e- 7 for the clip parameters and 1e-4 for the non-clip param- eters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Following previous works [11, 21, 23], we train model for 5 epochs with Adam [16] optimizer and adopt a warmup [12] strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' All learning rates follow the cosine learning rate schedule with a linear warmup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The number C of generated captions per video is set to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The number L of transformer layers is set to 4 for VATEX and MSR-VTT, and 1 for Didemo and MSVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Comparison with State-of-the-arts Here we compare our Cap4Video with recent state-of- the-art methods on the four benchmarks, MSR-VTT [39], MSVD [38], VATEX [36] and DiDeMo [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Table 1 lists the comparisons on DiDeMo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We can see that our Cap4Video significantly surpasses CLIP4Clip [23] by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2% R@1 and exceeds DRL [34] by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0%, which proves the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Method R@1 R@5 R@10 MdR MnR CE [19] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 ClipBERT [17] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 Frozen [2] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 TMVM [18] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 CLIP4Clip [23] 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 TS2-Net [21] 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 HunYuan [27] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 DRL [34] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 Cap4Video (Ours) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Results of text-to-video retrieval on the DiDeMo [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Comparisons with recent state-of-the-art models on MSR-VTT are provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Our approach signif- icantly outperforms previous works and achieves new state- of-the-art performance with both ViT-B/32 and ViT-B/16 5 Method Date Text → Video Video → Text R@1 R@5 R@10 MdR↓ MnR↓ R@1 R@5 R@10 MdR↓ MnR↓ ClipBERT [17] CVPR’20 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 MMT [8] ECCV’20 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 SupportSet [28] ICLR’21 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 Frozen [2] ICCV’21 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 BridgeFormer [10] CVPR’22 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 TMVM [18] NeurIPS’22 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 CLIP-ViT-B/32 CLIP4Clip [23] ArXiv’21 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 8 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Retrieval results on the validation set of MSR-VTT 1K [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Here we report results without any post-processing operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', DSL [6] or QB-Norm [3]) during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Method R@1 R@5 R@10 MdR MnR CE [19] 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 SUPPORT [28] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 CLIP [29] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 Frozen [2] 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 TMVM [18] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 CLIP4Clip [23] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 X-Pool [11] 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 Cap4Video (Ours) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Results of text-to-video retrieval on the MSVD [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' For example, on text-to-video retrieval, we achieve +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8% higher R@1 than CLIP4Clip with the same ViT-B/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Also, our Cap4Video outperforms the brand-new method TS2-Net [21] by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0% with ViT-B/32 and ViT-B/16, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Table 3 and Table 4 show results for the MSVD dataset and VATEX dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We use the ViT-B/16 as our backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' For MSVD, our model achieves remarkable performance 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8% R@1 and a performance improvement of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6% on text-to-video retrieval when compared to CLIP-based models CLIP4Clip [23] and X-Pool [24], re- Method R@1 R@5 R@10 MdR MnR HGR [5] 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 CLIP [29] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 SUPPORT [28] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 CLIP4Clip [23] 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 Clip2Video [7] 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 QB-Norm [3] 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 TS2-Net [21] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5 Cap4Video (Ours) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Results of text-to-video retrieval on the VATEX [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' For VATEX, our approach also outperforms the recent state-of-the-art methods and achieves +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='5% R@1 improvement over TS2-Net [21] for text-to-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Recently, a few methods post-process the similarity gener- ated by the model to significantly improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' It should be noted that in all of the Tables, the results are re- ported without any unfair post-processing processes, such as DSL [6] and QB-Norm [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Overall, the consistent state-of-the-art performance across four benchmarks demonstrates the effectiveness of our Cap4Video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 6 Method Global embedding matching Fine-grained embedding matching R@1 R@5 R@10 MdR↓ MnR↓ R@1 R@5 R@10 MdR↓ MnR↓ Baseline 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0 2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 +Different Sources of Caption as Data Augmentation Video Title from Source URL 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9 2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Component-wise evaluation of our framework on the MSR-VTT 1K validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' With the ViT-B/32 backbone, we report the text-to-video retrieval results for two representative Query-Video matching mechanisms: global matching and fine-grained matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The consistent improvement on two typical matching mechanisms demonstrates the generalization ability and effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Ablation Study In this section, we provide detailed ablation studies to clarify the effects of each part of our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Results are obtained using the ViT-B/32 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Auxiliary caption as data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We begin by in- vestigating the impact of captions on data augmentation for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In a real-world scenario, we believe that the orig- inal video title would naturally be an additional auxiliary caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' As a result, we use the annotation of the dataset to manually extract the title from the video’s original webpage, ignoring the expired source link, and compare it to the cap- tion generated by the GPT-2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The results of different sources of the caption are shown in Table 5, from which we can see that using the captions generated by the web-scale model as data augmentation for training can directly bring additional R@1 improvements (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4%, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6%) under both matching mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' For global matching, using video titles can also bring a 1% improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Then we explore how many generated captions to use as augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We use the caption filtering mechanism mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='3 to rank the relevancy of captions and ground-truth query, then select different numbers of cap- tions for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Results show that one caption is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Video-Caption feature interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' As stated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4, we design four ways for Video-Caption feature interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' From Table 5, we summarize the following observations: 1) The most basic feature interaction approach, Sum, can en- hance fine-grained matching by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='9% R@1, but there is no discernible gain in global matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2) MLP is difficult to optimize and perform poorly in both matching settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We conjecture that, while the MLP provides a nonlinear metric space, its operation in a black-box environment may cause degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 3) Cross Transformer brings +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4%, +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6% improvements in two matching settings, respectively, which may be attributed to the self-attention mechanism that can capture the inter-modal relationship between video and cap- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 4) Furthermore, Co-attention Transformer can boost the performance by +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='1% and +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='2% for these two match- ing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In summary, the experimental results show that proper interaction between video and the gen- erated caption can produce better video representation for improving Query-Video matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Query-Caption matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We further investigate the Query-Caption matching branch for text-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' We use the mean pooling to aggregate caption embed- dings to yield a global embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' As shown in Table 5, the single Query-Caption matching branch can achieve 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='7% R@1 on text-to-video retrieval, outperforming sev- eral previous query-video matching methods such as Clip- BERT [17](22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='0%) and MMT [17](26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' This encour- ages us to combine its score with Query-Video match- ing branch, further improving the performance (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 7 Query7765：a person is discussing a car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Rank + Caption video of a car camera recording the driver’s voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Video Rank 2 1 3 6 4 5 Query9616：person is recording the brown horse which is having fun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Rank + Caption Video Rank 2 1 3 2 1 4 video showing the car in a parking spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' video of SUV in the video below shows a salesman talking to an audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' video of the horse jumping over a fence at Ranch in Nevada was captured on camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' video showing animation of a horse’s simulation, which simulates the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' video showing a horse simulation video game in which you could see your avatar being animated by the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The text-video results on the MSR-VTT 1K-A test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The left is the ranking results of query-video matching model, and the right is the ranking results of Cap4Video which involve gener- ated caption to enhance retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Please zoom in for best view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The results demonstrate that the Query-Caption matching branch can complement the regular Query-Video matching branch for enhanced text-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Overall, Our Cap4Video employs the generated captions from three aspects: input data augmentation, intermedi- ate feature interaction, and output score fusion, resulting in consistent improvements (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='8% and +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='6%) for both matching mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Visualization We present two examples of videos retrieved by our method and the model without auxiliary captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' As shown in Figure 4, with the help of the caption, our Cap4Video successfully retrieves the ground-truth video, whereas the video-only model returns several videos that are somewhat relevant to the query but are not precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' See more qualita- tive results in Supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Related Works Zero-shot Image Captioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In NLP, OpenAI pre- sented the transformer-based [33] GPT models [4,30] which are trained on large-scale text corpora and can then gener- ate text given a prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In computer vision, CLIP [29] has emerged as a successful vision-language alignment model by training on 400M noisy web-collected image-text pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' On tasks such as image classification and image-text re- trieval, the learned joint model shows impressive zero-shot performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' However, the research on transferring web- scale models to zero-shot image captioning is still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' ZeroCap [32] first proposes to employ CLIP together with the GPT-2 language model to generate a textual description of the input image, benefiting from the knowledge in both web-scale models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' ZeroCap is truly zero-shot, where the optimization is performed “ex post facto” in the activation space without re-training or fine-tuning the model parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' More recently, MAGIC [31] also employs CLIP scores to shift GPT-2 logits towards image correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' De- spite this, it is necessary to fine-tune the GPT-2 on the text corpus of MS-COCO captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In this paper, we expand the zero-shot capability of ZeroCap [32] to the video domain to generate auxiliary captions without additional training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Text-Video Retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Text-Video Retrieval aims to find relevant video content based on natural language de- scriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Early studies [5, 8, 20, 35, 37] focus on knowl- edge transfer from “expert” models and capture intra-modal and cross-modal interaction based on pre-extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' However, the performance of these methods is limited since they cannot perform end-to-end optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Recently, more methods involve end-to-end training for text-video re- trieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' One typical line [2, 25, 26] is to first do large-scale text-video pre-training, then transfer the model to the down- stream text-video retrieval tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Meanwhile, mismatch- ing noise in text-video retrieval datasets may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' An- other training-efficient line is to directly expand the pre- trained image-text model to the text-video retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' CLIPBERT [17] enables affordable pioneering end-to-end training with a sparse sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' After that, more recent works [3, 7, 9, 11, 21, 23, 43] focus on transferring knowledge from publicly available CLIP models that have been pre-trained on 400M image-text pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' CLIP4Clip [23] firstly provides a strong baseline with CLIP’s initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' The research path has evolved from the most direct global matching (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', video-sentence alignment [9, 23]) to fine- grained matching (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=', frame-word alignment [34], video- word alignment [11], multi-hierarchical alignment [7, 27], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Unlike these previous efforts on query-video match- ing, we propose to generate auxiliary captions from offline videos to improve text-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Thus our method is compatible with both global and fine-grained matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Conclusion We present a new framework Cap4Video which use cap- tions generated by web-scale language models to benefit the text-video matching in three aspects: 1) Input data augmen- tation for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 2) Intermediate video-caption feature interaction for compact video representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 3) Output score fusion for enhance text-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Cap4Video achieves consistent improvements on four standard text- video retrieval benchmarks and outperform the state-of-the- art by a clear margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' 8 MOTORMOTORFoal (F)AdultReferences [1] Lisa Anne Hendricks, Oliver Wang, Eli Shechtman, Josef Sivic, Trevor Darrell, and Bryan Russell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' Localizing mo- ments in video with natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} +page_content=' In ICCV, pages 5803– 5812, 2017.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AyT4oBgHgl3EQfXfd-/content/2301.00184v1.pdf'} diff --git a/cNFQT4oBgHgl3EQfiDZt/content/tmp_files/2301.13348v1.pdf.txt b/cNFQT4oBgHgl3EQfiDZt/content/tmp_files/2301.13348v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d5ac1887cfd39917f676de3ece484c5ef96738cf --- /dev/null +++ b/cNFQT4oBgHgl3EQfiDZt/content/tmp_files/2301.13348v1.pdf.txt @@ -0,0 +1,5465 @@ +A REINFORCEMENT LEARNING FRAMEWORK FOR DYNAMIC +MEDIATION ANALYSIS +A PREPRINT +Lin Ge1, Jitao Wang2, Chengchun Shi3, Zhenke Wu2, and Rui Song1 +1North Carolina State University +2University of Michigan, Ann Arbor +3London School of Economics and Political Science +ABSTRACT +Mediation analysis learns the causal effect transmitted via mediator variables between treatments and +outcomes and receives increasing attention in various scientific domains to elucidate causal relations. +Most existing works focus on point-exposure studies where each subject only receives one treatment +at a single time point. However, there are a number of applications (e.g., mobile health) where the +treatments are sequentially assigned over time and the dynamic mediation effects are of primary +interest. Proposing a reinforcement learning (RL) framework, we are the first to evaluate dynamic +mediation effects in settings with infinite horizons. We decompose the average treatment effect into +an immediate direct effect, an immediate mediation effect, a delayed direct effect, and a delayed +mediation effect. Upon the identification of each effect component, we further develop robust and +semi-parametrically efficient estimators under the RL framework to infer these causal effects. The +superior performance of the proposed method is demonstrated through extensive numerical studies, +theoretical results, and an analysis of a mobile health dataset. +1 +Introduction +Mediation analysis aims to understand the causal pathway from an exposure (e.g., treatment or action) to an outcome +variable of interest. It is gaining increasing popularity recently and has been frequently employed in a number of +domains including epidemiology (Richiardi et al., 2013; Rijnhart et al., 2021), psychology (Rucker et al., 2011), genetics +(Chakrabortty et al., 2018; Zeng et al., 2021; Djordjilovi´c et al., 2022), and economics (Celli, 2022). +Our paper is motivated by the need to learn the dynamic mediation effects in sequential decision making. One motivating +example is given by the Intern Health Study (IHS, NeCamp et al., 2020), which focuses on sequential mobile health +interventions to help improve the mental health of medical interns who work in stressful environments. Participants +were randomly assigned to receive notifications (e.g., tips and insights) throughout the study. For example, some +notifications remind participants to take a break or enjoy a tasty treat, while others summarize the trends of recent +physical activity and sleep. All the notifications are designed to improve participants’ mood scores (self-reported via a +custom-made study App) either directly or indirectly through increased activity or sleep hours. In addition, it is essential +to note that participants’ recent behavior will not only influence their proximal mood but will also influence their +behavior and mood scores in the following days. To design a more effective intervention policy in IHS, it is necessary +to understand how mobile prompts impact mood scores. In particular, the mobile prompts may directly impact the +mood scores or encourage more physical activity and sleep, which may then impact the mood scores. In addition, an +individual’s past treatment sequence and behavior trajectory may impact the mood score. Teasing out these distinct +sources of causal impacts on mood scores and their relative magnitudes needs new definitions, identification results, +and inferential methods. +A fundamental question considered in this paper is how to infer the dynamic mediation effects in the aforementioned +applications. Solving this question raises at least three challenges. First, the mediator at a given time affects both +the current and future outcomes, inducing temporal carryover effects. As demonstrated in the case study in Section +8, the delayed direct effect (DDE) and the delayed mediator effect (DME) are significant and dominate the average +arXiv:2301.13348v1 [stat.ML] 31 Jan 2023 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Figure 1: Mediated MDP. +treatment effect for the intervention policy used in the IHS (Sen et al., 2010; NeCamp et al., 2020). In contrast, the +immediate direct effect (IDE) and immediate mediator effect (IME) are both insignificant. Nonetheless, most existing +mediation analyses focus on estimating the indirect effect on the immediate reward and are hence inappropriate to our +application. Second, the horizon (e.g., number of decision stages) in the aforementioned applications is typically very +long or diverges with the sample size. Existing solutions developed in finite horizon settings typically suffer from the +curse of horizon in the sense that the variances of the proposed estimators grow exponentially fast with respect to the +horizon (Liu et al., 2018) and are hence inapplicable; see Section 2 for details. Third, regardless of how the dynamic +effects may change during the sequential treatments (or lack thereof), most works focus on examining the causal effects +on the final outcome obtained at the end of the treatment process. However, in the context of behavioral change, the +goal is to encourage and maintain small improvements to nudge individuals into generating sustained improvements in +outcomes like mood scores. Currently, there is a dearth of methods to analyze causal effects for outcomes measured at +every decision point in the sequence. +To address these limitations, we propose formulating the evaluation of dynamic mediation effects as a reinforcement +learning (RL) problem. In particular, we use the Markov decision process (MDP) that is commonly employed in RL +to model the mediated dynamic decision process over an infinite time horizon. Building upon the standard MDP, we +introduce four additional sets of causal relationships, including state-mediator, action-mediator, mediator-state, and +mediator-reward, as shown in Figure 1. To evaluate the effects of different treatment policies, we consider using the +off-policy evaluation (OPE, Dudík et al., 2014; Uehara et al., 2022), which is widely used to avoid the difficulty of +rerunning trials by evaluating treatment policies based on observational data. +Contributions. The main contributions are as follows. Motivated by the mobile health applications, we first construct +the mediation analysis within the framework of RL over an infinite time horizon. Second, we propose to decompose +the average treatment effect between a target policy and a control policy into IDE, IME, DDE, and DME. While +IDE and IME have been extensively studied in single-stage settings, we introduce the DDE and DME to quantify +the carryover effects of past actions and mediators. Third, upon the identification result of each effect component, +multiply-robust estimators are developed. In particular, each proposed estimator is consistent even when models such +as mediator distribution and reward distribution are misspecified (See Section 7.1). Furthermore, we theoretically +show the semiparametric efficiency of the proposed estimators and confirm the theoretical prediction using numerical +studies. Lastly, we conclude by analyzing the IHS data and providing new insights into guiding future designs of these +behavioral interventions. +2 +Related work +Mediation analysis is widely studied in point-exposure studies under the classical structure consisting of a treatment, a +mediator, and an outcome (Robins & Greenland, 1992; Pearl, 2022; Petersen et al., 2006; van der Laan & Petersen, 2008; +Imai et al., 2010; Tchetgen & Shpitser, 2012; Tchetgen Tchetgen & Shpitser, 2014; VanderWeele, 2015), decomposing +the average treatment effect into direct effect and indirect effect. Recently, to address commonly observed intermediate +confounders that would be affected by the exposure and then affect both mediator and outcome, multiple methods have +been developed to extend the classical mediation analysis (Robins & Richardson, 2010; Tchetgen & VanderWeele, +2014; VanderWeele et al., 2014; Vansteelandt & Daniel, 2017; Díaz et al., 2021; Díaz, 2022), among which the random +intervention (RI)-based approach (VanderWeele et al., 2014; Díaz, 2022) further sets the foundation for the recent +advancement of longitudinal mediation analysis. +There is a rich literature on longitudinal mediation analysis with no intermediate confounders (Selig & Preacher, 2009; +Roth & MacKinnon, 2013). See also Preacher (2015) for a detailed review. However, time-varying intermediate +confounders are ubiquitous in longitudinal data contexts. For example, in the IHS, doing exercises may result in a good +mood, which may, in turn, increase the likelihood of engaging in more activities the next day and then subsequently affect +the mood that follows. +2 + +Mt-1 +Rt-1 +Mt +Rt +At-1 +St +AtA Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +In the presence of time-varying intermediate confounders, there are two major RI-based approaches. VanderWeele & +Tchetgen Tchetgen (2017) and Díaz et al. (2022) proposed to intervene in the mediator sequence by randomly drawing +mediators from the corresponding marginal distribution and defined the longitudinal interventional indirect/direct effect, +which is different from the natural effect decomposition. Our work is primarily related to the work of Zheng & van der +Laan (2017), which proposed to intervene in the mediator by randomly drawing the mediator from its conditional +distribution and provided a natural decomposition of the total effect. Using the efficient influence function (EIF), +they developed a multiply-robust estimator with less reliance on the correct model specification. However, all the +aforementioned methods only focused on the treatment impact on the final outcome in finite horizons and did not +consider immediate outcomes or infinite horizon settings. In addition, the estimator developed by Zheng & van der +Laan (2017) is based on the product of importance sampling ratios at all time points and suffers from the curse of +horizon. Zheng & van der Laan (2012) also analyzed the longitudinal mediation effect by drawing mediators from +conditional distribution but with a focus on single-exposure settings. +Using an RL framework for dynamic mediation analysis over an infinite horizon, our work is also connected to the line +of research on OPE. Existing OPE-related research evaluates the discounted sum of rewards or average rewards for a +target policy using observational data gained by following a different behavior policy. In general, there are three types +of estimation procedures. The first is known as the direct method (DM, Le et al., 2019; Feng et al., 2020; Luckett et al., +2020; Hao et al., 2021; Liao et al., 2021; Chen & Qi, 2022; Shi et al., 2022a), which directly learns Q-functions and +obtains value estimates based on their estimators. The second category of approaches utilizes importance sampling (IS, +Precup, 2000; Thomas et al., 2015; Hallak & Mannor, 2017; Hanna et al., 2017; Liu et al., 2018; Xie et al., 2019; Dai +et al., 2020; Zhang et al., 2020), which re-weights the rewards to eliminate the bias due to distributional shift. The third +category develops doubly robust (DR) estimators by appropriately integrating DM with IS estimators (Jiang & Li, 2016; +Thomas & Brunskill, 2016; Farajtabar et al., 2018; Liao et al., 2020; Tang et al., 2020; Uehara et al., 2020; Kallus & +Uehara, 2022). DR estimators are also known to achieve the semiparametric efficiency bound (Bickel et al., 1993). +However, none of the above papers studied mediation analysis. Recently, Shi et al. (2022b) proposed a consistent DR +estimator for OPE in the presence of unmeasured confounders with the help of a mediator variable, which is used to +intercept each directed path from treatments to reward/state. Our paper differs from theirs in that we decompose the +off-policy value into the sum of IDE, IME, DDE, and DME and focus on settings without unmeasured confounding. +3 +Preliminaries +3.1 +Data Generating Process +We consider the observational data generated from a mediated Markov decision process (MMDP), as illustrated in +Figure 1. Suppose there exists an agent that tries to learn from the data and interact with a given environment. At each +time t, the environment arrives at a state St ∈ S, and the agent selects an action At ∈ A = {0, 1, · · · , K −1} according +to a behavior policy πb(•|St). Building upon the usual MDP, to further analyze the mediation effect, we consider +an immediate mediator variable Mt ∈ M drawn according to pm(•|St, At), which mediates the effect of At on the +environment. Accordingly, the agents would receive an immediate Rt and the the environment transits to a next-state +St+1 according to ps′,r(•, •|St, At, Mt). Both S and M are finite dimensional vector spaces. To summarize, the +observed data sequences consist of the state-action-mediator-reward tuples (St, At, Mt, Rt)t≥0 satisfying the following +Markov assumption: (Mt, Rt, St+1) ⊥⊥ (Sj, Aj, Mj, Rj)j 1/4, respectively. The MR estimators are asymptotically +normal with an asymptotic variance achieving the semiparametric efficiency bound. +To save the space, the proof of this theorem is differed to the Appendix F with a sketch of the proof at the beginning. A +Wald-type Confidence Interval (CI) for each MR estimator can be derived from Theorem 6.2. +7 +Numerical Examples +In this section, we evaluate the estimation performance of the proposed methods through three simulation studies. +Specifically, we demonstrate the robustness of the proposed MR estimator to model misspecification in the first +simulation. In the second simulation, we compare the DM, MIS, and MR estimators to the classic direct/indirect +estimator (Pearl, 2022) to demonstrate the importance of longitudinal mediation analysis, considering the policy effect +on state transition. The final simulation is a semi-synthetic study that simulates the generation process of real data and +demonstrates the superiority of the proposed MR estimators. For any effect X, let ˆX be an estimator. We define the +logbias as log |E( ˆX − X)| and logMSE as E[log( ˆX − X)2]. +7.1 +Toy Example I +We consider a simplified MMDP setting with binary states, actions, mediators, and rewards. See Appendix H.1 for +specific data generation settings. To investigate the robustness of the MR estimator, we test its performance in four +scenarios: i) M1, M2, and M3 are all correctly specified; ii) only M1 is correctly specified; iii) only M2 is correctly +specified; iv) only M3 is correctly specified; and v) all the models in M1, M2, and M3 are incorrectly specified by +injecting non-negligible random noises. As shown in Fig 3, MR-IDE(πe, π0) and MR-IME(πe, π0) are consistent when +either M1, M2, or M3 is correctly specified, and MR-DDE(πe, π0) and MR-DME(πe, π0) are consistent when either +M2 or M3 is correctly specified. +7.2 +Toy Example II +As discussed in Section 2, most existing works focus on a two-way decomposition of immediate treatment effects +under the setting with a single stage. In this section, we compare the proposed estimators of IDE and IME to baseline +estimators assuming i.i.d. samples (See Appendix I for details). We first repeat the data generation process from Section +7.1, in which the states are affected by the history observations for each trajectory. Then, by modifying the distribution +of the next state, St+1, as Pr(St+1 = 1) = .2, we consider a second scenario in which all observations of states are i.i.d +sampled. Note that there are two versions of MIS estimators for IDE and IME. Let MIS2 denote the MIS estimators +using the MIS2 to estimate ηGe. According to Fig 4, when states are i.i.d. sampled, all estimators produce consistent +8 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Figure 3: Bias and the logMSE of MR estimators, aggregated over 200 random seeds. The error bars represent the 95% +CI. +estimates. However, when policy-induced state transitions occur, the baseline estimators yield biased estimates, whereas +the proposed estimators continue to provide consistent estimates, implying the necessity of accounting for the policy +effect on the state transition. +Figure 4: Bias and the logMSE of estimators, under different data generation scenarios. The results are aggregated over +200 random seeds. The error bars represent the 95% CI. Nuisance functions are estimated as discussed in Appendix D. +7.3 +Semi-Synthetic Data +In this section, we evaluate empirical performance of estimators using a semi-synthetic dataset structured similarly to +the real dataset analyzed in Section 8. Specifically, we consider an MMDP setting with continuous reward, state, and +mediator spaces and a binary action space. See Appendix H.2 for more information on the data-generation process. We +compared the MR estimators to the DM estimators, the MIS estimators, and the baseline estimators. As shown in Fig 5 +and Fig 6, the MR estimators outperform all other estimators for all components of ATE, especially when the sample +size is large. We first focus on IDE(πe, π0) and IME(πe, π0). On the one hand, the baseline and MIS estimators are +biased, whereas the bias and MSE of the proposed DM and MR estimators decay continuously as N or T increases. On +the other hand, the DM estimators yield relatively more significant bias and MSE than MR estimators. Considering the +DDE(πe, π0) and DME(πe, π0), both the DM and MIS estimators are biased with non-decreasing MSE, whereas the +9 + +IDE +IME +DDE +DME +0.D. +0.B +1.D +D.2 +0.6 +D.5 +se! +D.4 +1.0 ++.4 +0.5 +D.6 +0.2 +D.B +1.5 +0.0 ++.口 +2 +2 +MSE +E +-2 +60 +6 +6 +100 +DDZ +300 +100 +DDZ +300 +100 +20G +30-0 +100 +DOZ +O-DE +N +N +N +N +All Correct +M1 +M2 +M3 +All IncorrectIDE fi.i.d. SJ +IME (i.i.d. S) +IDE (w. S transit.) IME (w. S transit.) +0.2D +0.DDD +0.0 +栏轻栏栏轻栏 +0.15 ++.DD - +D.025 +0.10 +D.1 +D.050 +D.05 +D.2 +0.05 +D.075 +D.10 +D.3 +0.DD +D.100 +D.4 +4 +E +- +5 +: +200 +100150 +200 +50100150 +2F0 +05 +100150200 +N +N +N +N +MR +DM +MIS1 +MIS2 ++ +BaselineA Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +MR estimators continue to provide estimates with low bias and low MSE that decrease with N and T. The results are +in line with our theoretical findings. +Figure 5: The logbias and logMSE of various estimators, aggregated over 100 random seeds. The error bars represent +the 95% CI. Fix T = 25. +Figure 6: The logbias and logMSE of various estimators, aggregated over 100 random seeds. The error bars represent +the 95% CI. Fix N = 50. +8 +Real Data Application +In this section, we apply the proposed MR estimators to analyze the real dataset from the IHS (NeCamp et al., 2020), +which was discussed as a motivating example in Section 1. The study involved 1565 interns and lasted six months. Every +day, the participant would either receive a notification (At = 1) or no notification (At = 0). Meanwhile, participants’ +mood score (Rt), step count (Mt,1), and hours of sleep (Mt,2) were recorded. At each time step, we consider the +previous time step’s mood score as the current state (i.e., St = Rt−1). +Using the control policy π0 of no intervention, we are interested in evaluating the treatment effects of two target policies. +One is the behavior policy πb used throughout the study, which sends notifications to individuals randomly with a +constant probability of .75. According to NeCamp et al. (2020), pushing notifications has a negative impact on the +10 + +IDE +IME +DDE +DME +0.5 +D.25 ++1 +-1 +0.50 +0.D +2 +D.75 +D.5 +60 +DD'T- +1.0 +1.25 +1.5 +6 +MSE +E +-2 +-3 +E- +10-00 +DDSL +10-00 +DDSL +DDSL +10-00 +DDSL +N +N +N +N +MR +DM +MIS1 +MIS2 +BaselineIDE +IME +DDE +DME ++. +0.0 +sei +D.5 +1 +D.5 +601 +-1.0 +1.0 +1.5 +1.5 +2.0 +0 +E +-1 +MSE +-1 +-2 +60 +-2 +2 +E- +DDSL +O5 +1000 +DDSL +DDSL +1000 +DDSL +T +T +T +MR +DM +MIS1 +MIS2 +BaselineA Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +mood condition when participants are already in a good mood (i.e., St > 6). Given that the majority of observations in +the data have St > 6, the ATE of πb is expected to be negative. Another is an estimated optimal policy, ˆπopt, that takes +actions based on the current state of individuals to optimize the reward received. ˆπopt is estimated using single-stage +policy estimation based on the observed data (See Appendix J for more information). +As summarized in Table 1, the ATE of πb is significantly negative with an effect size of .1, which is consistent with our +expectations. Further investigation of the ATE composition reveals that the immediate effects are all negligible. In +contrast, the DDE and DME are both significant and account for the majority of the treatment effect, indicating the +importance of learning the delayed effects and mediator effects to understand the entire mechanism from actions to +outcomes. Furthermore, in contrast to πb, the estimated ATE of ˆπopt is .090, with significantly positive direct effects, +demonstrating the benefit of a data-adaptive policy. +πe +IDE +IME +DDE +DME +ATE +πb +-.007(.007) +-.000(.001) +-.085(.034) +-.008(.004) +-.100 (.041) +ˆπopt +.018(.006) +-.001(.001) +.077(.030) +-.005(.005) +.090 (.037) +Table 1: Estimated treatments effects (standard error) for πb and ˆπopt, compared to π0 with no intervention. +11 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +References +Bickel, P. J., Klaassen, C. A., Bickel, P. J., Ritov, Y., Klaassen, J., Wellner, J. A., and Ritov, Y. Efficient and adaptive +estimation for semiparametric models, volume 4. Springer, 1993. +Breiman, L. Random forests. Machine learning, 45(1):5–32, 2001. +Breiman, L., Friedman, J. H., Olshen, R. A., and Stone, C. J. Classification and regression trees. Routledge, 2017. +Celli, V. Causal mediation analysis in economics: Objectives, assumptions, models. 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Causal mediation in a survival setting with time-dependent mediators. 2012. +14 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +A +More Details about Effect Decomposition +A.1 +Effect Decomposition in the Framework of Potential Outcomes +Let ¯at = (a0, · · · , at) denote a fixed treatment sequence up to time t. Let W ∗ +t (¯at) denote the potential covariates that +would be observed at t if ¯at were taken, and ¯W ∗ +t (¯at) = (W ∗ +0 (¯a0), · · · , W ∗ +t (¯at)) for W ∈ {M, R}. We remark that the +potential states observed at t are defined analogously, but depend on the action sequence ¯at−1 instead of ¯at. Replacing +the fixed action sequence by any random policy π, W ∗ +t (π) denotes the potential covariates if the actions were taken +under π. +We first focus on the effects of action and mediator on their proximal outcome. Denotes πt +e,0 a policy where the first +t − 1 steps follow πe and then follow π0 at t. For X ∈ {S, R}, X∗ +t (π1, ¯ +M ∗ +t (π2)) denotes the potential covariate if π1 +were used to determine actions and the mediators were set to levels as if π2 were used. IDEt and IMEt are defined as +IDEt(πe, π0) = E +� +R∗ +t (πe, ¯ +M ∗ +t (πe)) − R∗ +t (πt +e,0, ¯ +M ∗ +t (πe)) +� +, +IMEt(πe, π0) = E[R∗ +t (πt +e,0, ¯ +M ∗ +t (πe)) − R∗ +t (πt +e,0, ¯ +M ∗ +t (πt +e,0))]. +Given that both ¯At−1 and ¯ +Mt−1 were set to levels as if πe were used, IDEt(πe, π0) contrasts the impact of At generated +by πe and π0 on the proximal outcome Rt, fixing Mt to M ∗ +t (πe). IMEt(πe, π0) compares the effect of Mt at levels +M ∗ +t (πe) and M ∗ +t (πt +e,0) on Rt, when At is set by π0. +Next, we focus on the delayed effects of the historical action sequence ¯At−1 and mediator sequence ¯ +Mt−1 on Rt. Within +the MMDP framework, ¯At−1 and ¯ +Mt−1 affect Rt through St. Noticing that E[R∗ +t (π0, ¯ +M ∗ +t (πt +e,0))] is unidentifiable due +to the presence of intermediate confounders ¯St (Tchetgen & VanderWeele, 2014), we adopt the RI-based approach +proposed in Zheng & van der Laan (2017). +We first define the conditional probability density of mediator at t, +G¯a′ +t +t (·|¯st, ¯mt−1, ¯rt−1) = pM ∗ +t (¯a′ +t)| ¯S∗ +t (¯a′ +t), ¯ +M ∗ +t−1(¯a′ +t−1), ¯ +R∗ +t−1(¯a′ +t−1)(·|¯st, ¯mt−1, ¯rt−1), +if ¯a′ +t is assigned. At time t, given the historical trajectories ¯st, ¯mt−1, and ¯rt−1, we intervene in the mediator by randomly +drawing Mt ∼ G¯a′ +t +t (·|¯st, ¯mt−1, ¯rt−1). For brevity, we omit the conditionality and let ¯G¯a′ +t +t += (G¯a′ +0 +0 , · · · , G¯a′ +t +t ) denote +the process by which the mediator is set to a conditional random draw at each time t. Using a two-stage interventional +process as an illustration, we set ¯A1 = ¯a1 and ¯ +M1 ∼ ¯G¯a′ +1 +1 . The generating process of R∗ +1(¯a1, ¯G¯a′ +1 +1 ) is as follows: After +observing an initial state s0, we would first assign a treatment a0 and set M0 by randomly drawing m0 ∼ Ga′ +0 +0 (·|s0), +and then measure the resulting R∗ +0(a0, ¯Ga′ +0 +0 ) = r0 and S∗ +1(a0, ¯Ga′ +0 +0 ) = s1. At t = 1, we then take action a1 and set +M1 by randomly drawing m1 ∼ G¯a′ +1 +1 (·|s0, s1, m0, r0), and finally observe R∗ +1(¯a1, ¯G¯a′ +1 +1 ) as the outcome. Analogously, +R∗ +t (π1, ¯Gπ2 +t ) is the potential reward if π1 were used to determine ¯At and ¯ +Mt were set to have the π2-driven conditional +distributions ¯Gπ2 +t . We then define the delayed effects as +DDEt(πe, π0) = E[R∗ +t (πt +e,0, ¯ +M ∗ +t (πt +e,0)) − R∗ +t (π0, ¯G +πt +e,0 +t +)], +DMEt(πe, π0) = E[R∗ +t (π0, ¯G +πt +e,0 +t +) − R∗ +t (π0, ¯ +M ∗ +t (π0))]. +Setting At and Mt to levels as if policy π0 were used at t, DDEt(πe, π0) compares the effects of ¯At−1 generated by +πe and π0 on Rt when ¯ +Mt−1 is generated by πe, while DMEt(πe, π0) contrasts the effects of ¯ +Mt−1 generated by πe +and π0 on Rt when ¯At−1 is set by π0. See Appendix A.3 for more discussion about the non-identifiability issue and +Appendix A.2 for graphical representations of each component. +Remark As suggested in Robins & Greenland (1992), there are two ways to decompose the total effect. The above +definitions of direct effects and mediator effects are analogous to the Total Direct Effect (TDE) and the Pure Indirect +Effect (PIE) (Robins & Greenland, 1992), while an alternative decomposition is provided in Appendix B. By replacing +πe and π0 with ¯a′ +t and ¯at, IDE and IME are equivalent to TDE and PIE. Let ˜¯at = {¯a′ +t−1, at}, we further replace πt +e,0 +with ˜¯at to define DDE and DME. When t > 0, if we set ˜¯at = ¯a′ +t, DDE and DME are analogous to the effect components +defined in Zheng & van der Laan (2017). +A.2 +Graphical Representation of Potential Outcomes +In Table 2 and Table 3, using causal graphs, we explicitly depict the process generating the potential reward terms +involved in the effect decomposition. Specifically, R∗ +t (πe, ¯ +M ∗ +t (πe)) is the potential reward that would be observed if +15 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +R∗ +t (πe, ¯ +M ∗ +t (πe)) +R∗ +t (πt +e,0, ¯ +M ∗ +t (πe)) +R∗ +t (πt +e,0, ¯ +M ∗ +t (πt +e,0)) +Table 2: Potential Outcomes Related to Immediate Effects. +R∗ +t (πt +e,0, ¯ +M ∗ +t (πt +e,0)) +R∗ +t (π0, ¯G +πt +e,0 +t +) +R∗ +t (π0, ¯ +M ∗ +t (π0)) +Table 3: Potential Outcomes Related to Delayed Effects. +πe were used to determine ¯At and ¯ +Mt; R∗ +t (πt +e,0, ¯ +M ∗ +t (πe)) is the potential reward that would be observed if πe were +used to determine the historical sequences ¯At−1 and ¯ +Mt−1, while At were determined by π0 and Mt were set to +M ∗ +t (πe); R∗ +t (πt +e,0, ¯ +M ∗ +t (πt +e,0)) is the potential reward if πe were used to determine ¯At−1 and ¯ +Mt−1, while At and Mt +are generated by π0; R∗ +t (π0, ¯G +πt +e,0 +t +) is the potential reward if π0 were used to determine At and Mt, while the historical +sequences ¯At−1 and ¯ +Mt−1 were determined by π0 and πe respectively; and R∗ +t (π0, ¯ +M ∗ +t (π0)) is the potential reward +that would be observed if π0 were used to determine ¯At and ¯ +Mt. +By definition, IDE(πe, π0) is the contrast between causal structures of R∗ +t (πe, ¯ +M ∗ +t (πe)) and R∗ +t (πt +e,0, ¯ +M ∗ +t (πe)); +IME(πe, π0) is the contrast between causal structures of R∗ +t (πt +e,0, ¯ +M ∗ +t (πe)) and R∗ +t (πt +e,0, ¯ +M ∗ +t (πt +e,0)); DDE(πe, π0) is +the contrast between causal structures of R∗ +t (πt +e,0, ¯ +M ∗ +t (πt +e,0)) and R∗ +t (π0, ¯G +πt +e,0 +t +); and DME(πe, π0) is the contrast +between causal structures of R∗ +t (π0, ¯G +πt +e,0 +t +) and R∗ +t (π0, ¯ +M ∗ +t (π0)). +A.3 +Non-identifiability Issue +To understand the non-identifiability issue, we consider two fixed action sequences ¯at and ¯a′ +t and discuss the identifica- +tion of E[R∗ +t (¯at, ¯ +M ∗ +t (˜¯at))]. For simplicity, let the mediator and state be discrete values. Based on the definition, let +˜¯at = (¯a∗ +t−1, at) and t = 1, we have that +E[R∗ +1(¯a1, ¯ +M ∗ +1 (˜¯a1))] = +� +a0,a1,a∗ +0,m0,m1,s0,s1,s∗ +1 +E(R|a0, a1, m0, m1, s0, s1)Pr(M ∗ +1 = m1|a∗ +0, a1, m0, s0, s∗ +1) +× Pr(S∗ +1(a0) = s1, S∗ +1(a∗ +0) = s∗ +1|m0, s0)Pr(M ∗ +0 = m0|a∗ +0, s0)Pr(S0 = s0). +While E(R|a0, a1, m0, m1, s0, s1), Pr(M ∗ +1 = m1|a∗ +0, a1, m0, s0, s∗ +1), Pr(M ∗ +0 = m0|a∗ +0, s0), and Pr(S0 = s0) are +identifiable from the observational data, the joint distribution of Pr(S∗ +1(a0) = s1, S∗ +1(a∗ +0) = s∗ +1|m0, s0) is not identified, +leading to the non-identifiability of E[R∗ +1(¯a1, ¯ +M ∗ +1 (˜¯a1))]. The non-identifiability of E[R∗ +t (π0, ¯ +M ∗ +t (πt +e,0))] is then +followed. +16 + +Mt +Rt- +R +Mt +Rt +A +St +At- +St +5 +AtA +Mt +Rt +2 +A +Mt +R, +M +Rt +St +TO +At-Mt +Rt- +Rt +Mt +Rt +St +At +St +70 +5 +At +AtMt-1 +Rt +-1 +IM: +Rf +At +SMt-1 +At +Mt +Rt +St +At- +St +AtMt +Rt- +Mt +Rt +St +St +A. +AtA Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +B +Alternative Decomposition of ATE +In this section, we provide an alternative decomposition of ATE(πe, π0). Let ˜G denote the stochastic process selecting +actions according to πe and drawing mediators assuming π0 was applied. Adopting the notations used in the main text, +we have that +ATE(πe, π0) = ηπe − η +˜ +Ge +� +�� +� +DME(2)(πe,π0) ++ η +˜ +Ge − ηπ0,e +� +�� +� +DDE(2)(πe,π0) ++ ηπ0,e − η +˜ +G0 +� +�� +� +IME(2)(πe,π0) ++ η +˜ +G0 − ηπ0 +� +�� +� +IDE(2)(πe,π0) +. +In the following subsections, we further written the alternative decomposition in the framework of potential outcomes +along with the corresponding MR estimators. +B.1 +Decomposition in the Framework of Potential Outcomes +We follow the notations used in the Appendix A. Another classic decomposition of the total effect is well-known as +natural effect decomposition, which divides the total effect into Natural Direct Effect (NDE) (also named as Pure Direct +Effect) and Natural Indirect Effect (NIE) (also named as Total Indirect Effect) (Robins & Greenland, 1992; Pearl, 2022; +VanderWeele, 2013). Denotes πt +0,e a policy where the first t − 1 steps follow π0 and then follow πe at t. Following the +natural effect decomposition , we alternatively decompose the TEt(πe, π0) as follows: +TEt(πe, π0) = DME(2) +t (πe, π0) + DDE(2) +t (πe, π0) + IME(2) +t (πe, π0) + IDE(2) +t (πe, π0), +where +DME(2) +t (πe, π0) = E[R∗ +t (πe, ¯ +M ∗ +t (πe)) − R∗ +t (πe, ¯G +πt +0,e +t +)], +DDE(2) +t (πe, π0) = E[R∗ +t (πe, ¯G +πt +0,e +t +) − R∗ +t (πt +0,e, ¯ +M ∗ +t (πt +0,e))], +IME(2) +t (πe, π0) = E[R∗ +t (πt +0,e, ¯ +M ∗ +t (πt +0,e)) − R∗ +t (πt +0,e, ¯ +M ∗ +t (π0))], +IDE(2) +t (πe, π0) = E[R∗ +t (πt +0,e, ¯ +M ∗ +t (π0)) − R∗ +t (π0, ¯ +M ∗ +t (π0))]. +Then, for X ∈ {IDE(2), IME(2), DDE(2), DME(2)}, we have that +X = lim +T →∞ +1 +T +T −1 +� +t=0 +Xt. +(7) +By replacing πe and π0 with ¯a′ +t and ¯at, IDE(2) and IME(2) are equivalent to NDE and NIE derived in Pearl (2022). Let +˜¯at = {¯at−1, a′ +t}, we further replace πt +0,e with ˜¯at to define DDE(2) and DME(2) for fixed action sequnces. When t > 0, +if we set ˜¯at = ¯at, DDE(2) and DME(2) are equivalent to NDE/NIE defined in Zheng & van der Laan (2017). +B.2 +MR Estimators of the Alternative Decomposition +Similar to Section 5.3, we first define three additional Q functions: +Q +˜ +G0(s, a, m) = +� +t≥0 +Eπ0[Eπe +a∗r(St, a∗, Mt) − η +˜ +G0|S0 = s, A0 = a, M0 = m], +Qπ0,e(s, a, m) = +� +t≥0 +Eπ0[Eπe +a∗,m∗r(St, a∗, m∗) − ηπ0,e|S0 = s, A0 = a, M0 = m], +Q +˜ +Ge(s, a, m) = +� +t≥0 +E +˜ +G[Eπe +a∗,m∗r(St, a∗, m∗) − η +˜ +Ge|S0 = s, A0 = a, M0 = m], +where η ˜ +G0 is the expected value of Eπe +a∗r(St, a∗, Mt) under policy π0, ηπ0,e is the expectation of Eπe +a∗,m∗r(St, a∗, m∗) +under π0, and η ˜ +Ge is the expectation of Eπe +a∗,m∗r(St, a∗, m∗) under the treatment process and the intervened mediator +process of ˜G. +17 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Next, we construct three additional augmentation terms similar to the augmentation terms defined in the main text. Let +ρ(2)(S, A, M) = +� +a π0(a|S)p(M|S,a) +p(M|S,A) +. We define that +I6(O) = ωπ0(S)π0(A|S) +πb(A|S) +� +Eπe +a′ r(S, a′, M) + Eπ0 +a,mQ +˜ +G0(S′, a, m) − EmQ +˜ +G0(S, A, m) − η +˜ +G0� ++ ωπ0(S)πe(A|S) +πb(A|S)ρ(2)(S, A, M){R − r(S, A, M)}, +I7(O) = ωπ0(S)π0(A|S) +πb(A|S) +� +Eπe +a′,mr(S, a′, m) + Eπ0 +a,mQπ0,e(S′, a, m) − EmQπ0,e(S, A, m) − ηπ0,e� ++ ωπ0(S)πe(A|S) +πb(A|S){R − Emr(S, A, m)}, +I8(O) = ω +˜ +G(S)πe(A|S) +πb(A|S)ρ(2)(S, A, M) +� +Eπe +a′,mr(S, a′, m) + E +˜ +G +a,mQ +˜ +Ge(S′, a, m) − Q +˜ +Ge(S, A, M) − η +˜ +Ge� ++ ω +˜ +G(S)πe(A|S) +πb(A|S) +� +R − Emr(S, A, m) +� ++ ω +˜ +G(S)π0(A|S) +πb(A|S) × +� +a +πe(a|S) +� +Q +˜ +Ge(S, a, M) − +� +m +p(m|A, S)Q +˜ +Ge(S, a, m) +� +Then the MR estimator of IDE(2)(πe, π0) is +MR-IDE(2)(πe, π0) = +1 +NT +� +i,t +η +˜ +G0 − ηπ0 + I6(Oi,t) − I5(Oi,t). +The MR estimator of IME(2)(πe, π0) is +MR-IME(2)(πe, π0) = +1 +NT +� +i,t +ηπ0,e − η +˜ +G0 + I7(Oi,t) − I6(Oi,t). +The MR estimator of DDE(2)(πe, π0) is +MR-DDE(2)(πe, π0) = +1 +NT +� +i,t +η +˜ +Ge − ηπ0,e + I8(Oi,t) − I7(Oi,t). +The MR estimator of DME(2)(πe, π0) is +MR-IDE(2)(πe, π0) = +1 +NT +� +i,t +ηπe − η +˜ +Ge + I1(Oi,t) − I8(Oi,t). +Following Theorem 6.1 and Theorem 6.2, we can show that MR-IDE(2), MR-IME(2), MR-DDE(2), and MR-DME(2) +are multiply robust and achieve the semi-parametric efficiency bound. +C +Proof of Theorem 4.1 +This proof adheres strictly to the definitions of potential outcomes discussed in Appendix A. +We first clarify three standard assumptions, and then identify the potential rewards E +� +R∗ +t (π, ¯ +M ∗ +t (π)) +� +for any arbitrary +policy π and E +� +R∗ +t (π0, ¯G +πt +e,0 +t +) +� +using the observed data distribution, followed by the identification function for each of +the IDE(πe, π0), IME(πe, π0), DDE(πe, π0), and DME(πe, π0). +C.1 +Standard Assumptions +The decomposed effects are identifiable under three standard assumptions (Zheng & van der Laan, 2017; Luckett et al., +2019): +18 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Assumption 1 (Consistency). St = S∗ +t ( ¯At−1), Mt = M ∗ +t ( ¯At), Rt = R∗ +t ( ¯At), St = S∗ +t ( ¯At−1, ¯ +Mt−1), and Rt = +R∗ +t ( ¯At, ¯ +Mt) for all t. +Assumption 2 (Sequential Randomization). For all j ≥ t: i) (S∗ +j+1(¯aj), M ∗ +j (¯aj), R∗ +j(¯aj)) ⊥⊥ At| ¯At−1, ¯ +Mt−1, ¯St; +ii) (S∗ +j+1(¯aj, ¯mj), R∗ +j(¯aj, ¯mj)) ⊥⊥ At| ¯At−1, ¯ +Mt−1, ¯St; and iii) (S∗ +j+1(¯aj, ¯mj), R∗ +j(¯aj, ¯mj)) ⊥⊥ Mt| ¯At−1, ¯ +Mt−1, ¯St +Assumption 3 (Positivity). Let ht = ( ¯mt, ¯rt, ¯st+1). For all t ≥ 0 and all (ht, ¯at, ¯a′ +t): i) if pπb(¯at, ht) > 0, then +pπb(at+1|¯at, ht) > 0; ii) if pπb(¯a′ +t, ht) > 0, then pπb(a′ +t+1|¯a′ +t, ht) > 0; iii) if pπb(rt, st+1|¯at, ht−1, mt) > 0, then +pπb(rt, st+1|¯a′ +t, ht−1, mt) > 0; and iv) if pπb(mt|¯a′ +t, ht−1) > 0, then pπb(mt|¯at, ht−1) > 0. +Assumption 1 states that the observed mediator, state, and reward are equivalent to their counterfactuals, which would +be observed if the observed actions were carried out, and that the observed reward and state are consistent with the +potential reward and state if the observed sequences of actions and mediators were taken. Assumption 2 requires that +there are no unmeasured confounders between At and all of its subsequent covariates and between Mt and all of its +subsequent covariates. Lastly, assumption 3 ensures that treatments and covariates are not exclusive to a specific stratum +of covariates. The identification result is summarized as follows. +C.2 +Identification of E +� +R∗ +t (π, ¯ +M ∗ +t (π)) +� +Without loss of generality, we first consider the states and mediators in discrete values. By definition, we have that +E +� +R∗ +t (π, ¯ +M ∗ +t (π)) +� += +� +¯at, ¯mt,¯st+1,¯rt +rtPr(S0 = s0) +t� +j=0 +π(aj|S∗ +j (¯aj−1, ¯ +M ∗ +j−1(¯aj−1)) = sj) +× Pr[M ∗ +j (¯aj) = mj| ¯S∗ +j (¯aj−1, ¯ +M ∗ +j−1(¯aj−1)) = ¯sj, ¯ +M ∗ +j−1(¯aj−1) = ¯mj−1] +× Pr[S∗ +j+1(¯aj, ¯ +M ∗ +j (¯aj)) = sj+1, R∗ +j(¯aj, ¯ +M ∗ +j (¯aj)) = rj| ¯S∗ +j (¯aj−1, ¯ +M ∗ +j−1(¯aj−1)) = ¯sj, ¯ +M ∗ +j (¯aj) = ¯mj]. +To identify the potential reward, we first consider t = 0 and observe that +π(at|S∗ +t (¯at−1, ¯ +M ∗ +t−1(¯at−1)) = st) = π(a0|S0 = s0). +Next, we show that +Pr[M ∗ +0 (a0) = m0|S0 = s0] = Pr[M ∗ +0 (a0) = m0|A0 = a0, S0 = s0] += Pr[M0 = m0|A0 = a0, S0 = s0], +where the first equality holds by Assumption 2 and the second equality follows from the Assumption 1. Similarly, using +the same arguments, we can show that +Pr[S∗ +1(a0, M ∗ +0 (a0)) = s1, R∗ +0(a0, M ∗ +0 (a0)) = r0|S0 = s0, M ∗ +0 (a0) = m0] += Pr[S∗ +1(a0, M ∗ +0 (a0)) = s1, R∗ +0(a0, M ∗ +0 (a0)) = r0|A0 = a0, S0 = s0, M ∗ +0 (a0) = m0] += Pr[S1 = s1, R0 = r0|A0 = a0, S0 = s0, M0 = m0]. +Applying the same arguments for the subsequent potential covariates repeatedly, we can show that +E +� +R∗ +t (π, ¯ +M ∗ +t (π)) +� += +� +¯at, ¯mt,¯st+1,rt +rtPr(S0 = s0) +t� +j=0 +π(aj|Sj = sj)Pr[Mj = mj| ¯Aj = ¯aj, ¯Sj = ¯sj, ¯ +Mj−1 = ¯mj−1] +× Pr[Sj+1 = sj+1, Rj = rj| ¯Aj = ¯aj, ¯Sj = ¯sj, ¯ +Mj = ¯mj]. +Finally, under the assumption that the data generating process satisfied the Markov property, such that i) the distribution +of At is independent of all the past history observations given St, ii) the distribution of Mt is independent of all the +past history observations given (St, At), and iii) the distributions of Rt and St+1 are independent of all the past history +observations given (St, At, Mt), we have that +E +� +R∗ +t (π, ¯ +M ∗ +t (π)) +� += +� +¯at, ¯mt,¯st+1,rt +rtPr(S0 = s0) +t� +j=0 +π(aj|Sj = sj)Pr[Mj = mj|Aj = aj, Sj = sj] +× Pr[Sj+1 = sj+1, Rj = rj|Aj = aj, Sj = sj, Mj = mj]. +19 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Let τt denote the data trajectory {(sj, aj, mj, rj, sj+1)}0≤j≤t. Replacing the probability mass functions by probability +density functions, we have that +E +� +R∗ +t (π, ¯ +M ∗ +t (π)) +� += +� +τt +rt +t� +j=0 +p(sj+1, rj|sj, aj, mj)p(mj|sj, aj)π(aj|sj)ν(s0) += +� +τt +rtp(st+1, rt|st, at, mt)p(mt|st, at)π(at|st) +t−1 +� +j=0 +pπ(sj+1, rj, mj, aj|sj)ν(s0), +the identifiability of which is guaranteed by Assumption 3. +When π = πe, +E +� +R∗ +t (πe, ¯ +M ∗ +t (πe)) +� += +� +τt +rtp(st+1, rt|st, at, mt)p(mt|st, at)πe(at|st) +t−1 +� +j=0 +pπe(sj+1, rj, mj, aj|sj)ν(s0). +When π = π0, +E +� +R∗ +t (π0, ¯ +M ∗ +t (π0)) +� += +� +τt +rtp(st+1, rt|st, at, mt)p(mt|st, at)π0(at|st) +t−1 +� +j=0 +pπ0(sj+1, rj, mj, aj|sj)ν(s0). +When π = πt +e,0, +E +� +R∗ +t (πt +e,0, ¯ +M ∗ +t (πt +e,0)) +� += +� +τt +rtp(st+1, rt|st, at, mt)p(mt|st, at)π0(at|st) +t−1 +� +j=0 +pπe(sj+1, rj, mj, aj|sj)ν(s0). +Following the same arguments, we can show that +E +� +R∗ +t (πt +e,0, ¯ +M ∗ +t (πe)) +� += +� +τt +� +s∗,r∗,a′ +r∗p(s∗, r∗|st, a′, mt)π0(a′|st)p(mt|st, at)πe(at|st) +t−1 +� +j=0 +pπe(sj+1, rj, mj, aj|sj)ν(s0). +C.3 +Identification of E[R∗ +t (π0, ¯G +πt +e,0 +t +)] +Without loss of generality, we first consider the states and mediators in discrete values. Let ˜¯at = (¯a′ +t−1, at). By +definition, we have that +E[R∗ +t (π0, ¯G˜¯at +t )] = +� +¯at,¯a′ +t−1, ¯mt,¯st+1,¯rt +rtPr(S0 = s0) +t−1 +� +j=0 +π(aj|S∗ +j (¯aj−1, ¯G˜¯at +j−1)) = sj)π(a′ +j|S∗ +j (¯aj−1, ¯G˜¯at +j−1)) = sj) +(8) +× Pr[G˜¯at +j = mj| ¯S∗ +j (¯aj−1, ¯G˜¯at +j−1) = ¯sj, ¯G˜¯at +j−1 = ¯mj−1] +(9) +× Pr[S∗ +j+1(¯aj, ¯G˜¯at +j ) = sj+1, R∗ +j(¯aj, ¯G˜¯at +j ) = rj| ¯S∗ +j (¯aj−1, ¯G˜¯at +j−1) = ¯sj, ¯G˜¯at +j = ¯mj] +(10) +× π(at|S∗ +t (¯at−1, ¯G˜¯at +t−1)) = st)Pr[G˜¯at +t = mt| ¯S∗ +t (¯at−1, ¯G˜¯at +t−1) = ¯st, ¯G˜¯at +t−1 = ¯mt−1] +(11) +× Pr[S∗ +t+1(¯at, ¯G˜¯at +t ) = st+1, R∗ +t (¯at, ¯G˜¯at +t ) = rt| ¯S∗ +t (¯at−1, ¯G˜¯at +t−1) = ¯st, ¯G˜¯at +t = ¯mt]. +(12) +For j < t, By the definition of ¯G˜¯at +j , we have that +Pr[G˜¯at +j = mj| ¯S∗ +j (¯aj−1, ¯G˜¯at +j−1) = ¯sj, ¯G˜¯at +j−1 = ¯mj−1] = Pr[M ∗ +j (¯a′ +j) = mj| ¯S∗ +j (¯a′ +j) = ¯sj, ¯ +M ∗ +j−1(¯a′ +j−1) = ¯mj−1]. +(13) +Using the same arguments in C.2, we can show that equation (13) equals +Pr[Mj = mj| ¯Aj = ¯a′ +j, ¯Sj = ¯sj, ¯ +Mj−1 = ¯mj−1], +which is identifiable under Assumption 3. +20 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Further, to show the identification of equation (10), we prove it at j = 0 as follows: +Pr[S∗ +1(a0, G˜¯at +0 ) = s1, R∗ +0(a0, G˜¯at +0 ) = r0|S0 = s0, G˜¯at +0 = m0] += Pr[S∗ +1(a0, m0) = s1, R∗ +0(a0, m0) = r0|S0 = s0, G˜¯at +0 = m0] += Pr[S∗ +1(a0, m0) = s1, R∗ +0(a0, m0) = r0|S0 = s0] += Pr[S1 = s1, R0 = r0|A0 = a0, S0 = s0, M0 = m0]. +The second equality holds by the definition of the process G˜¯at +t , in which we randomly draw M0 from G˜¯at +0 . Specifically, +given S0 = s0, G˜¯at +0 is independent of S∗ +1(a0, m0) and R∗ +1(a0, m0). The last equality follows from Assumption 1 and 2. +A similar proof can be found in Zheng & van der Laan (2017). +Then, following the steps in C.2, we can show that +E +� +R∗ +t (π0, ¯G +πt +e,0 +t +) +� += +� +τt,¯a∗ +t−1 +rtp(st+1, rt, mt|st, at)π0(at|st) +t−1 +� +j=0 +p(sj+1, rj|sj, aj, mj)π0(aj|sj)p(mj|sj, a∗ +j)πe(a∗ +j|sj)ν(s0), +the identifiability of which is guaranteed by Assumption 3. +C.4 +Identification of IDE(πe, π0), IME(πe, π0), DDE(πe, π0), DME(πe, π0) +Using the above identification results, the identification functions of IDE(πe, π0), IME(πe, π0), DDE(πe, π0), +DME(πe, π0) are directly induced. Specifically, +IDE(πe, π0) = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +� +rtp(st+1, rt|st, at, mt) − +� +s∗,r∗,a′ +r∗p(s∗, r∗|st, a′, mt)π0(a′|st) +� +× p(mt|st, at)πe(at|st) +t−1 +� +j=0 +pπe(sj+1, rj, mj, aj|sj)ν(s0), +IME(πe, π0) = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtp(st+1, rt|st, at, mt)π0(at|st)[ +� +a′ +p(mt|st, a′)πe(a′|st) − p(mt|at, st)] +× +t−1 +� +j=0 +[pπe(sj+1, rj, mj, aj|sj)] ν(s0), +DDE(πe, π0) = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtp(st+1, rt|st, at, mt)p(mt|st, at)π0(at|st) +× +� t−1 +� +j=0 +pπe(sj+1, rj, mj, aj|sj) − +� +¯a∗ +t−1 +t−1 +� +j=0 +p(sj+1, rj|sj, aj, mj)π0(aj|sj)p(mj|sj, a∗ +j)πe(a∗ +j|sj) +� +ν(s0), +and +DME(πe, π0) = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtp(st+1, rt|st, at, mt)p(mt|st, at)π0(at|st) +× +� � +¯a∗ +t−1 +t−1 +� +j=0 +p(sj+1, rj|sj, aj, mj)π0(aj|sj)p(mj|sj, a∗ +j)πe(a∗ +j|sj) − +t−1 +� +j=0 +pπ0(sj+1, rj, mj, aj|sj) +� +ν(s0). +The proof of Theorem 4.1 is thus completed. +21 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +D +Learning Nuisance Functions +Recall that the MR estimators requires estimation of nuisance functions including πb, r, pm, ω, Q, and η. While πb, r, +and pm can be estimated efficiently using state-of-the-art nonparametric methods (i.e., regression/classification tree +(Breiman et al., 2017), random forest (Breiman, 2001)), we focus on the methods used to learn ω, Q, and η. +We first consider the estimation of ωπ for any stationary policy π. Following the arguments in Liu et al. (2018) and +Uehara et al. (2020), we can show that for any function f +E +� +ωπ(S){f(S) − π(A|S) +πb(A|S)f(S′)} +� += 0, +subjected to the constraint that E[ +1 +� +i Ti ωπ(Si,t)] = 1, where the expectation is taken over the observed stationary +distribution of (S, A, S′). Therefore, estimating the ωπ is equivalent to solving a mini-max problem such that +min +ωπ∈Ω max +f∈F E2 +� +ωπ(S){f(S) − π(A|S) +πb(A|S)f(S′)} +� +(14) +for some function classes Ω and F. In our implementation, we consider linear function classes Ω and F, which yields +closed-form expressions. Specifically, let ωπ(s) = ξT (s)β for some dω-dimensional β ∈ Rdω, where ξ(s) is the feature +vector generated by RBF sampler (Rahimi & Recht, 2007). Then (14) is equivalent to obtain β by solving the equation +1 +NT +� +i,t +� +ξ(Si, t) − π(Ai,t|Si,t) +πb(Ai,t|Si,t)ξ(Si,t+1) +� +ξT (Si,t)β = 0, +subjected to that +1 +NT +� +i,t ξT (Si,t)β = 1. Similarly, considering ωG, we can show that +E +� +ωG(S){f(S) − ρ(S, A, M)π0(A|S) +πb(A|S)f(S′)} +� += 0, +where the expectation is taken over the distribution of (S, A, M, S′). ωG can then be estimated following the same +steps. +We next consider the estimation of pairs of (Q, η). Taking (Qπe, ηπe) as an example, the estimation procedure is +motivated by the Bellman equation model, such that: +Qπe(St, At, Mt) = Eπe[Rt + Eπe +a,mQπe(St+1, a, m) − ηπ]. +(15) +Similar to the work of Shi et al. (2022a), we approximate the Q function using linear sieves. Specifically, we assume +that +Qπe(s, a, m) ≈ ΦT +L(s, m)βa, ∀s ∈ S, a ∈ A, m ∈ M, +where ΦT +L(s, m) is a L-dimensional feature vector derived using L sieve basis functions, such as splines (Huang, 1998). +Let β∗ = (βT +0 , · · · , βT +K−1, ηπ)T . Let U(s, a, m) denotes +[ΦT +L(s, m)1(a = 0), · · · , ΦT +L(s, m)1(a = K − 1), 1]T , +and V (s) denotes +[Em|s,a=0ΦT +L(s, m)πe(0|s), · · · , Em|s,a=K−1ΦT +L(s, m)πe(K − 1|s), 0]T , +where Em|S′,aΦT +L(S′, m) = +� +m ΦT +L(S′, m)p(m|S′, a) can be approximated by Monte Carlo sampling in practice. The +equation (15) can be rewritten as +EU(S, A, M)[R + V (S′)T β∗ − U(S, A, M)T β∗] = 0. +Let Ui,t = U(Si,t, Ai,t, Mi,t) and Vi,t = V (Si,t). Based on the observational data, the closed-form solution of β∗ is +� +� 1 +NT +� +i,t +Ui,t(Ui,t − Vi,t+1)T +� +� +−1 +1 +NT +� +i,t +Ui,tRi,t. +In practice, we add ridge penalty to the term within the bracket to avoid overfitting, and let L grow as the sample size +grows to improve the approximation precision. +22 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +E +Proof of Theorem 6.1 +The proof of the triply robustness property of the proposed estimator is similar for IDE(πe, π0), IME(πe, π0), +DDE(πe, π0) and DME(πe, π0). Here, we take the estimator of IDE as an example. Let O denote a data tuple +(S, A, M, R, S′), ρ(S, A, M) = +� +a πe(a|S)p(M|S,a) +p(M|S,A) +, and δπ(S, A) = ωπ(S) π(A|S) +πb(A|S) for any policy π. Without loss of +generality, we let Ti = T, ∀i = 1, · · · , N. We first reorganize the estimator of IDE into four parts. Recall that ηd = η. +Let +φ1(O) = ηπe − ηGe, +φ2(O) = δπe(S, A) +� +R + E a∼πe(•|S′) +m∼p(•|a,S′) +Qπe(S′, a, m) − Em∼p(•|A,S)Qπe(S, A, m) − ηπe +� +, +φ3(O) = δπe(S, A)ρ(S, A, M)π0(A|S) +πe(A|S){R − r(S, A, M)}, +φ4(O) = δπe(S, A) +� +Ea∼π0(•|S)r(S, a′, M) + E a∼πe(•|S′) +m∼p(•|S′,a) +QGe(S′, a, m) − Em∼p(•|S,A)QGe(S, A, m) − ηGe +� +. +Then the proposed MR estimator of IDE is +MR-IDE(πe, π0) = +1 +NT +� +i,t +[ˆφ1(Oi,t) + ˆφ2(Oi,t) − ˆφ3(Oi,t) − ˆφ4(Oi,t)]. +The proof of robustness can be divided into four parts. In part I, we show that when ˆπb and ˆωπe are consistent, the sum +of terms involving Qπe, QGe, ηπe, and ηGe converges to zero by the stationary property. Then, the remaining part of +MR-IDE(πe, π0) is +1 +NT +� +i,t +ˆδπe(Si,t, Ai,t) +� +Ri,t − Ea∼π0(•|S)ˆr(Si,t, a′, Mi,t) +� +� +�� +� +ˆφ5(Oi,t) +−ˆφ3(Oi,t). +(16) +In part II, we consider the condition M1, where ˆπb, ˆωπe, and ˆr are consistent. We show that +1 +NT +� +i,t ˆφ3(Oi,t) +converged to 0, and +1 +NT +� +i,t ˆφ5(Oi,t) is unbiased to the IS estimator with correctly specified πb, ωπe, and r and thus +unbiased and consistent to IDE(πe, π0), using the arguments used in part I. Together with the results from part I, the +consistency of our estimator is proved. +In part III, we focus on the condition M2, where ˆπb, ˆωπe, and ˆpm are consistent. We show that (16) is consistent to +IDE(πe, π0). The consistency is then completed, together with part I. +Finally, in part IV, applying similar arguments in part I, we observe that +1 +NT +� +i,t ˆφ2(Oi,t), +1 +NT +� +i,t ˆφ3(Oi,t), and +1 +NT +� +i,t ˆφ4(Oi,t) converge to 0 respectively, when ˆQπe, ˆQGe, ˆηπe, ˆηGe, ˆr, and ˆpm are consistent. Then, we show that +MR-IDE(πe, π0) = ˆφ1 is consistent to IDE(πe, π0), with consistent ˆηπe and ˆηGe. The consistency of the proposed +estimator is thus proved, and the proof of triply-robustness is thus completed. +We next detail the proof for each part. +Part I. Condition: ˆπb and ˆωπe are consistent. +First, we focus on the terms involving Qπe. Let f1(O; ωπe, πb, pm, Qπe) denotes +δπe(A|S) +� +E a∼πe(•|S′) +m∼p(•|a,S′) +Qπe(S′, a, m) − Em∼p(•|A,S)Qπe(S, A, m) +� +. +23 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +To show that +1 +NT +� +i,t f1(Oi,t; ˆωπe, ˆπb, ˆpm, ˆQπe) converges to 0, when πb and ωπe are consistent, we decompose it +into +1 +NT +� +i,t +f1(Oi,t; ˆωπe, ˆπb, ˆpm, ˆQπe) − +1 +NT +� +i,t +f1(Oi,t; ωπe, ˆπb, ˆpm, ˆQπe) +� +�� +� +Γ1 ++ +1 +NT +� +i,t +f1(Oi,t; ωπe, ˆπb, ˆpm, ˆQπe) − +1 +NT +� +i,t +f1(Oi,t; ωπe, πb, ˆpm, ˆQπe) +� +�� +� +Γ2 ++ +1 +NT +� +i,t +f1(Oi,t; ωπe, πb, ˆpm, ˆQπe) +� +�� +� +Γ3 +. +It suffices to show that Γ1, Γ2, and Γ3 all converge to zero in probability. +Let us focus on Γ1 first. Under the assumptions that Ωπe, Qπe, Hm, and Πb are all bounded function classes and +ˆπb(Ai,t|Si,t) is uniformly bounded away from zero, |Γ1| is upper bounded by +O(1) +NT +� +i,t +|ˆωπe(Si,t) − ωπe(Si,t)|, +(17) +where O(1) is some positive constant. By Markov’s inequality, to prove (17) converges to zero in probability, it suffices +to show that +1 +NT E +� +i,t +|ˆωπe(Si,t) − ωπe(Si,t)| = o(1). +(18) +For any sufficient small constant ϵ > 0, let Ωπe(ϵ) defines a set of function ω, such that, +Es∼p∞|ω(s) − ωπe(s)|2 ≤ ϵ2, +(19) +where p∞ denotes the limiting distribution of state under behavior policy. Since ˆωπe is consistent and converge to ωπe +in L2-norm, we can show that ˆωπe ∈ Ωπe(ϵ) with probability approaching to 1 (wpa1) for large NT, by Markov’s +inequality. Therefore, the right-hand side (RHS) of (18) is upper bounded by +1 +NT E +sup +ω∈Ωπe(ϵ) +� +i,t +|ω(Si,t) − ωπe(Si,t)|, +(20) +wpa1. Then, it suffices to show that (20) is op(1). +Implementing the empirical process theory (Van Der Vaart & Wellner, 1996), we first decompose (20) into +1 +NT E +sup +ω∈Ωπe(ϵ) +� +� +� +� +i,t +|ω(Si,t) − ωπe(Si,t)| − E +� +i,t +|ω(Si,t) − ωπe(Si,t)| +� +� +� +� +�� +� +Γ4 ++ +1 +NT +sup +ω∈Ωπe(ϵ) +� +� +�E +� +i,t +|ω(Si,t) − ωπe(Si,t)| +� +� +� +� +�� +� +Γ5 +. +By the definition of Ωπe(ϵ) and the Cauchy Schwartz inequality, E|ω(Si,t) − ωπe(Si,t)| ≤ ϵ for any ω ∈ Ωπe(ϵ). Thus, +Γ5 is upper bounded by ϵ and converges to zero when ϵ → 0 (i.e., Γ5 = o(1)). +Next, we show that Γ4 converges to zero as well. Under the assumption that Ωπe(ϵ) is a VC-type classes with VC +indices upper bounded by O(N k) for k < 1 +2 and ϵ is sufficiently small, using the maximal inequality (See Section 4.2 +in Dedecker & Louhichi (2002) and Corollary 5.1 in Chernozhukov et al. (2014)), we can show that +√ +NTΓ4 converges +to zero (i.e., +√ +NTΓ4 = op(1)). Therefore, we have that, Γ4 = op( +1 +√ +NT ). The proof of Γ1 = op(1) is then completed. +Similarly, following the steps to prove Γ1 = op(1), we can show that Γ2 = op(1). Then, it remains to show that +Γ3 = op(1). By Markov’s inequality, it suffices to show that E(Γ3) = o(1). By the definition of Γ3, E(Γ3) is upper +bounded by +1 +NT E +sup +˜p∈Hm,Q∈Q +� +i,t +f1(Oi,t; ωπe, πb, ˜p, Q). +(21) +24 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +We first observe that, for any Q ∈ Qπe and ˜p ∈ Hm, the expectation of Γ3 is zero. Specifically, +E +� +ωπe(S)πe(A|S) +πb(A|S)E a∼πe(•|S′) +m∼˜p(•|a,S′) +Q(S′, a, m) − Em∼˜p(•|A,S)Q(S, A, m) +� += +� +a +� +s,m,s′ p(m, s′|a, s)pπe(s)πe(a|s) +� +a′ +� +m′ Q(s′, a′, m′)˜p(m′|a′, s′)πe(a′|s′) +− +� +a +� +s,m,s′ p(m, s′|a, s)pπe(s)πe(a|s) +� +m′ Q(s, a, m′)˜p(m′|a, s) += +� +a′ +� +s′,m′ pπe(s′)πe(a′|s′)Q(s′, a′, m′)˜p(m′|a′, s′) +− +� +a +� +s,m′ pπe(s)πe(a|s)Q(s, a, m′)˜p(m′|a, s) +=0. +Then, following the same steps we used to bound (20), we can show that (21) is o(1). Thus, Γ3 = op(1). Together with +Γ1 = op(1) and Γ2 = op(1), we finish the proof of +1 +NT +� +i,t f1(Oi,t; ˆωπe, ˆπb, ˆpm, ˆQπe) = op(1). +Then we focus on the terms involving QGe. Let f2(O; ωπe, πb, pm, QGe) denotes +δπe(A|S) +� +E a∼πe(•|S′) +m∼p(•|a,S′) +QGe(S′, a, m) − Em∼p(•|A,S)QGe(S, A, m) +� +. +Replacing Qπe(S, A, m) with QGe(S, A, m) in the proof of +1 +NT +� +i,t f1(Oi,t; ˆωπe, ˆπb, ˆpm, ˆQπe) = op(1), we can +directly show that +1 +NT +� +i,t f2(Oi,t; ˆωπe, ˆπb, ˆpm, ˆQGe) = op(1) as well. +Finally, we need to show that the sum of terms involving ηπe and ηGe converges to zero. Let f3(O; ωπe, πb, ηπe, ηGe) +denotes +� +1 − ωπe(S)πe(A|S) +πb(A|S) +� +(ηπe − ηGe). +For any η1 ∈ R and η2 ∈ R, +1 +NT +� +i,t f3(Oi,t; ωπe, πb, η1, η2) has mean zero. Specifically, +E[η1 − η2 − ωπe(S)πe(A|S) +πb(A|S)(η1 − η2)] += +� +1 − +� +a +� +s +pπb(a, s)ωπe(s)πe(a|s) +πb(a|s) +� +(η1 − η2) +=0 × (η1 − η2) +=0. +Applying the same arguments in showing that Γ3 = op(1), we can show that +1 +NT +� +i,t f3(Oi,t; ωπe, πb, ˆηπe, ˆηGe) = +op(1). Then, following the same steps proving that Γ1 = op(1), we can show that +1 +NT +� +i,t +� +f3(Oi,t; ˆωπe, ˆπb, ˆηπe, ˆηGe) − f3(Oi,t; ωπe, ˆπb, ˆηπe, ˆηGe) +� += op(1), +and +1 +NT +� +i,t +� +f3(Oi,t; ωπe, ˆπb, ˆηπe, ˆηGe) − f3(Oi,t; ωπe, πb, ˆηπe, ˆηGe) +� += op(1). +Therefore, +1 +NT +� +i,t f3(Oi,t; ˆωπe, ˆπb, ˆηπe, ˆηGe) = op(1). The proof of part I is thus completed. +Part II. Condition: ˆπb(A|S), ˆωπe(S), and ˆr are consistent. +With true r, ωπe, and πb, we can show that Eφ3(Oi,t; ωπe, πb, ˆpm, r) has a mean of zero, as E[R − r(s, a, m)|S = +s, A = a, M = m] = 0. Then, using the same arguments in showing that Γ3 = op(1) in part I, we can show that +25 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +1 +NT +� +i,t ˆφ3(Oi,t; ωπe, πb, ˆpm, r) = op(1). Next, following the same steps proving that Γ1 = op(1), we can show that +1 +NT +� +i,t +ˆφ3(Oi,t; ˆωπe, ˆπb, ˆpm, ˆr) − +1 +NT +� +i,t +ˆφ3(Oi,t; ωπe, πb, ˆpm, r) = op(1). +Therefore, we finish the proof showing that +1 +NT +� +i,t ˆφ3(Oi,t; ˆωπe, ˆπb, ˆpm, ˆr) = op(1). Then, it remains to show that +1 +NT +� +i,t ˆφ5(Oi,t; ˆωπe, ˆπb, ˆpm, ˆr) is consistent to IDE(πe, π0). Again, applying the arguments used in showing that +Γ1 = op(1), we can show that +1 +NT +� +i,t +ˆφ5(Oi,t; ˆωπe, ˆπb, ˆpm, ˆr) − +1 +NT +� +i,t +ˆφ5(Oi,t; ωπe, πb, ˆpm, r) = op(1). +Then, it suffices to show that +1 +NT +� +i,t ˆφ5(Oi,t; ωπe, πb, ˆpm, r) is consistent to IDE(πe, π0). Specifically, +1 +NT +� +i,t +ˆφ5(Oi,t; ωπe, πb, ˆpm, r) = +1 +NT +� +i,t +ωπe πe(Ai,t|Si,t) +πb(Ai,t|Si,t) +� +Ri,t − +� +a +π0(a|Si,t)r(Si,t, a, Mi,t) +� +. +(22) +Under the assumption of stationary state process, since the action space is finite, it suffices to show that, +Es∼ˆpπe +m∼ˆpm +ωπe πe(a|s) +πb(a|s) +� +r − +� +a′ +π0(a′|s)r(s, a′, m) +� +P−→ Es∼pπe +m∼pm +ωπe πe(a|s) +πb(a|s) +� +r − +� +a′ +π0(a′|s)r(s, a′, m) +� +(23) +for any a. By the weak law of large number, we can show that (23) holds when NT is sufficiently large. Together with +the results in part I, we thus complete the proof of Part II. +Part III. Condition: ˆπb(A|S), ˆωπe(S), and ˆpm are consistent. +Applying the same arguments used in showing that Γ1 = op(1), we can show that +1 +NT +� +i,t +ˆφ5(Oi,t; ˆωπe, ˆπb, ˆpm, ˆr) − +1 +NT +� +i,t +ˆφ5(Oi,t; ωπe, πb, pm, ˆr) = op(1), +and +1 +NT +� +i,t +ˆφ3(Oi,t; ˆωπe, ˆπb, ˆpm, ˆr) − +1 +NT +� +i,t +ˆφ3(Oi,t; ωπe, πb, pm, ˆr) = op(1). +Then, it suffices to show that +1 +NT +� +i,t +ˆφ5(Oi,t; ωπe, πb, pm, ˆr) − +1 +NT +� +i,t +ˆφ3(Oi,t; ωπe, πb, pm, ˆr) +p→ IDE(πe, π0). +(24) +The LHS of (24) can be decomposed into two parts. Specifically, it suffices to show that +1 +NT +� +i,t +δπe(Si,t, Ai,t) +� +Ea′∼π0(•|Si,t)ˆr(Si,t, a′, Mi,t) − ρ(Si,t, Ai,t, Mi,t)π0(Ai,t|Si,t) +πe(Ai,t|Si,t) ˆr(Si,t, Ai,t, Mi,t) +� += op(1), +(25) +and +1 +NT +� +i,t +δπe(Si,t, Ai,t) +� +Ri,t − ρ(Si,t, Ai,t, Mi,t)π0(Ai,t|Si,t) +πe(Ai,t|Si,t)Ri,t +� +P−→ IDE(πe, π0). +(26) +Following the steps showing that Γ3 = op(1) in part I, since the expectation of the LHS of (25) is 0, we can show that +(25) holds. Furthermore, applying the arguments used in showing (23) in part II, we can show that (26) holds. Together +with the results in part I, we thus complete the proof of Part III. +Part IV. Condition: ˆQπe, ˆQGe, ˆηπe, ˆηGe, ˆr, and ˆpm are consistent. +As we discussed in the main context, with true Qπe, QGe, ηπe, ηGe, r, and pm, we can show that +Eˆφj(Oi,t; Qπe, QGe, ηπe, ηGe, r, pm, ˆωπe, ˆπb) = 0 for j = 2, 3, 4. Then, using the same arguments in showing +that Γ3 = op(1) in part 1, we can show that +1 +NT +� +i,t +ˆφj(Oi,t; Qπe, QGe, ηπe, ηGe, r, pm, ˆωπe, ˆπb) = op(1), for j = 2, 3, 4. +26 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Then, applying the arguments used in showing that Γ1 = op(1), we can further show that +1 +NT +� +i,t +� +ˆφj(Oi,t; ˆQπe, ˆQGe, ˆηπe, ˆηGe, ˆr, ˆpm, ˆωπe, ˆπb) − ˆφj(Oi,t; Qπe, QGe, ηπe, ηGe, r, pm, ˆωπe, ˆπb) +� += op(1), +for j = 2, 3, 4. These two results further yields that +1 +NT +� +i,t +ˆφj(Oi,t; ˆQπe, ˆQGe, ˆηπe, ˆηGe, ˆr, ˆpm, ˆωπe, ˆπb) = op(1) +for j = 2, 3, 4. Then, it remains to show that ˆφ1(ˆηπe, ˆηGe) is consistent to IDE(πe, π0). Applying the arguments used +to show Γ1 = op(1) again, under the assumption that we have that ˆηπe and ˆηGe are consistent, +ˆφ1(ˆηπe, ˆηGe) +P−→ ˆφ1(ηπe, ηGe) = IDE(πe, π0), +where the equation holds by definition. The proof of part IV is thus completed. +F +Proof of Theorem 6.2 +First, we clarify the assumption of convergence. We required that each of ˆQ(·), ˆω(·), ˆpm, ˆr, ˆπb, and ˆη(·) converges to its +corresponding oracle value in L2-norm at a rate of N −k∗, for some k∗ > 1/4. Specifically, taking ˆωπe as an example, +we assume that +� +Es∼p∞|ˆωπe(s) − ωπe(s)| = Op(N −k∗). +The proof of the efficiency of the proposed estimator is similar for IDE(πe, π0), IME(πe, π0), DDE(πe, π0), and +DME(πe, π0). Here, we take the MR estimator of IDE as an example. Adopting the notation used in the Appendix E, +we have the proposed multiply robust estimator of IDE as +MR-IDE(πe, π0) = +1 +NT +� +i,t +[ˆφ1(Oi,t) + ˆφ2(Oi,t) − ˆφ3(Oi,t) − ˆφ4(Oi,t)]. +Taking the oracle values of the estimators (i.e., Qπe, QGe, ηπe, ηGe, r, pm, ωπe, πb), we define the oracle estimator as +MR-IDE∗(πe, π0) = +1 +NT +� +i,t[ˆφ∗ +1(Oi,t) + ˆφ∗ +2(Oi,t) − ˆφ∗ +3(Oi,t) − ˆφ∗ +4(Oi,t)]. +We decompose the proof into two parts. In part I, we show that the proposed estimator is asymptotically equivalent +to the oracle estimator, such that MR-IDE(πe, π0) − MR-IDE∗(πe, π0) = op( +1 +√ +NT ). In part II, we show that the +oracle estimator is asymptotically normal such that +√ +N[MR-IDE∗(πe, π0) − IDE(πe, π0)] +d−→ N(0, σ2 +T ), where σ2 +T +is the semiparametric efficiency bound. Noticing that ψ2(Oi,t), ψ3(Oi,t), and ψ4(Oi,t) are the martingale difference +sequence with respect to {Oi,t}0≤t≤T −1, under the assumption of stationarity, we have that +σ2 +T = 1 +T V ar [φ2(Ot) − φ3(Ot) − φ4(Ot)] . +Therefore, we have that +√ +NT[MR-IDE∗(πe, π0) − IDE(πe, π0)] +d−→ N(0, σ2), +where σ2 = V ar[φ2(Ot) − φ3(Ot) − φ4(Ot)]. Finally, by Slutsky’s theorem, the proposed estimator is asymptotically +normally distributed with mean 0 and a variance achieving the semiparametric efficiency bound. Specifically, +√ +NT +� +MR-IDE(πe, π0) − IDE(πe, π0) +� d−→ N(0, σ2). +In the following, we detail the proof of each part. +Part I. Let ˆψ = { ˆQπe, ˆQGe, ˆηπe, ˆηGe, ˆpm}. We first decompose the MR-IDE(πe, π0) − MR-IDE∗(πe, π0) in to three +parts, such that MR-IDE(πe, π0) − MR-IDE∗(πe, π0) = MR-IDE(1)( ˆψ) + MR-IDE(2)( ˆψ) + MR-IDE(3)( ˆψ, ˆr), where +MR-IDE(1)( ˆψ) = +1 +NT +� +i,t +� +2 +� +j=1 +[ˆφ1( ˆψ, ωπe, πb, r) − ˆφ∗ +1(Oi,t)] − +4 +� +j=3 +[ˆφj(Oi,t; ˆψ, ωπe, πb, r) − ˆφ∗ +j(Oi,t)] +� +, +27 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +MR-IDE(2)( ˆψ) = +1 +NT +� +i,t +� +2 +� +j=1 +[ˆφ1( ˆψ, ωπe, πb, ˆr) − ˆφ1( ˆψ, ωπe, πb, r)] − +4 +� +j=3 +[ˆφj(Oi,t; ˆψ, ωπe, πb, ˆr) − ˆφj(Oi,t; ˆψ, ωπe, πb, r)] +� +, +and +MR-IDE(3)( ˆψ, ˆr) = +1 +NT +� +i,t +� +2 +� +j=1 +[ˆφ1( ˆψ, ˆωπe, ˆπb, ˆr) − ˆφ1( ˆψ, ωπe, πb, ˆr)] +− +4 +� +j=3 +[ˆφj(Oi,t; ˆψ, ˆωπe, ˆπb, ˆr) − ˆφj(Oi,t; ˆψ, ωπe, πb, ˆr)] +� +. +Following the arguments in part I and part II of the proof of Robustness in Appendix E, the expectation of MR-IDE(2)( ˆψ) +is zero. Then, applying the same arguments used in showing that Γ3 = op( +1 +√ +NT ) in part I of the proof of Robustness, +we can show that MR-IDE(1)( ˆψ) = op( +1 +√ +NT ) under the assumption that each component in ˆφ converges to its oracle +value in L2 norm at a rate of N −k∗ for k∗ > 1 +4. +Then, we focus on showing that MR-IDE(2)( ˆψ) = op( +1 +√ +NT ). Noticing that MR-IDE(2)( ˆψ) can be further decomposed +as +MR-IDE(2)( ˆψ) − MR-IDE(2)(ψ) + MR-IDE(2)(ψ), +it suffices to show that MR-IDE(2)( ˆψ) − MR-IDE(2)(ψ) = op( +1 +√ +NT ) and MR-IDE(2)(ψ) = op( +1 +√ +NT ). First, similar +to the part III of the proof of Theorem 6.1, the expectation of MR-IDE(2)(ψ) is 0, for any ˆr ∈ Hr. Then, applying the +arguments used in showing that Γ3 = op( +1 +√ +NT ), we can show that MR-IDE(2)(ψ, r) = op( +1 +√ +NT ) under the assumption +that ˆωπe and ˆπb converge to their oracle values. Then, it remains to show that MR-IDE(2)( ˆψ) − MR-IDE(2)(ψ) = +op( +1 +√ +NT ). It suffices to show that +1 +NT +� +i,t +[ˆφj(Oi,t; ˆψ, ωπe, πb, ˆr) − ˆφj(Oi,t; ˆψ, ωπe, πb, r)] − [ˆφj(Oi,t; ψ, ωπe, πb, ˆr) − ˆφj(Oi,t; ψ, ωπe, πb, r)] = op( +1 +√ +NT +), +(27) +for j = 1, 2, 3, 4. Here, we prove that the above equation holds for j = 3 as an example. For j = 1, 2, 4, the proof can +be completed using similar arguments. +We first observe that the LHS of (27) is upper bounded by +1 +NT +� +i,t +|[ˆφj(Oi,t; ˆψ, ωπe, πb, ˆr) − ˆφj(Oi,t; ˆψ, ωπe, πb, r)] − [ˆφj(Oi,t; ψ, ωπe, πb, ˆr) − ˆφj(Oi,t; ψ, ωπe, πb, r)]| += 1 +NT +� +i,t +|δπe(Si,t, Ai,t)||ˆρ(Si,t, Ai,t, Mi,t) − ρ(Si,t, Ai,t, Mi,t)|π0(Ai,t|Si,t) +πe(Ai,t|Si,t)|r(Si,t, Ai,t, Mi,t) − ˆr(Si,t, Ai,t, Mi,t)| +≤ C +NT +� +i,t +|ˆρ(Si,t, Ai,t, Mi,t) − ρ(Si,t, Ai,t, Mi,t)||r(Si,t, Ai,t, Mi,t) − ˆr(Si,t, Ai,t, Mi,t)| +≤ C +2NT +� +i,t +|ˆρ(Si,t, Ai,t, Mi,t) − ρ(Si,t, Ai,t, Mi,t)|2 + +C +2NT +� +i,t +|r(Si,t, Ai,t, Mi,t) − ˆr(Si,t, Ai,t, Mi,t)|2 +=op( +1 +√ +NT +), +where C is some positive constant. The first inequality holds under the assumption that Ωπe and Πb are bounded +function classes of ωπe and πb, respectively. The second inequality holds by applying the Cauchy-Schwartz inequality +such that ab ≤ a2+b2 +2 +. Using the similar arguments used to bound (18) in part I of the proof of Theorem 6.1, under +the assumption that ˆpm and ˆr converge to their oracle values respectively in L2 norm at a rate of Op(N −k∗) for some +k∗ > 1/4, we can show that the final equality holds. Similarly, we can show that MR-IDE(3)( ˆψ, ˆr) = op( +1 +√ +NT ) as +well. The proof of part I is thus completed. +28 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Part II. By Central Limit Theorem, when N → ∞, we can show that +√ +N[MR-IDE∗(πe, π0) − IDE(πe, π0)] +d−→ N(0, σ2 +T ), +for some variance σ2 +T . Then it remains to show that σ2 +T achieves the asymptotic semiparametric efficiency bound, which +is the supreme of the Cramer-Rao lower bounds for all parametric submodels (Newey, 1990). +We first introduce some additional notations. Let πb,θ, pm,θ and ps′,r,θ, and νθ be some parametric models parameterized +by θ for πb, pm and ps′,r, and ν, and M denotes the set of all such parametric models. Then, by Theorem 1, IDE(πe, π0) +can be represented as a function of θ. We denote the IDE(πe, π0) parameterized by θ as IDEθ(πe, π0). By definition, +the Cramer-Rao lower bound for an unbiased estimator is +CR(πb,θ, pm,θ, ps′,r,θ, νθ) = ∂IDEθ(πe, π0) +∂θ +� +E +�∂l({Ot}0≤t≤T −1; θ) +∂θ +∂lT ({Ot}0≤t≤T −1; θ) +∂θ +��−1 ∂IDEθ(πe, π0) +∂θ +T +, +where l({Ot}0≤t≤T −1; θ) is the log-likelihood function. +Suppose that there exists some parameter θ0 such that πb,θ0, pm,θ0 and ps′,r,θ0, and νθ0 are the corresponding true +models. Then the semiparametric efficiency bound is +sup +M +CR = +sup +πb,pm,ps′,r,ν∈M +CR(πb, pm, ps′,r, ν) = CR(πb,θ0, pm,θ0, ps′,r,θ0, νθ0). +(28) +It suffices to show that σ2 +T = supM CR. +On the one hand, from Appendix G, we have that +∂IDEθ0(πe, π0) +∂θ += E[(ηπe − ηGe)S( ¯OT −1)] + D1(θ0) − D2(θ0), +(29) +where ¯OT −1 is the sequence of observations such that ¯OT −1 = {O1, O2, · · · , OT −1}, S(·) is the gradient of the +log-likelihood function evaluated at θ = θ0 (i.e., ∂l({Ot}0≤t≤T −1;θ0) +∂θ +), +D1(θ0) = E +� +1 +T +T −1 +� +t=0 +ωπe(St) πe(At|St) +πb,θ0(At|St){Rt + Eπe +a∗,m∗;θ0Qπe(St+1, a, m) − Em;θ0Qπe(St, At, m) − ηπe}S( ¯OT −1) +� +, +and +D2(θ0) = E +� 1 +T +T −1 +� +t=0 +ωπe(St) +�� +a pθ0(Mt|St, a)πe(a|St) +pθ0(Mt|St, At) +π0(At|St) +πb,θ0(At|St)[Rt − rθ0(St, At, Mt)] + +πe(At|St) +πb,θ0(At|St) +× { +� +a′ +rθ0(St, a′, Mt)π0(a′|St) − ηGe + Eπe +a,m;θ0QGe(St+1, a, m) − Em;θ0QGe(St, At, m)} +� +S( ¯OT −1) +� +. +Adopting the notation used in Appendix F, (29) can be rewritten as +E +�� +1 +T +� +t +[φ1(Ot) + φ2(Ot) − φ3(Ot) − φ4(Ot)] +� +S( ¯OT −1) +� +. +Furthermore, since the expectation of a score function is 0, we can show that E[IDEθ0(πe, π0) × S( ¯OT −1)] = +IDEθ0(πe, π0) × E[S( ¯OT −1)] = 0. Therefore, +∂IDEθ0(πe,π0) +∂θ +can be further represented as +E +�� +1 +T +� +t +[φ1(Ot) + φ2(Ot) − φ3(Ot) − φ4(Ot)] − IDEθ0(πe, π0) +� +S( ¯OT −1) +� +. +By Cauchy-Schwartz inequality (Tripathi, 1999), we have that +sup +M +CR ≤E +� +� +� +1 +T +� +t +[φ1(Ot) + φ2(Ot) − φ3(Ot) − φ4(Ot)] − IDEθ0(πe, π0) +�2� +� +=V ar +� +1 +T +� +t +[φ1(Ot) + φ2(Ot) − φ3(Ot) − φ4(Ot)] − IDEθ0(πe, π0) +� +=σ2 +T . +On the other hand, by Lemma 20 in Kallus & Uehara (2022), there exists model Mθ′ ∈ M with sufficiently large +number of parameters, having CR(πb,θ′, pm,θ′, ps′,r,θ′, νθ′) = σ2 +T . Therefore, we have that σ2 +T = supM CR. The +proof is thus completed. +29 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +G +Derivation of Efficient Influence Functions (EIF) +In this section, we focus on deriving the efficient influence function for each component of the average treatment effect. +Without loss of generality, we assume that the state, action, mediator and reward are all discrete. While adopting the +notations used in the Appendix F, we omit the subscript in pm and ps′,r when there is no confusion. Let τt denote the +data trajectory {(sj, aj, mj, rj, sj+1)}0≤j≤t. +G.1 +EIF for Immediate Direct Effect +Let us first focus on the immediate direct effect (IDE). IDEθ0(πe, π0) can be represented as +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +� +rtpθ0(st+1, rt|st, at, mt) − +� +s∗,r∗,a′ +r∗pθ0(s∗, r∗|st, a′, mt)π0(a′|st) +� +pθ0(mt|st, at)πe(at|st) +× +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj)νθ0(s0), +(30) +where ν denotes the initial state distribution, and +pπe +θ0 (sj+1, rj, mj, aj|sj) = pθ0(sj+1, rj|sj, aj, mj)pθ0(mj|sj, aj)πe(aj|sj). +Taking the derivative of (30), we have +∂IDEθ0(πe, π0) +∂θ += C1 + D1 − D2, +where +C1 = (30) × ▽θ log(νθ0(s0)), +D1 = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rt +t� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +t +� +j=0 +� +▽θ log pπe +θ0 (sj+1, rj, mj, aj|sj)] × νθ0(s0), +and +D2 = lim +T →∞ +1 +T +T −1 +� +t=0 +� +a′,τt +rtpθ0(st+1, rt|st, at, mt)π0(at|st)pθ0(mt|st, a′)πe(a′|st) +× +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +▽θ log pπe +θ0 (st+1, rt|mt, st, at) + ▽θ log pθ0(mt|st, a′) ++ +t−1 +� +j=0 +� +▽θ log pπe +θ0 (sj+1, rj, mj|sj, aj)] +� +× νθ0(s0). +(31) +In the following sections, we will derive C1, D1, and D2, respectively. +G.1.1 +C1 +We first focus on C1. Since the expectation of a score function is zero, we have that +C1 = E[IDEθ0(πe, π0) × ▽θ log(νθ0(s0))] = E[IDEθ0(πe, π0) × S( ¯OT −1)] = E[(ηπe − ηGe) × S( ¯OT −1)]. +G.1.2 +D1 +We then focus on the derivation of D1. Notice that +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +ηπe +t� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +t +� +j=0 +� +▽θ log pπe +θ0 (sj+1, rj, mj, aj|sj)] νθ0(s0), += lim +T →∞ +1 +T +T −1 +� +t=0 +ηπeE[ +t +� +j=0 +▽θ log pπe +θ0 (sj+1, rj, mj, aj|sj)] × νθ0(s0), +=0, +30 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +where the last equation holds using the fact that the expectation of a score function is 0. Therefore, +D1 += +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +[r − ηπe] +t� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +t +� +j=0 +� +▽θ log pπe +θ0 (sj+1, rj, mj, aj|sj)] νθ0(s0). +Together with the trick of the equality ⋆ (See Appendix G.5 for a complete proof of it), we have that +D1 +⋆= lim +T →∞ +1 +T +T −1 +� +j=0 +� +τj +[r − ηπe + Eπe +a∗,m∗Qπe(sj+1, a∗, m∗)] +j� +k=0 +pπe +θ0 (sk+1, rk, mk, ak|sk) +× ▽θ log pπe +θ0 (sj+1, rj, mj, aj|sj) × νθ0(s0). +(32) +Then, we note that +� +s0 +j� +k=0 +pπe +θ0 (sk+1, rk, mk, ak|sk)νθ0(s0) +⋆⋆= pπe +θ0 (sj+1, rj, mj, aj|sj)pπe(sj), +which is the probability of {Sj+1 = sj+1, Rj = rj, Mj = mj, Aj = aj} under the target polity πe. Further, we notice +that +▽θ log pπe +θ0 (sj+1, rj, mj, aj|sj) = ▽θ log pθ0(sj+1, rj, mj|aj, sj) +Using the fact that the expectation of a score function is 0, we have +� +sj+1,rj,mj +[pθ0(sj+1, rj, mj|aj, sj)▽θ log pπe +θ0 (sj+1, rj, mj|aj, sj)] = 0 +for any j, which follows that +lim +T →∞ +1 +T +T −1 +� +j=0 +� +τj +Em∗Qπe(sj, aj, m∗)pπe +θ0 (sj+1, rj, mj, aj|sj)pπe(sj) × ▽θ log pπe +θ0 (sj+1, rj, mj|aj, sj) += +lim +T →∞ +1 +T +T −1 +� +j=0 +� +τj−1,aj,sj +Em∗Qπe(sj, aj, m∗)πe(aj|sj)pπe(sj) +× +� +sj+1,rj,mj +[pθ0(sj+1, rj, mj|aj, sj)▽θ log pπe +θ0 (sj+1, rj, mj|aj, sj)] +=0 +Thus, combined with the D1 in equation (32), we have that +D1 = lim +T →∞ +1 +T +T −1 +� +j=0 +� +τj +[rj − ηπe + Eπe +a∗,m∗Qπe(sj+1, a∗, m∗) − Em∗Qπe(sj, aj, m∗)] +×pπe +θ0 (sj+1, rj, mj, aj|sj)pπe(sj)▽θ log pθ0(sj+1, rj, mj|aj, sj), += lim +T →∞ +1 +T +T −1 +� +j=0 +� +τj +[rj − ηπe + Eπe +a∗,m∗Qπe(sj+1, a∗, m∗) − Em∗Qπe(sj, aj, m∗)] +× πe(aj|sj)pπe(sj) +πb,θ0(aj|sj)pπb(sj)pθ0(sj+1, rj, mj|aj, sj)πb,θ0(aj|sj)pπb(sj)▽θ log pθ0(sj+1, rj, mj|aj, sj), += lim +T →∞ +1 +T +T −1 +� +j=0 +� +τj +[rj − ηπe + Eπe +a∗,m∗Qπe(sj+1, a∗, m∗) − Em∗Qπe(sj, aj, m∗)] +× πe(aj|sj)pπe(sj) +πb,θ0(aj|sj)pπb(sj)pθ0(sj+1, rj, mj|aj, sj)πb,θ0(aj|sj)pπb(sj)▽θ log pπb +θ0 (sj+1, rj, mj, aj, sj). +31 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +The second equation holds by substituting pπe(sj) with pπe(sj) +pπb(sj)pπb(sj) = ωπe(sj)pπb(sj) and πe(aj|sj) with +πe(aj|sj) +πb,θ0(aj|sj)πb,θ0(aj|sj). The last equation holds, using the definition of Qπe(s, a, m), +lim +T →∞ +1 +T +T −1 +� +j=0 +� +τj−1,aj,sj +πe(aj|sj)pπe(sj) +πb,θ0(aj|sj)pπb(sj)πb,θ0(aj|sj)pπb(sj) × {▽θ log πb,θ0(aj|sj) + log pπb(sj)} +× +� +sj+1,rj,mj +[rj − ηπe + Eπe +a∗,m∗Qπe(sj+1, a∗, m∗) − Em∗Qπe(sj, aj, m∗)]pθ0(sj+1, rj, mj|aj, sj) = 0 +Therefore, implementing the fact that the expectation of a score function is zero and utilizing the Markov property, we +obtain that, +D1 = E +� +ωπe(S) πe(A|S) +πb,θ0(A|S){R + Eπe +a,mQπe(S′, a, m) − EmQπe(S, A, m) − ηπe}S( ¯OT −1) +� +. +Since (S, A, M, R, S′) is any arbitrary transaction tuple follows the corresponding distribution, we have that +D1 = E +� +1 +T +T −1 +� +t=0 +ωπe(St) πe(At|St) +πb,θ0(At|St){Rt + Eπe +a,mQπe(St+1, a, m) − EmQπe(St, At, m) − ηπe}S( ¯OT −1) +� +. +G.1.3 +D2 +Finally, we focus on the derivation of D2. Note that in equation (31), D2 can be divided into two parts, where +D(1) +2 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtpθ0(st+1, rt|st, at, mt)π0(at|st) +� +a′ +pθ0(mt|st, a′)πe(a′|st) +× +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj)▽θ log pπe +θ0 (st+1, rt|mt, st, at) × νθ0(s0), +and +D(2) +2 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,mt,τt−1 +pθ0(mt|st, at)πe(at|st) +� +st+1,rt,a′ +rtpθ0(st+1, rt|st, a′, mt)π0(a′|st) +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +× +� +▽θ log pθ0(mt|st, at) + +t−1 +� +j=0 +� +▽θ log pπe +θ0 (sj+1, rj, mj|sj, aj)] +� +× νθ0(s0), +note that here we switch the summation of a and a′ and change the subscript of the summation accordingly. +Part I (D(1) +2 ). Using the fact that the expectation of a score function is zero, we first obtain that +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rθ0(st, at, mt)pθ0(st+1, rt|st, at, mt)π0(at|st) +� +a′ +pθ0(mt|st, a′)πe(a′|st) +× +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj)▽θ log pπe +θ0 (st+1, rt|mt, st, at) × νθ0(s0) += +lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,mt,τt−1 +rθ0(st, at, mt)π0(at|st) +� +a′ +pθ0(mt|st, a′)πe(a′|st) +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +× +� +st+1,rt +pθ0(st+1, rt|st, at, mt)▽θ log pπe +θ0 (st+1, rt|mt, st, at) × νθ0(s0) +=0 +32 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Therefore, it follows that +D(1) +2 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +[rt − rθ0(st, at, mt)]pθ0(st+1, rt|st, at, mt)π0(at|st) +� +a′ +pθ0(mt|st, a′)πe(a′|st) +× +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj)▽θ log pπe +θ0 (st+1, rt|mt, st, at) × νθ0(s0). +Furthermore, since +lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,mt,τt−1 +[ +� +st+1,rt +rtpθ0(st+1, rt|st, at, mt) − rθ0(st, at, mt)]π0(at|st) +� +a′ +pθ0(mt|st, a′)πe(a′|st) +× +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +t−1 +� +j=0 +▽θ log pπe +θ0 (sj+1, rj|mj, sj, aj) × νθ0(s0) = 0, +D(1) +2 +can be further written as +D(1) +2 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +[rt − rθ0(st, at, mt)]pθ0(st+1, rt|st, at, mt)π0(at|st) +� +a′ +pθ0(mt|st, a′)πe(a′|st) +× +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +t +� +j=0 +▽θ log pπe +θ0 (sj+1, rj|mj, sj, aj) × νθ0(s0). +Then, following the same steps in the proof of the equality D1, we obtain that +D(1) +2 +⋆= lim +T →∞ +1 +T +T −1 +� +j=0 +� +τj +[rj − rθ0(sj, aj, mj)]pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj) +� +a′ +pθ0(mj|sj, a′)πe(a′|sj) +× +j−1 +� +k=0 +pπe +θ0 (sk+1, rk, mk, ak|sk) νθ0(s0)▽θ log pθ0(sj+1, rj|sj, aj, mj). +Similarly, using the equality ⋆⋆, +D(1) +2 +⋆⋆= lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj,aj,mj,rj,sj+1 +[rj − rθ0(sj, aj, mj)]pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj) +× +� +a′ +pθ0(mj|sj, a′)πe(a′|sj)pπe(sj)▽θ log pθ0(sj+1, rj|sj, aj, mj). +Replacing π0(aj|sj) with +π0(aj|sj) +πb,θ0(aj|sj)πb,θ0(aj|sj), +pπe(sj) with +pπe(sj) +pπb(sj)pπb(sj) += +ωπe(sj)pπb(sj), +and +� +a′ pθ0(mj|sj, a′)πe(a′|sj) with +� +a′ pθ0(mj|sj,a′)πe(a′|sj) +pθ0(mj|sj,aj) +pθ0(mj|sj, aj), we obtain that +D(1) +2 += lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj,aj,mj,rj,sj+1 +ωπe(sj) +� +a′ pθ0(mj|sj, a′)πe(a′|sj) +pθ0(mj|sj, aj) +π0(aj|sj) +πb,θ0(aj|sj)[rj − rθ0(sj, aj, mj)] +× pθ0(sj+1, rj|sj, aj, mj)pθ0(mj|sj, aj)πb,θ0(aj|sj)pπb(sj)▽θ log pθ0(sj+1, rj|sj, aj, mj). +Further, since +lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj,aj,mj +ωπe(sj) +� +a′ pθ0(mj|sj, a′)πe(a′|sj) +pθ0(mj|sj, aj) +π0(aj|sj) +πb,θ0(aj|sj)pθ0(mj|sj, aj)πb,θ0(aj|sj)pπb(sj) +× ▽θ log pπb +θ0 (mj, aj, sj) +� +rj,sj+1 +[rj − rθ0(sj, aj, mj)]pθ0(sj+1, rj|sj, aj, mj) = 0, +33 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +we have that +D(1) +2 += lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj,aj,mj,rj,sj+1 +ωπe(sj) +� +a′ pθ0(mj|sj, a′)πe(a′|sj) +pθ0(mj|sj, aj) +π0(aj|sj) +πb,θ0(aj|sj)[rj − rθ0(sj, aj, mj)] +× pθ0(sj+1, rj|sj, aj, mj)pθ0(mj|sj, aj)πb,θ0(aj|sj)pπb(sj)▽θ log pπb +θ0 (sj+1, rj, mj, aj, sj). +Then, combining the fact that the expectation of a score function is zero and the Markov property, we have that +D(1) +2 += E +� +ωπe(S) +� +a pθ0(M|S, a)πe(a|S) +pθ0(M|S, A) +π0(A|S) +πb,θ0(A|S)[R − r(S, A, M)]S( ¯OT −1) +� +. +(33) +Part II (D(2) +2 ). Note that, taking the sum over rt and st+1, D(2) +2 +can be equally represented as +lim +T →∞ +1 +T +T −1 +� +t=0 +� +a′,at,mt,τt−1 +rθ0(st, a′, mt)π0(a′|st)pθ0(mt|st, at)πe(at|st) +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +× +� +▽θ log pθ0(mt|st, at) + +t−1 +� +j=0 +� +▽θ log pπe +θ0 (sj+1, rj, mj|sj, aj)] +� +× νθ0(s0). +Taking the average over rt and st+1, and noticing that +lim +T →∞ +1 +T +T −1 +� +t=0 +� +a′,at,mt,τt−1 +rθ0(st, a′, mt)π0(a′|st)pθ0(mt|st, at)πe(at|st) +t−1 +� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj) +× +� +st+1,rt +pθ0(st+1, rt|st, at, mt)▽θ log pθ0(st+1, rt|st, at, mt)νθ0(s0) = 0, +we can rewrite the D(2) +2 +as +lim +T →∞ +1 +T +T −1 +� +t=0 +� +a′,τt +rθ0(st, a′, mt)π0(a′|st) +t� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj)νθ0(s0) +t +� +j=0 +▽θ log pπe +θ0 (sj+1, rj, mj|sj, aj). +Then, following the steps as we did in deriving D1, we first show that +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +ηGe +t� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj)νθ0(s0) +t +� +j=0 +▽θ log pπe +θ0 (sj+1, rj, mj|sj, aj) = 0, +which follows that +D(2) +2 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +a′,τt +[rθ0(st, a′, mt)π0(a′|st) − ηGe] +t� +j=0 +pπe +θ0 (sj+1, rj, mj, aj|sj)νθ0(s0) +t +� +j=0 +▽θ log pπe +θ0 (sj+1, rj, mj|sj, aj). +Next, similarly, given the definition of QGe, following the steps in deriving D1 and combining with the trick of score +function together with the Markov property, we obtain that +D(2) +2 += E[ωπe(S) πe(A|S) +πb,θ0(A|S){ +� +a′ +rθ0(S, a′, M)π0(a′|S) − ηGe + Eπe +a,mQGe(S′, a, m) − EmQGe(S, A, m)}S( ¯OT −1)]. +(34) +Combining equation(33) and equation(34), we have that +D2 = E +� +ωπe(S) +�� +a pθ0(M|S, a)πe(a|S) +pθ0(M|S, A) +π0(A|S) +πb,θ0(A|S)[R − rθ0(S, A, M)] + +πe(A|S) +πb,θ0(A|S) +× { +� +a′ +rθ0(S, a′, M)π0(a′|S) − ηGe + Eπe +a,mQGe(S′, a, m) − EmQGe(S, A, m)} +� +S( ¯OT −1) +� +. +34 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Since (S, A, M, R, S′) is any arbitrary transaction tuple follows the corresponding distribution, we have that +D2 = E +� 1 +T +T −1 +� +t=0 +ωπe(St) +�� +a pθ0(Mt|St, a)πe(a|St) +pθ0(Mt|St, At) +π0(At|St) +πb,θ0(At|St)[Rt − rθ0(St, At, Mt)] + +πe(At|St) +πb,θ0(At|St) +× { +� +a′ +rθ0(St, a′, Mt)π0(a′|St) − ηGe + Eπe +a,mQGe(St+1, a, m) − EmQGe(St, At, m)} +� +S( ¯OT −1) +� +. +G.1.4 +Derivative of IDEθ0(πe, π0) +Given C1, D1, and D2, the derivative of IDEθ0(πe, π0) is ηπe − ηGe + I1 − I2, where +I1 = E[ωπe(S) πe(A|S) +πb,θ0(A|S){R − ηπe + Eπe +a,mQπe(S′, a, m) − EmQπe(S, A, m)}], +and +I2 = E +� +ωπe(S) +�� +a pθ0(M|S, a)πe(a|S) +pθ0(M|S, A) +π0(A|S) +πb,θ0(A|S)[R − rθ0(S, A, M)] + +πe(A|S) +πb,θ0(A|S) +× { +� +a′ +rθ0(S, a′, M)π0(a′|S) + Eπe +a,mQGe(S′, a, m) − EmQGe(S, A, m) − ηGe} +�� +. +G.2 +EIF for Immediate Mediator Effect +Immediate Mediator Effect (IME) can be represented as +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtpθ0(st+1, rt|st, at, mt)π0(at|st)[ +� +a′ +pθ0(mt|st, a′)πe(a′|st) − pθ0(mt|at, st)] +× +t−1 +� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +νθ0(s0). +(35) +Taking the derivative of IMEθ0(πe, π0), we get that +∂IMEθ0(πe, π0) +∂θ0 += C2 + D2 − D3, +where +C2 = (35) × ▽θ log(νθ0(s0)) = E[IMEθ0(πe, π0) × S( ¯OT −1)] = E[(ηGe − ηπe,0) × S( ¯OT −1)], +D2 is derived in Appendix G.1.3, and +D3 = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtpθ0(st+1, rt|st, at, mt)pθ0(mt|st, at)π0(at|st) +t−1 +� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +× +� t−1 +� +j=0 +[▽θ log pθ0(sj+1, rj, mj|aj, sj)] + ▽θ log pθ0(st+1, rt, mt|st, at) +� +νθ0(s0), +which can be represented as the sum of two parts. Specifically, D3 = D(1) +3 ++ D(2) +3 , where +D(1) +3 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtpθ0(st+1, rt|st, at, mt)pθ0(mt|st, at)π0(at|st) +t−1 +� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +× ▽θ log pθ0(st+1, rt, mt|st, at)νθ0(s0), +and +D(2) +3 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtpθ0(st+1, rt|st, at, mt)pθ0(mt|st, at)π0(at|st) +t−1 +� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +× +t−1 +� +j=0 +▽θ log pθ0(sj+1, rj, mj|aj, sj)νθ0(s0). +35 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +G.2.1 +D3 +Part I (D(1) +3 ). First, using the fact that the expectation of a score function is 0, we notice that, +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +Em∗rθ0(st, at, m∗)pθ0(st+1, rt|st, at, mt)pθ0(mt|st, at)π0(at|st) +× +t−1 +� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +▽θ log pθ0(st+1, rt, mt|st, at)νθ0(s0), += +lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,τt−1 +Em∗rθ0(st, at, m∗)π0(at|st) +t−1 +� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +νθ0(s0) +× +� +st+1,rt,mt +pθ0(st+1, rt, mt|st, at)▽θ log pθ0(st+1, rt, mt|st, at), +=0, +which follows that +D(1) +3 += +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +[rt − Em∗rθ0(st, at, m∗)]pθ0(st+1, rt|st, at, mt)pθ0(mt|st, at)π0(at|st) +× +t−1 +� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +νθ0(s0)▽θ log pθ0(st+1, rt, mt|st, at), += lim +T →∞ +1 +T +T −1 +� +t=0 +� +st+1,rt,at,mt,st +[rt − Em∗rθ0(st, at, m∗)]pθ0(st+1, rt, mt|st, at)π0(at|st) +×pπe(st)▽θ log pθ0(st+1, rt, mt|st, at). +Replacing the π0(at|st) with +π0(at|st) +πb,θ0(at|st)πb,θ0(at|st), and pπe(st) with pπe(st) +pπb(st)pπb(st) = ωπe(st)pπb(st), we obtain +that +D(1) +3 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +st+1,rt,at,mt,st +ωπe(st) π0(at|st) +πb,θ0(at|st)[rt − Em∗rθ0(st, at, m∗)] +× pθ0(st+1, rt, mt|st, at)πb,θ0(at|st)pπb(st)▽θ log pθ0(st+1, rt, mt|st, at). +Further, since +lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,st +ωπe(st) π0(at|st) +πb,θ0(at|st)πb,θ0(at|st)pπb(st)▽θ log pπb +θ0 (at, st) +× +� +st+1,rt,mt +[rt − Em∗rθ0(st, at, m∗)]pθ0(st+1, rt, mt|st, at) = 0, +we have that +D(1) +3 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +st+1,rt,at,mt,st +ωπe(st) π0(at|st) +πb,θ0(at|st)[rt − Em∗rθ0(st, at, m∗)] +× pθ0(st+1, rt, mt|st, at)πb,θ0(at|st)pπb(st)▽θ log pπb +θ0 (st+1, rt, mt, at, st). +Lastly, combining the fact that the expectation of a score function is 0 and the Markov property, we finalize the +derivation of D(1) +3 +with +D(1) +3 += E +� +ωπe(S) π0(A|S) +πb,θ0(A|S)[R − Emrθ0(S, A, m)]S( ¯OT −1) +� +. +(36) +36 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Part II (D(2) +3 ). We first rewrite the D(2) +3 +as +lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,mt,τt−1 +rθ0(st, at, mt)pθ0(mt|st, at)π0(at|st) +t−1 +� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +× +t−1 +� +j=0 +▽θ log pθ0(sj+1, rj, mj|aj, sj)νθ0(s0). +Taking the additional average over s∗, r∗, m∗, and a∗, and noticing that +lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,mt,τt−1 +rθ0(st, at, mt)pθ0(mt|st, at)π0(at|st) +t−1 +� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +× +� +s∗,r∗,m∗,a∗ +pθ0(s∗, r∗, m∗|st, a∗)πe(a∗|st)▽θ log pθ0(s∗, r∗, m∗|st, a∗)νθ0(s0) = 0, +we further represent D(2) +3 +as +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +� +m∗,a∗ +pθ0(m∗|st, a∗)rθ0(st, a∗, m∗)π0(a∗|st) +t� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +× +t +� +j=0 +▽θ log pθ0(sj+1, rj, mj|aj, sj)νθ0(s0). +Note that we change the subscript of the summations accordingly. Then, following the steps we processed to derive the +D1, we first show that +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +ηπe,0 +t� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +t +� +j=0 +▽θ log pθ0(sj+1, rj, mj|aj, sj)νθ0(s0) = 0. +Therefore, +D(2) +3 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +� � +m∗,a∗ +rθ0(st, a∗, m∗)pθ0(m∗|st, a∗)π0(a∗|st) − ηπe,0� +t� +j=0 +� +pπe +θ0 (sj+1, rj, mj, aj|sj) +� +× +t +� +j=0 +▽θ log pθ0(sj+1, rj, mj|aj, sj)νθ0(s0). +Next, using the equality properties ⋆ and ⋆⋆, together with the definition of Qπe,0(s, a, m) and the trick of score +functions, we can show that +D(2) +3 += lim +T →∞ +1 +T +T −1 +� +j=0 +� +τj +� � +m∗,a∗ +rθ0(st, a∗, m∗)pθ0(m∗|st, a∗)π0(a∗|st) − ηπe,0 + Eπe +a∗,m∗Qπe,0(sj+1, a∗, m∗) +− Em∗Qπe,0(sj, aj, m∗) +� πe(aj|sj)pπe(sj) +πb,θ0(aj|sj)pπb(sj)pθ0(sj+1, rj, mj|aj, sj) +× πb,θ0(aj|sj)pπb(sj)▽θ log pπb +θ0 (sj+1, rj, mj, aj, sj). +Implementing the fact that the expectation of a score function is zero and utilizing the Markov property, we finally +obtain that, +D(2) +3 += E +� +ωπe(S) πe(A|S) +πb,θ0(A|S){ +� +a′ +Em∼pθ0(•|S,a′)rθ0(S, a′, m)π0(a′|S) ++ Eπe +a,mQπe,0(S′, a, m) − EmQπe,0(S, A, m) − ηπe,0}S( ¯OT −1) +� +. +(37) +37 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Combining equation (36) and equation (37), we have that +D3 = E +� +ωπe(S) +� π0(A|S) +πb,θ0(A|S)[R − Emrθ0(S, A, m)] + +πe(A|S) +πb,θ0(A|S){ +� +a′ +Em∼pθ0(•|S,a′)rθ0(S, a′, m)π0(a′|S) ++ Eπe +a,mQπe,0(S′, a, m) − EmQπe,0(S, A, m) − ηπe,0} +� +S( ¯OT −1) +� +. +Since (S, A, M, R, S′) is any arbitrary transaction tuple follows the corresponding distribution, we have that +D3 = E +� 1 +T +T −1 +� +t=0 +ωπe(St) +� π0(At|St) +πb,θ0(At|St)[Rt−Emrθ0(St, At, m)]+ πe(At|St) +πb,θ0(At|St){ +� +a′ +Em∼p(•|St,a′)rθ0(St, a′, m)π0(a′|St) ++ Eπe +a,mQπe,0(St+1, a, m) − EmQπe,0(St, At, m) − ηπe,0} +� +S( ¯OT −1) +� +. +G.2.2 +Efficient Function +Given C2, D2, and D3, the efficient influence function for IMEθ0(πe, π0) is ηGe − ηπe,0 + I2 − I3, where +I2 = E +� +ωπe(S) +�� +a pθ0(M|S, a)πe(a|S) +pθ0(M|S, A) +π0(A|S) +πb,θ0(A|S)[R − rθ0(S, A, M)] + +πe(A|S) +πb,θ0(A|S) +× { +� +a′ +rθ0(S, a′, M)π0(a′|S) + Eπe +a,mQGe(S′, a, m) − EmQGe(S, A, m) − ηGe} +�� +, +and +I3 = E +� +ωπe(S) +� π0(A|S) +πb,θ0(A|S)[R − Emrθ0(S, A, m)] + +πe(A|S) +πb,θ0(A|S){ +� +a′ +Em∼pθ0(•|S,a′)rθ0(S, a′, m)π0(a′|S) ++ Eπe +a,mQπe,0(S′, a, m) − EmQπe,0(S, A, m) − ηπe,0} +�� +. +G.3 +EIF for Delayed Direct Effect +Delayed Direct Effect (DDE) can be represented as +DDE(πe, π0) = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtp(st+1, rt|st, at, mt)p(mt|st, at)π0(at|st) +× +� t−1 +� +j=0 +pπe(sj+1, rj, mj, aj|sj) − +� +¯a∗ +t−1 +t−1 +� +j=0 +p(sj+1, rj|sj, aj, mj)π0(aj|sj)p(mj|sj, a∗ +j)πe(a∗ +j|sj) +� +ν(s0). +(38) +Taking the derivative of DDEθ0(πe, π0), we get that +∂DDEθ0(πe, π0) +∂θ0 += C3 + D3 − D4, +where +C3 = (38) × ▽θ log(νθ0(s0)) = E[DDEθ0(πe, π0) × S( ¯OT −1)] = E[(ηπe,0 − ηG0) × S( ¯OT −1)], +D3 is derived in Appendix G.2.1, and +D4 = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt,¯a∗ +t−1 +rtpθ0(st+1, rt, mt|st, at)π0(at|st) +t−1 +� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj) +� +▽θ log pθ0(st+1, rt, mt|st, at) +� +�� +� +D(1) +4 ++ +t−1 +� +j=0 +▽θ log pθ0(sj+1, rj|sj, aj, mj) +� +�� +� +D(2) +4 ++ +t−1 +� +j=0 +▽θ log pπe +θ0 (mj|sj, a∗ +j) +� +�� +� +D(3) +4 +� +νθ0(s0). +38 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +G.3.1 +D4 +Part I (D(1) +4 ). First, using the fact that the expectation of a score function is 0, we notice that, +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt,¯a∗ +t−1 +Em′rθ0(st, at, m′)pθ0(st+1, rt, mt|st, at)π0(at|st)▽θ log pθ0(st+1, rt, mt|st, at) +t−1 +� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj)νθ0(s0), += +lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,τt−1,¯a∗ +t−1 +Em′rθ0(st, at, m′)π0(at|st) +t−1 +� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj) +� +st+1,rt,mt +pθ0(st+1, rt, mt|st, at)▽θ log pθ0(st+1, rt, mt|st, at)νθ0(s0), +=0, +which follows that +D(1) +4 += +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt,¯a∗ +t−1 +[rt − Em′rθ0(st, at, m′)]pθ0(st+1, rt, mt|st, at)π0(at|st)▽θ log pθ0(st+1, rt, mt|st, at) +t−1 +� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj)νθ0(s0), += lim +T →∞ +1 +T +T −1 +� +t=0 +� +st+1,rt,mt,at,st +[rt − Em′rθ0(st, at, m′)]pθ0(st+1, rt, mt|st, at)π0(at|st)pG +θ0(st)▽θ log pθ0(st+1, rt, mt|st, at). +The last equation holds, since +� +s0,τt−1,¯a∗ +t−1 +t−1 +� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj)νθ0(s0) = pG +θ0(st). +Replacing the π0(at|st) with +π0(at|st) +πb,θ0(at|st)πb,θ0(at|st), and pG +θ0(st) with +pG +θ0(st) +pπb(st)pπb(st) = ωG +θ0(st)pπb(st), we obtain +that +D(1) +4 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +st+1,rt,mt,at,st +ωG +θ0(st) π0(at|st) +πb,θ0(at|st)[rt − Em′rθ0(st, at, m′)] +pθ0(st+1, rt, mt|st, at)πb,θ0(at|st)pπb(st)▽θ log pθ0(st+1, rt, mt|st, at). +Further, since +lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,st +ωG +θ0(st) π0(at|st) +πb,θ0(at|st)πb,θ0(at|st)pπb(st)▽θ log pπb +θ0 (at, st) +× +� +st+1,rt,mt +[rt − Em′rθ0(st, at, m′)]pθ0(st+1, rt, mt|st, at) = 0, +we have that +D(1) +4 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +st+1,rt,at,mt,st +ωG +θ0(st) π0(at|st) +πb,θ0(at|st)[rt − Em′rθ0(st, at, m′)] +× pθ0(st+1, rt, mt|st, at)πb,θ0(at|st)pπb(st)▽θ log pπb +θ0 (st+1, rt, mt, at, st). +39 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Lastly, combining the fact that the expectation of a score function is 0 and the Markov property, we finalize the +derivation of D(1) +4 +with +D(1) +4 += E +� +ωG +θ0(S) π0(A|S) +πb,θ0(A|S)[R − Emrθ0(S, A, m)]S( ¯OT −1) +� +. +(39) +Part II (D(2) +4 ). Taking the additional average over s′, r′, a′, m′, and ˜a, and noticing that +lim +T →∞ +1 +T +T −1 +� +t=0 +� +at,mt,τt−1,¯a∗ +t−1 +rθ0(st, at, mt)pθ0(mt|st, at)π0(at|st) +t−1 +� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj) +νθ0(s0) × +� +s′,r′,a′,m′,˜a +pθ0(s′, r′|st, a′, m′)π0(a′|st)pθ0(m′|st, ˜a)πe(˜a|st)▽θ log pθ0(s′, r′|st, a′, m′) = 0, +we further represent D(2) +4 +as +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt,¯a∗ +t +� +m′,a′ +pθ0(m′|st, a′)rθ0(st, a′, m′)π0(a′|st) +t� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj) +× +t +� +j=0 +▽θ log pθ0(sj+1, rj|sj, aj, mj)νθ0(s0). +Note that we change the subscript of the summations accordingly. Then, following the steps we processed to derive the +D1, we first show that +lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt,¯a∗ +t +ηG0 +t� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj) +t +� +j=0 +▽θ log pθ0(sj+1, rj|aj, sj, mj)νθ0(s0) = 0. +Therefore, +D(2) +4 += lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt,¯a∗ +t +� � +m′,a′ +pθ0(m′|st, a′)rθ0(st, a′, m′)π0(a′|st) − ηG0� +t� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj) +t +� +j=0 +▽θ log pθ0(sj+1, rj|aj, sj, mj)νθ0(s0). +Similar to ⋆⋆, we have that +� +s0,τt−1,¯a∗ +t−1 +t−1 +� +j=0 +pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj)νθ0(s0) +⋆⋆⋆ += pG +θ0(st). +Next, using the equality properties ⋆ and ⋆ ⋆ ⋆, together with the definition of QG0(s, a, m) and the trick of score +functions, we can show that +D(2) +4 += lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj+1,rj,aj,sj,mj,a∗ +j +� � +m′,a′ +pθ0(m′|sj, a′)rθ0(sj, a′, m′)π0(a′|sj)−ηG0+EG +a,mQG0(sj+1, a, m) +� +× pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj)pG(sj)▽θ log pθ0(sj+1, rj|aj, sj, mj). +(40) +Then, following the steps in deriving D1, we have that +D(2) +4 += lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj+1,rj,aj,sj,mj +ωG +θ0(sj) +� +a′ pθ0(mj|sj, a′)πe(a′|sj) +pθ0(mj|sj, aj) +π0(aj|sj) +πb,θ0(aj|sj) +× +� � +m′,a′ +pθ0(m′|sj, a′)rθ0(sj, a′, m′)π0(a′|sj) − ηG0 + EG +a,mQG0(sj+1, a, m) − QG0(sj, aj, mj) +� +× pθ0(sj+1, rj|sj, aj, mj)pθ0(mj|sj, aj)πb,θ0(aj|sj)pπb(sj)▽θ log pπb +θ0 (sj+1, rj, mj, aj, sj). +40 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +Finally, combining the fact that the expectation of a score function is zero and the Markov property, we have that +D(2) +4 += E +�� � +m′,a′ +pθ0(m′|S, a′)rθ0(S, a′, m′)π0(a′|S) − ηG0 + EG +a,mQG0(S′, a, m) − QG0(S, A, M) +� +× ωG +θ0(S) +� +a′ pθ0(M|S, a′)πe(a′|S) +pθ0(M|S, A) +π0(A|S) +πb,θ0(A|S)S( ¯OT −1) +� +. +(41) +Part III (D(3) +4 ). Following the same steps used in deriving the equation (40), we can show that, +D(3) +4 += lim +T →∞ +1 +T +T −1 +� +j=0 +� +aj,sj,mj,a∗ +j +� +sj+1,rj +� � +m′,a′ +pθ0(m′|sj, a′)rθ0(sj, a′, m′)π0(a′|sj)−ηG0+EG +a,mQG0(sj+1, a, m) +� +× pθ0(sj+1, rj|sj, aj, mj)π0(aj|sj)pθ0(mj|sj, a∗ +j)πe(a∗ +j|sj)pG(sj)▽θ log pθ0(mj|sj, a∗ +j). +(42) +Based on the definition of QG0 and the corresponding Bellman equation, we have that +E +� � +m′,a′ +pθ0(m′|sj, a′)rθ0(sj, a′, m′)π0(a′|sj) − ηG0 + EG +a,mQG0(sj+1, a, m)|sj, aj, mj +� += QG0(sj, aj, mj). +Therefore, (42) can be rewritten as +lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj,mj,aj +� � +a′ +QG0(sj, a′, mj)π0(a′|sj) +� +pθ0(mj|sj, aj)πe(aj|sj)pG(sj)▽θ log pθ0(mj|sj, aj). +Notice that +lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj,mj,aj +� � +a′,m∗ +j +QG0(sj, a′, m∗ +j)pθ0(m∗ +j|sj, aj)π0(a′|sj) +� +pθ0(mj|sj, aj)πe(aj|sj)pG(sj)▽θ log pθ0(mj|sj, aj) += lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj,aj +� � +a′,m∗ +j +QG0(sj, a′, m∗ +j)pθ0(m∗ +j|sj, aj)π0(a′|sj) +� +πe(aj|sj)pG(sj) +� +mj +pθ0(mj|sj, aj)▽θ log pθ0(mj|sj, aj) +=0. +Therefore, we have that +D(3) +4 += lim +T →∞ +1 +T +T −1 +� +j=0 +� +sj,mj,aj +�� � +a′ +QG0(sj, a′, mj)π0(a′|sj) +� +− +� +a′,m∗ +j +QG0(sj, a′, m∗ +j)pθ0(m∗ +j|sj, aj)π0(a′|sj) +� +× pθ0(mj|sj, aj)πe(aj|sj)pG(sj)▽θ log pθ0(mj|sj, aj). +Following the same steps we used in getting the final expression of D(1) +4 , we can show that +D(3) +4 += E +� +ωG +θ0(S) πe(A|S) +πb,θ0(A|S) +� +[ +� +a′ +QG0(S, a′, M)π0(a′|S) +� +− +� +a′,m′ +QG0(S, a′, m′)pθ0(m′|S, A)π0(a′|S) +� +S( ¯OT −1) +� +. +(43) +Combining D(1) +4 , D(2) +4 , and D(3) +4 , we have that +D4 = E +� +ωG(S)π0(A|S) +πb(A|S) +�� +R − Emr(S, A, m) +� ++ ρ(S, A, M) +� +Eπ0 +a′,mr(S, a′, m) + EG +a,mQG0(S′, a, m) +− QG0(S, A, M) − ηG0�� ++ ωG(S)πe(A|S) +πb(A|S) +� +a +π0(a|S) +� +QG0(S, a, M) − +� +m +p(m|S, A)QG0(S, a, m) +�� +. +Since (S, A, M, R, S′) is any arbitrary transaction tuple follows the corresponding distribution, we have that +D4 = E +� 1 +T +T −1 +� +t=0 +ωG(St)π0(At|St) +πb(At|St) +�� +Rt−Emr(St, At, m) +� ++ρ(St, At, Mt) +� +Eπ0 +a′,mr(St, a′, m)+EG +a,mQG0(St+1, a, m) +−QG0(St, At, Mt)−ηG0�� ++ωG(St)πe(At|St) +πb(At|St) +� +a +π0(a|St) +� +QG0(St, a, Mt)− +� +m +p(m|St, At)QG0(St, a, m) +�� +. +41 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +G.3.2 +Efficient Function +Given C3, D3, and D4, the efficient influence function for DDEθ0(πe, π0) is ηπe,0 − ηG0 + I3 − I4, where +I3 = E +� +ωπe(S) +� π0(A|S) +πb,θ0(A|S)[R − Emrθ0(S, A, m)] + +πe(A|S) +πb,θ0(A|S){ +� +a′ +Em∼pθ0(•|S,a′)rθ0(S, a′, m)π0(a′|S) ++ Eπe +a,mQπe,0(S′, a, m) − EmQπe,0(S, A, m) − ηπe,0} +�� +, +and +I4 = E +� +ωG(S)π0(A|S) +πb(A|S) +�� +R − Emr(S, A, m) +� ++ ρ(S, A, M) +� +Eπ0 +a′,mr(S, a′, m) + EG +a,mQG0(S′, a, m) +− QG0(S, A, M) − ηG0�� ++ ωG(S)πe(A|S) +πb(A|S) +� +a +π0(a|S) +� +QG0(S, a, M) − +� +m +p(m|S, A)QG0(S, a, m) +�� +. +G.4 +EIF for Delayed Mediator Effect +Delayed Mediator Effect (DME) can be represented as +DME(πe, π0) = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rtp(st+1, rt|st, at, mt)p(mt|st, at)π0(at|st) +× +� � +¯a∗ +t−1 +t−1 +� +j=0 +p(sj+1, rj|sj, aj, mj)π0(aj|sj)p(mj|sj, a∗ +j)πe(a∗ +j|sj) − +t−1 +� +j=0 +pπ0(sj+1, rj, mj, aj|sj) +� +ν(s0). +(44) +Taking the derivative of DMEθ0(πe, π0), we get that +∂DMEθ0(πe, π0) +∂θ0 += C4 + D4 − D5, +where +C4 = (44) × ▽θ log(νθ0(s0)) = E[DMEθ0(πe, π0) × S( ¯OT −1)] = E[(ηG0 − ηπ0) × S( ¯OT −1)], +D4 is derived in Appendix G.3.1, and +D5 = lim +T →∞ +1 +T +T −1 +� +t=0 +� +τt +rt +t� +j=0 +pπ0 +θ0 (sj+1, rj, mj, aj|sj) +t +� +j=0 +� +▽θ log pπ0 +θ0 (sj+1, rj, mj, aj|sj)] × νθ0(s0), +Notice that D5 is similar as D1, and can be derived similarly as D1 by replacing the πe in D1 with π0. Therefore, with +the definition of Qπ0(s, a, m), we can show that +D5 = E +� +ωπ0(S) π0(A|S) +πb,θ0(A|S){R + +� +a′ +EmQπ0(S′, a′, m)π0(a′|S′) − EmQπ0(S, A, m) − ηπ0}S( ¯OT −1) +� +. +Since (S, A, M, R, S′) is any arbitrary transaction tuple follows the corresponding distribution, we have that +D5 = E +� 1 +T +T −1 +� +t=0 +ωπ0(St) π0(At|St) +πb,θ0(At|St){Rt + +� +a′ +EmQπ0(St+1, a′, m)π0(a′|St+1) − EmQπ0(St, At, m) − ηπ0}S( ¯OT −1) +� +. +G.4.1 +Efficient Function +Given C4, D4, and D5, the efficient influence function for DMEθ0(πe, π0) is ηG0 − ηπ0 + I4 − I5, where +I4 = E +� +ωG(S)π0(A|S) +πb(A|S) +�� +R − Emr(S, A, m) +� ++ ρ(S, A, M) +� +Eπ0 +a′,mr(S, a′, m) + EG +a,mQG0(S′, a, m) +− QG0(S, A, M) − ηG0�� ++ ωG(S)πe(A|S) +πb(A|S) +� +a +π0(a|S) +� +QG0(S, a, M) − +� +m +p(m|S, A)QG0(S, a, m) +�� +. +and +I5 = E +� +ωπ0(S) π0(A|S) +πb,θ0(A|S){R + +� +a′ +EmQπ0(S′, a′, m)π0(a′|S′) − EmQπ0(S, A, m) − ηπ0} +� +. +42 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +G.5 +Proof of the Equality ⋆ +The equality can be proved with the following three steps: +Step 1. We first exchange the summation of t and j in the first line of the equation D1, which yields that +D1 = lim +T →∞ +1 +T +T −1 +� +j=0 +T −1 +� +t=j +� +τt +[rt − ηπe] +t� +k=0 +pπe +θ0 (sk+1, rk, mk, ak|sk) +× +� +▽θ log pπe +θ0 (sj+1, rj, mj, aj|sj)] × νθ0(s0). +(45) +Step 2. Then we split the summation �T −1 +t=j into t = j and �T −1 +t=j+1, and split the product �t +k=0 into �j +k=0 and +�t +k=j+1, which leads to +D1 = lim +T →∞ +1 +T +T −1 +� +j=0 +� � +τj +[rj − ηπe] + +T −1 +� +t=j+1 +� +τt +[rt − ηπe] +t� +k=j+1 +pπe +θ0 (sk+1, rk, mk, ak|sk) +� +× +j� +k=0 +pπe +θ0 (sk+1, rk, mk, ak|sk) +� +▽θ log pπe +θ0 (sj+1, rj, mj, aj|sj)] × νθ0(s0). +(46) +Step 3. By the definition of Qπe (see equation (2)), we have that +T −1 +� +t=j+1 +� +τt +[rt − ηπe] +t� +k=j+1 +pπe +θ0 (sk+1, rk, mk, ak|sk) += Eπe +mj+1,aj+1Qπe(sj+1, aj+1, mj+1) += Eπe +a∗,m∗Qπe(sj+1, a∗, m∗). +Substituting this equation, we conclude the proof of ⋆ with that +D1 = lim +T →∞ +1 +T +T −1 +� +j=0 +� � +τj +[rj − ηπe] + Eπe +a∗,m∗Qπe(sj+1, a∗, m∗) +� +× +j� +k=0 +pπe +θ0 (sk+1, rk, mk, ak|sk) +� +▽θ log pπe +θ0 (sj+1, rj, mj, aj|sj)] νθ0(s0). +H +Settings for Numerical Examples +H.1 +Toy Example 1 & Toy Example 2 +Settings. We consider a scenario with discrete states, actions, mediators, and rewards. We set time T = 50, and +S0 for each trajectory is sampled from a Bernoulli distribution with a mean probability of 0.5. Denote the sigmoid +function as expit(·). Following the behavior policy, the action At ∈ {0, 1} is sampled from a Bernoulli distribution, +where Pr(At = 1|St) = expit(1.0 − 2.0St). Observing St and At, the mediator Mt ∈ {0, 1} is drawn from a +Bernoulli distribution with Pr(Mt = 1|St, At) = expit(1.0 − 1.5St + 2.5At). The distributions of Rt and St+1 are +both Bernoulli and conditional on St, At, and Mt. Specifically, the reward distribution of Rt ∈ {0, 10} satisfies that +Pr(Rt = 10|St, At, Mt) = expit(1.0 + 2.0St − 1.0At − 2.5Mt), while the distribution of next state St+1 ∈ {0, 1} +satisfies that Pr(St+1 = 1|St, At, Mt) = expit(.5 + 3.0St − 2.5At − .5Mt). We are interested in estimating the +treatment effect of the target policy πe, which applies a treatment with Pr(At = 1|St) = expit(1.5 + 1.0St), compared +to the control policy π0, which always applies no treatment (i.e., Pr(At = 1) = 0). Monte Carlo (MC) simulations +are used to calculate the oracle distributions of ω(·) and Q(·), and the oracle values of η(·), IDE(πe, π0), IME(πe, π0), +DDE(πe, π0), and DME(πe, π0). Based on 40K simulated trajectories with 1K observations each, we obtained that +IDE = −1.277, IME = −1.222, DDE = −2.982, and DME = −.085. Considering the true distributions of Qπe, +Qπe,a0, Qπe,a0∗, QG, and Qa0, we approximate each of them by assuming linear equation models (Shi et al., 2022a). +Misspecification. To misspecify the ωπe, we add .25 to the ωπe(St = 1) and subtract .25 from the ωπe(St = 0). +Similarly, we subtract .3 from the ωa0(St = 1) (ωG(St = 1)) and add .3 to the ωa0(St = 0) (ωG(St = 0)). For Q(·), +43 + +A Reinforcement Learning Framework for Dynamic Mediation Analysis +A PREPRINT +and r functions, we inject Gaussian noises into each parameter involved in the true model. For the misspecification of +pm and πb, we multiply the true value by a random variable drawn from a bounded uniform distribution and then clip +the probabilities to ensure that they are within the range of .01 and .99. +H.2 +Semi-Synthetic Data +The spaces for reward, state, and mediator are continuous, and the action space is binary. Specifically, the semi-synthetic +data is generated as follows. The initial states are i.i.d. sampled from the standard normal distribution. πb follows a +Bernoulli distribution, satisfying that Pr(At = 1) = Pr(At = 0) = .5. We consider a 2-dimensional mediator, where +Mt,1 and Mt,2 are independent and normally distributed with a standard deviation of 2. While the mean of Mt,1 is +� +|St| + (At − .5), the mean of Mt,2 is .5(At − .5) ∗ +� +|St| − .5St. We set Rt = St+1, and Rt is drwan from a normal +distribution with a mean of .75[St + +� +|St| + (1 + +� +|Mt1| + |Mt2|)(At − .5)] + 1.5(Mt1 + Mt2) and a standard +deviation of 2. The control policy always takes action At = 0, while the target policy follows a Bernoulli distribution +with Pr(At = 1) = expit(.7 ∗ St). Monte Carlo (MC) simulations are used to calculate the oracle the oracle values +of η(·), IDE(πe, π0), IME(πe, π0), DDE(πe, π0), and DME(πe, π0). Based on 20K simulated trajectories with 6400 +observations each, we obtained that IDE = 2.680, IME = 3.654, DDE = 1.244, and DME = .689. +I +Baseline Estimators +Following the definitions of direct and indirect effect in Pearl (2022), we consider a baseline estimator for IDE(πe, π0) +as +1 +NT +� +i,t,a,m +� +r(Si,t, a, m) − +� +a′ +r(Si,t, a′, m)π0(a′|Si,t) +� +p(m|Si,t, a)πe(a|Si,t), +and a baseline estimator for IME(πe, π0) as +1 +NT +� +i,t,a,m +r(Si,t, a, m) +� � +a′ +p(m|Si,t, a′)πe(a′|Si,t) − p(m|Si,t, a) +� +π0(a|Si,t). +J +Estimating Optimal Policy +To estimate the optimal policy, we first estimate a Q function based on the observational data, following the same +methods described in Section D. Specifically, let +Q(s, a, m) = +� +t≥0 +E[Rt − η], +which leads to a Bellmen equation model, such that +Q(St, At, Mt) = +� +E[Rt + +� +a +� +m +EQ(St+1, a, m) − η]. +We then approximate the Q function using linear sieves. Finally, the estimated optimal policy is defined as +ˆπopt(s) = arg max +a∈A +� +a +� +m +E ˆQ(s, a, m). +It is worth noting that we used cross-validation to estimate the ATE of ˆπopt. To be more specific, we divide the observed +trajectories into two folds. In each round k, we first estimate the ˆπk +opt based on the trajectories within fold k, and then +estimate the ATE of ˆπk +opt on another fold of trajectories. +44 + diff --git a/cNFQT4oBgHgl3EQfiDZt/content/tmp_files/load_file.txt b/cNFQT4oBgHgl3EQfiDZt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..de542fc4b537f976a97ff021d7ddbe312c42c13e --- /dev/null +++ b/cNFQT4oBgHgl3EQfiDZt/content/tmp_files/load_file.txt @@ -0,0 +1,2284 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf,len=2283 +page_content='A REINFORCEMENT LEARNING FRAMEWORK FOR DYNAMIC MEDIATION ANALYSIS A PREPRINT Lin Ge1, Jitao Wang2, Chengchun Shi3, Zhenke Wu2, and Rui Song1 1North Carolina State University 2University of Michigan, Ann Arbor 3London School of Economics and Political Science ABSTRACT Mediation analysis learns the causal effect transmitted via mediator variables between treatments and outcomes and receives increasing attention in various scientific domains to elucidate causal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Most existing works focus on point-exposure studies where each subject only receives one treatment at a single time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' However, there are a number of applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', mobile health) where the treatments are sequentially assigned over time and the dynamic mediation effects are of primary interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Proposing a reinforcement learning (RL) framework, we are the first to evaluate dynamic mediation effects in settings with infinite horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' We decompose the average treatment effect into an immediate direct effect, an immediate mediation effect, a delayed direct effect, and a delayed mediation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Upon the identification of each effect component, we further develop robust and semi-parametrically efficient estimators under the RL framework to infer these causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' The superior performance of the proposed method is demonstrated through extensive numerical studies, theoretical results, and an analysis of a mobile health dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' 1 Introduction Mediation analysis aims to understand the causal pathway from an exposure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', treatment or action) to an outcome variable of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' It is gaining increasing popularity recently and has been frequently employed in a number of domains including epidemiology (Richiardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Rijnhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2021), psychology (Rucker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2011), genetics (Chakrabortty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Djordjilovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2022), and economics (Celli, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Our paper is motivated by the need to learn the dynamic mediation effects in sequential decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' One motivating example is given by the Intern Health Study (IHS, NeCamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2020), which focuses on sequential mobile health interventions to help improve the mental health of medical interns who work in stressful environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Participants were randomly assigned to receive notifications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', tips and insights) throughout the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' For example, some notifications remind participants to take a break or enjoy a tasty treat, while others summarize the trends of recent physical activity and sleep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' All the notifications are designed to improve participants’ mood scores (self-reported via a custom-made study App) either directly or indirectly through increased activity or sleep hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' In addition, it is essential to note that participants’ recent behavior will not only influence their proximal mood but will also influence their behavior and mood scores in the following days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' To design a more effective intervention policy in IHS, it is necessary to understand how mobile prompts impact mood scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' In particular, the mobile prompts may directly impact the mood scores or encourage more physical activity and sleep, which may then impact the mood scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' In addition, an individual’s past treatment sequence and behavior trajectory may impact the mood score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Teasing out these distinct sources of causal impacts on mood scores and their relative magnitudes needs new definitions, identification results, and inferential methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' A fundamental question considered in this paper is how to infer the dynamic mediation effects in the aforementioned applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Solving this question raises at least three challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' First, the mediator at a given time affects both the current and future outcomes, inducing temporal carryover effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' As demonstrated in the case study in Section 8, the delayed direct effect (DDE) and the delayed mediator effect (DME) are significant and dominate the average arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content='13348v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content='ML] 31 Jan 2023 A Reinforcement Learning Framework for Dynamic Mediation Analysis A PREPRINT Figure 1: Mediated MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' treatment effect for the intervention policy used in the IHS (Sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' NeCamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' In contrast, the immediate direct effect (IDE) and immediate mediator effect (IME) are both insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Nonetheless, most existing mediation analyses focus on estimating the indirect effect on the immediate reward and are hence inappropriate to our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Second, the horizon (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', number of decision stages) in the aforementioned applications is typically very long or diverges with the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Existing solutions developed in finite horizon settings typically suffer from the curse of horizon in the sense that the variances of the proposed estimators grow exponentially fast with respect to the horizon (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2018) and are hence inapplicable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' see Section 2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Third, regardless of how the dynamic effects may change during the sequential treatments (or lack thereof), most works focus on examining the causal effects on the final outcome obtained at the end of the treatment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' However, in the context of behavioral change, the goal is to encourage and maintain small improvements to nudge individuals into generating sustained improvements in outcomes like mood scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Currently, there is a dearth of methods to analyze causal effects for outcomes measured at every decision point in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' To address these limitations, we propose formulating the evaluation of dynamic mediation effects as a reinforcement learning (RL) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' In particular, we use the Markov decision process (MDP) that is commonly employed in RL to model the mediated dynamic decision process over an infinite time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Building upon the standard MDP, we introduce four additional sets of causal relationships, including state-mediator, action-mediator, mediator-state, and mediator-reward, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' To evaluate the effects of different treatment policies, we consider using the off-policy evaluation (OPE, Dudík et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Uehara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2022), which is widely used to avoid the difficulty of rerunning trials by evaluating treatment policies based on observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' The main contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Motivated by the mobile health applications, we first construct the mediation analysis within the framework of RL over an infinite time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Second, we propose to decompose the average treatment effect between a target policy and a control policy into IDE, IME, DDE, and DME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' While IDE and IME have been extensively studied in single-stage settings, we introduce the DDE and DME to quantify the carryover effects of past actions and mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Third, upon the identification result of each effect component, multiply-robust estimators are developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' In particular, each proposed estimator is consistent even when models such as mediator distribution and reward distribution are misspecified (See Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Furthermore, we theoretically show the semiparametric efficiency of the proposed estimators and confirm the theoretical prediction using numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Lastly, we conclude by analyzing the IHS data and providing new insights into guiding future designs of these behavioral interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' 2 Related work Mediation analysis is widely studied in point-exposure studies under the classical structure consisting of a treatment, a mediator, and an outcome (Robins & Greenland, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Pearl, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' van der Laan & Petersen, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Imai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Tchetgen & Shpitser, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Tchetgen Tchetgen & Shpitser, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' VanderWeele, 2015), decomposing the average treatment effect into direct effect and indirect effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Recently, to address commonly observed intermediate confounders that would be affected by the exposure and then affect both mediator and outcome, multiple methods have been developed to extend the classical mediation analysis (Robins & Richardson, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Tchetgen & VanderWeele, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' VanderWeele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Vansteelandt & Daniel, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Díaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Díaz, 2022), among which the random intervention (RI)-based approach (VanderWeele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Díaz, 2022) further sets the foundation for the recent advancement of longitudinal mediation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' There is a rich literature on longitudinal mediation analysis with no intermediate confounders (Selig & Preacher, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Roth & MacKinnon, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' See also Preacher (2015) for a detailed review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' However, time-varying intermediate confounders are ubiquitous in longitudinal data contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' For example, in the IHS, doing exercises may result in a good mood, which may, in turn, increase the likelihood of engaging in more activities the next day and then subsequently affect the mood that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' 2 Mt-1 Rt-1 Mt Rt At-1 St AtA Reinforcement Learning Framework for Dynamic Mediation Analysis A PREPRINT In the presence of time-varying intermediate confounders, there are two major RI-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' VanderWeele & Tchetgen Tchetgen (2017) and Díaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' (2022) proposed to intervene in the mediator sequence by randomly drawing mediators from the corresponding marginal distribution and defined the longitudinal interventional indirect/direct effect, which is different from the natural effect decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Our work is primarily related to the work of Zheng & van der Laan (2017), which proposed to intervene in the mediator by randomly drawing the mediator from its conditional distribution and provided a natural decomposition of the total effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Using the efficient influence function (EIF), they developed a multiply-robust estimator with less reliance on the correct model specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' However, all the aforementioned methods only focused on the treatment impact on the final outcome in finite horizons and did not consider immediate outcomes or infinite horizon settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' In addition, the estimator developed by Zheng & van der Laan (2017) is based on the product of importance sampling ratios at all time points and suffers from the curse of horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Zheng & van der Laan (2012) also analyzed the longitudinal mediation effect by drawing mediators from conditional distribution but with a focus on single-exposure settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Using an RL framework for dynamic mediation analysis over an infinite horizon, our work is also connected to the line of research on OPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Existing OPE-related research evaluates the discounted sum of rewards or average rewards for a target policy using observational data gained by following a different behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' In general, there are three types of estimation procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' The first is known as the direct method (DM, Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Luckett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Chen & Qi, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2022a), which directly learns Q-functions and obtains value estimates based on their estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' The second category of approaches utilizes importance sampling (IS, Precup, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Hallak & Mannor, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2020), which re-weights the rewards to eliminate the bias due to distributional shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' The third category develops doubly robust (DR) estimators by appropriately integrating DM with IS estimators (Jiang & Li, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Thomas & Brunskill, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Farajtabar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Uehara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Kallus & Uehara, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' DR estimators are also known to achieve the semiparametric efficiency bound (Bickel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=', 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' However, none of the above papers studied mediation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Recently, Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' (2022b) proposed a consistent DR estimator for OPE in the presence of unmeasured confounders with the help of a mediator variable, which is used to intercept each directed path from treatments to reward/state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Our paper differs from theirs in that we decompose the off-policy value into the sum of IDE, IME, DDE, and DME and focus on settings without unmeasured confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' 3 Preliminaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content='1 Data Generating Process We consider the observational data generated from a mediated Markov decision process (MMDP), as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Suppose there exists an agent that tries to learn from the data and interact with a given environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' At each time t, the environment arrives at a state St ∈ S, and the agent selects an action At ∈ A = {0, 1, · · · , K −1} according to a behavior policy πb(•|St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Building upon the usual MDP, to further analyze the mediation effect, we consider an immediate mediator variable Mt ∈ M drawn according to pm(•|St, At), which mediates the effect of At on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Accordingly, the agents would receive an immediate Rt and the the environment transits to a next-state St+1 according to ps′,r(•, •|St, At, Mt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' Both S and M are finite dimensional vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNFQT4oBgHgl3EQfiDZt/content/2301.13348v1.pdf'} +page_content=' To summarize, the observed data sequences consist of the state-action-mediator-reward tuples (St, At, Mt, Rt)t≥0 satisfying the following Markov assumption: (Mt, Rt, St+1) ⊥⊥ (Sj, Aj, Mj, Rj)j= 70 years old +-0.373∗∗∗ +0.114 +-0.569∗∗∗ +0.193 +-0.611∗∗ +0.249 +Male +0.177∗∗∗ +0.058 +0.297∗∗∗ +0.099 +0.175 +0.141 +Resident in Tokyo +-0.028 +0.064 +-0.096 +0.109 +-0.128 +0.166 +Free membership +0.415∗∗∗ +0.059 +0.751∗∗∗ +0.102 +0.767∗∗∗ +0.141 +Teleworker +0.102 +0.104 +0.168 +0.176 +0.402 +0.275 +Delivery box +0.157∗∗ +0.065 +0.331∗∗∗ +0.112 +0.436∗∗∗ +0.165 +Scheduled delivery +Baseline +-1.941∗∗∗ +0.120 +-5.740∗∗∗ +0.302 +-6.889∗∗∗ +0.571 +Age <= 29 years old +0.238 +0.210 +0.206 +0.406 +0.833 +0.567 +Age >= 70 years old +-0.261∗∗ +0.129 +-0.289 +0.241 +-0.932∗∗ +0.369 +Male +-0.070 +0.068 +-0.332∗∗ +0.140 +-0.562∗∗ +0.224 +Resident in Tokyo +0.153∗∗ +0.071 +0.259∗ +0.145 +0.610∗∗ +0.277 +Free membership +0.140∗∗ +0.067 +0.018 +0.137 +-0.060 +0.240 +Teleworker +-0.308∗∗ +0.123 +-0.670∗∗∗ +0.237 +-0.824∗ +0.494 +Delivery box +-0.494∗∗∗ +0.078 +-1.091∗∗∗ +0.160 +-1.284∗∗∗ +0.285 +Day choice (ref. = Weekday) +Holiday +Baseline +0.333∗∗∗ +0.092 +3.063∗∗∗ +0.252 +4.651∗∗∗ +0.450 +Slot choice (ref. = Other slots) +Morning slot +Baseline +0.191 +0.138 +0.077 +0.312 +-0.159 +0.511 +Live with children +-0.110 +0.097 +-0.247 +0.220 +-0.216 +0.384 +Live with family +-0.219 +0.166 +-0.359 +0.377 +-0.747 +1.097 +Couple with double income +0.213∗ +0.116 +0.463∗ +0.272 +0.496 +0.411 +Live alone +0.348∗∗∗ +0.111 +0.646∗∗ +0.265 +0.602 +0.530 +Live in a detached house +-0.270∗∗∗ +0.078 +-0.485∗∗∗ +0.176 +-0.550 +0.347 +Holiday +0.411∗∗∗ +0.115 +0.815∗∗∗ +0.176 +0.723∗∗∗ +0.243 +Public officer +-0.051 +0.165 +-0.009 +0.384 +0.244 +0.694 +Professional job +-0.053 +0.212 +-0.420 +0.404 +-0.526 +0.954 +Self employed +-0.144 +0.167 +-0.391 +0.378 +-0.053 +0.578 +Part-time job +-0.130 +0.112 +-0.426∗ +0.258 +-0.347 +0.358 +No job +0.095 +0.097 +0.205 +0.221 +0.306 +0.403 +Night slot +Baseline +0.520∗∗∗ +0.142 +1.169∗∗∗ +0.264 +1.765∗∗∗ +0.378 +Live with children +0.034 +0.113 +0.186 +0.208 +0.246 +0.271 +Live with family +-0.731∗∗∗ +0.211 +-1.127∗∗∗ +0.373 +-1.162∗∗ +0.539 +Couple with double income +0.172 +0.134 +0.528∗ +0.280 +0.379 +0.307 +Live alone +-0.128 +0.140 +-0.252 +0.255 +-0.475 +0.318 +Live in a detached house +-0.168∗ +0.090 +-0.289∗ +0.172 +-0.434∗∗ +0.193 +Holiday +0.028 +0.113 +-0.314∗∗ +0.157 +-0.405∗∗ +0.203 +Public officer +0.179 +0.171 +0.487 +0.305 +0.865∗ +0.511 +Professional job +0.438∗∗ +0.201 +0.857∗∗ +0.404 +0.220 +0.625 +Self employed +-0.629∗∗∗ +0.239 +-1.174∗∗∗ +0.452 +-0.982∗∗ +0.488 +Part-time job +-0.257∗∗ +0.127 +-0.619∗∗∗ +0.238 +-0.768∗∗∗ +0.272 +No job +-0.807∗∗∗ +0.127 +-1.413∗∗∗ +0.263 +-1.649∗∗∗ +0.316 +Scales of error components +Option choice +Next-day delivery +2.235∗∗∗ +0.076 +-2.082∗∗∗ +0.122 +Scheduled delivery +-0.713∗∗∗ +0.197 +2.108∗∗∗ +0.530 +Normal delivery +0.055 +0.226 +-0.239∗ +0.143 +Day choice +Holiday +2.953∗∗∗ +0.144 +3.588∗∗∗ +0.343 +Weekday +3.652∗∗∗ +0.208 +5.474∗∗∗ +0.414 +Slot choice +Morning slot +3.250∗∗∗ +0.161 +-4.241∗∗∗ +0.218 +Night slot +2.628∗∗∗ +0.170 +-2.231∗∗∗ +0.328 +Other slots +2.545∗∗∗ +0.195 +2.400∗∗∗ +0.259 +17 + +5.3. Willingness-to-pay for delivery +Finally, we discuss the distributions of WTPs for delivery obtained from the estimated +model. In the final model, all the distributional parameters of VODT, VOTS, and the sensitivity +to the delivery fee are estimated with statistical significance. The polynomial approximation +enabled estimating their distributions in a data-oriented manner, without imposing a parametric +distributional form a priori. The estimation results of γPI and γFR, capturing the observed +heterogeneity in the willingness-to-pay distributions, are consistent with our expectations. The +negative sign of the power γPI of the item price suggests that users who order more expensive +items are less sensitive to the delivery fee. The positive sign of the power γFR of the user’s online +shopping frequency indicates that the scale of VODT increases according to the frequency. +5.3.1. Value of delivery time savings +The distribution of VODT obtained from the estimation of MXL 4 is shown in Figure 4, +and its summary statistics are reported in Table 5. The VODT ranges from −47.937 to 219.445 +JPY/day7, with 3.6 % of users having negative values for delivery time savings. The mean and +median are 44.459 and 25.608 JPY/day, respectively, which are relatively smaller than the values +reported in the literature (e.g., Hsiao, 2009; Gawor and Hoberg, 2019). These results imply that, +in the delivery option choice context, some users do not necessarily need fast delivery, and rather +prefer delivery as per their convenience, given their schedule constraints. In other words, not +everyone wants next-day delivery; in the current e-commerce situation, many users choose the +delivery option just because it is free. They (more than 50% of users) would be willing to wait +an additional day if the delivery fee were increased by only 26 JPY. Nevertheless, we note that +some users would highly value saving the delivery time, as the maximum of VODT is 219.445 +JPY/day. Therefore, there would be a large variety (or heterogeneity) in VODT in the population. +It should also be noted that in the literature VODT was analyzed only separately in different +markets; e.g., $0.53 and $3.61 per day for books (Hsiao, 2009) and electronic items (Gawor +and Hoberg, 2019), respectively. Because our survey did not impose any restriction on the +item category, we can analyze the variation of VODT across different categories in a unified +framework. Based on the estimation result, we performed an ex-post segmentation analysis of +VODT, and the results in Figure 5(a) show its heterogeneity across different categories of ordered +items. For software and computer-related devices, VODT indicates the highest values, followed +by CD/DVDs, toys, and electric devices. A possible explanation for this is that these items are +not readily available in nearby stores, but users often want them urgently, thus requesting fast +delivery. In fact, groceries have a lower VODT value than these items, because groceries for +daily use can be obtained in nearby supermarkets. Moreover, users ordering books and cosmetics +have low VODT values, suggesting that such items are relatively inexpensive and are often not +needed immediately. This is also seen in the result of Figure 5(b), which shows that the more +expensive the ordered item, the higher the VODT. Finally, Figure 5(c) clearly shows that users +who frequently shop online have higher VODT values, that is, frequent users need fast delivery +more than low-frequent users. +Table 5: Distribution characteristics of VODT +Mean +Std +Min +25% +50% +75% +Max +P(v < 0) +VODT (JPY/day) +44.459 +43.308 +-47.937 +18.29 +25.608 +60.226 +219.445 +0.036 +7The average rate over 10 years of 2012–2021 is 106.1 JPY/$. Source: International Monetary Fund (IMF) Data +https://data.imf.org/. +18 + +VODT (JPY/day) +Relative frequency +Figure 4: Distribution of VODT among respondents +Books +Software +Computing devices +Grocery +Clothing +CD/DVD +Gifts +Daily necessaties +Toys +Electric devices +Healthcare goods +Cosmetics +Office supplies +Interior items +Others +< 1 / year +1 / year +1 / 6 months +1 / 2-3 months +1 / month +2-3 / month +1-2 / week +3-4 /week +max. +75% +25% +min. +mean +median +VODT (JPY/day) +VODT (JPY/day) +Item Category +(a) +(b) +(c) +E-shopping Frequency +Item Price (JPY) +<1000 +1000-2999 +3000-4999 +5000-7499 +7500-9999 +10000-15000 +15000-20000 +20000-30000 +30000-50000 +> 50000 +Figure 5: VODT for different segments: focusing on (a) ordered item category, (b) ordered item price, and (c) +e-shopping frequency of users. +19 + +5.3.2. Value of time slot shortening +The distribution of VOTS obtained from MXL 4 is shown in Figure 6, and its statistics +are reported in Table 6. The VOTS ranges from −3.827 to 27.086 JPY/hour, with 4.2% of +the respondents having negative values. The mean and median are 4.716 and 4.968 JPY/hour, +respectively. As such, although a majority of users would be willing to pay for the shortening of +the delivery time slot size, the payment would be small. This means that users do not highly value +the reduction in time slot size in monetary terms. Since the size of a time slot (i.e., time window +constraint) has a significant impact on last-mile delivery (e.g., Nockold, 2001), this result may +be an important finding for delivery demand management to improve logistics efficiency without +a serious reduction in user satisfaction8. +VOTS (JPY/hour) +Relative frequency +Figure 6: Distribution of VOTS among respondents +Table 6: Distribution characteristics of VOTS +Mean +Std +Min +25% +50% +75% +Max +P(v < 0) +VOTS (JPY/hour) +4.716 +2.792 +-3.827 +2.979 +4.968 +6.216 +27.086 +0.042 +6. Concluding remarks +This study analyzed e-commerce users’ preferences for delivery options. To this end, we designed +and implemented a stated choice survey, where users were asked to indicate which option of next, +scheduled, and normal delivery they would choose to deliver the ordered item, and if they chose +scheduled delivery, to jointly select the delivery date and time slot too. The stated choice data +of 4,062 users living in the three major metropolitan areas of Japan were analyzed by estimating +a mixed logit model, capturing users’ taste heterogeneity and substitution patterns. We also +applied a semi-nonparametric approach by Fosgerau and Mabit (2013) to flexibly estimate the +distributions of willingness-to-pay (WTP) for delivery attributes. +The results of this study contribute to advancing the understanding of e-commerce user +behavior, which plays a key role in delivery demand management as well as in designing urban +logistics policies (Neslin et al., 2006; Holguín-Veras et al., 2017). Specifically, the present results +8Note that in our survey the time slot size ranged from two to four hours, and a two-hour slot might not have been +sufficiently tight to increase the convenience of experienced e-commerce users. +20 + +suggest that delivery service attributes including fee, time, and time slot size are significant +determinants of the choice of a delivery option. We found that elderly people do not tend to +choose user-oriented delivery options such as next-day and scheduled delivery. The frequency +of teleworking and the presence of a delivery box also have a strong relationship with a low +propensity to request scheduled delivery. However, users who have a delivery box at home tend +to choose next-day delivery; that is, although installing a delivery box reduces the demand for +scheduled delivery and the risk of delivery failure, it can lead to an increased demand for fast +delivery. +The analysis also revealed that the value of delivery time savings (VODT) is widely distributed +among the respondents, and its maximum is 219.4 JPY per day. Nevertheless, the median VODT +is only 25.6 JPY per day, debunking the myth that everyone needs fast delivery and adding to the +statement and results of Rai et al. (2019). Today many e-commerce marketplaces offer users fast +delivery for free, imposing a strict time constraint on urban logistics, but our results suggest that +more than half of users would be willing to wait an additional day if the delivery fee increased +only by 26 JPY. In terms of heterogeneity, VODT has a high value for users who frequently shop +online and/or order expensive items, and varies according to the category of the item ordered. +Additionally, the value of time slot shortening (VOTS) was found to be low and distributed +with the median 5.0 JPY per hour, which means that users do not highly value the reduction in +time slot size in monetary terms. Since time windows are important constraints for last-mile +delivery, the result may suggest that lengthening the size of delivery time slots would be a way to +significantly improve its efficiency without a serious reduction in user satisfaction. The present +WTP measures were calculated purely with respect to the delivery fee, not including item price +(Gawor and Hoberg, 2019) or travel cost (Hsiao, 2009); therefore, the results of this study can be +used for the level-of-service design for last-mile delivery, independent of the retailer/marketplace +strategy. +Given the findings of e-commerce user behavior, future work includes a study of the impact +of the design of delivery attributes on the efficiency of last-mile delivery (e.g., Agatz et al., +2021). Since our model describes the choices of option, date and time slot for delivery, it would +be possible to analyze both day-to-day and within-day dynamics of delivery demand and their +impact on operational efficiency, combined with a multi-period vehicle routing problem (Archetti +et al., 2015) or agent-based simulation (Sakai et al., 2020, 2022). +Note that this study was carried out during the COVID-19 pandemic, under the declaration of +a state of emergency. Although we collected detailed information on users’ lifestyles, including +teleworking frequency, the results may still include the effects of unobserved variables specific +during the pandemic. Therefore, the comparison of users’ preferences for delivery options before, +during (and possibly after) the pandemic is considered another direction of future work that could +add an important contribution to the literature (Choi, 2021; Chowdhury et al., 2021). +Acknowledgements +We are grateful to Yamato Holdings Co., Ltd. for sending emails to their customers “Kuroneko +Members” for invitation to our survey. +CRediT author statement +Yuki Oyama: Conceptualization, Methodology, Software, Validation, Formal analysis, Investi- +gation, Data Curation, Writing - Original Draft, Visualization, Supervision. Daisuke Fukuda: +Methodology, Software, Validation, Formal analysis, Writing - Review & Editing. Naoto Imura: +Investigation, Project administration. Katsuhiro Nishinari: Resources, Funding acquisition. All +authors approved the final manuscript. +21 + +Appendix A. Screen image of the stated choice survey +①翌日配送 +②翌々日以降の日時指定配送 +③通常配送(日時指定なし) +料 +金 +○○○円 +○○○円 +○○○円 +追 +加 +料 +金 +なし +土日祝日:+○○○円 +夜間:+○○○円 +なし +配 +送 +日 +翌日 +指定日 +○○○ +時 +間 +指 +定 +不可 +○○○ +不可 +①翌日配送 +②翌々日以降の日時指定配送 +③通常配送(日時指定なし) +あなたは、下記の配送オプション①～③の中から、どれを選びますか。 +1 +2 +3 +②「翌々日以降の日時指定配送」を選択された方は、ご希望の +ここからは、あなたが直近のオンラインショッピングで購入された ○○○ +について、本日、再度購入すると仮定してお考えいただきます。 +配送の料金や配送日が異なる5つのパターンが順に表示されますので、各パターンにおいて +「翌日配送・日時指定配送・通常配送」の3つからどれを選択したいかお考えください。 +配達日を以下の中からお選 +びください。 +※『以下回答欄をクリックして現れるカレンダーから』お選びください。 +※本日を注文日としてご回答をお願いします。 +9時～11時 +11時～13時 +13時～15時 +15時～17時 +17時～19時 +②「翌々日以降の日時指定配送」を選択された方は、ご希望の配達時間帯を以下の中から +お選びください。 +※○○○時間ごとの選択肢になっております。 +19時～21時 +1 +2 +3 +4 +5 +6 +Figure A.7: Screen image of a stated choice task, i.e., original version of Figure 1 (in Japanese). Items with three +circles “◦ ◦ ◦” depend on respondent’s answers from RP survey or the attributes of the choice task. +References +Agatz, N., Campbell, A., Fleischmann, M., Savelsbergh, M., 2011. Time slot management in +attended home delivery. Transportation Science 45, 435–449. +Agatz, N., Campbell, A.M., Fleischmann, M., Van Nunen, J., Savelsbergh, M., 2013. Revenue +management opportunities for internet retailers. Journal of Revenue and Pricing Management +12, 128–138. +Agatz, N., Fan, Y., Stam, D., 2021. The impact of green labels on time slot choice and operational +sustainability. Production and Operations Management 30, 2285–2303. +Archetti, C., Jabali, O., Speranza, M.G., 2015. Multi-period vehicle routing problem with due +dates. Computers & Operations Research 61, 122–134. +Axhausen, K.W., Hess, S., König, A., Abay, G., Bates, J.J., Bierlaire, M., 2008. Income and +distance elasticities of values of travel time savings: New swiss results. 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Transportation science 50, 473–488. +25 + diff --git a/gtAyT4oBgHgl3EQfxfkk/content/tmp_files/load_file.txt b/gtAyT4oBgHgl3EQfxfkk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5feb65f42c44cf62fa188e274d012525f42e3167 --- /dev/null +++ b/gtAyT4oBgHgl3EQfxfkk/content/tmp_files/load_file.txt @@ -0,0 +1,1514 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf,len=1513 +page_content='E-commerce users’ preferences for delivery options Yuki Oyamaa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Daisuke Fukudab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Naoto Imurac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Katsuhiro Nishinaric aDepartment of Civil Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Shibaura Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Japan bDepartment of Civil Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Japan cResearch Center for Advanced Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Japan Abstract Many e-commerce marketplaces offer their users fast delivery options for free to meet the increasing needs of users,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' imposing an excessive burden on city logistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Therefore, understanding e-commerce users’ preference for delivery options is one of the core challenges faced while designing logistics policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To advance such understanding, this study designs and implements a stated choice survey in which respondents are faced with choice tasks among different delivery options and time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The survey was completed by 4,062 users from the three major metropolitan areas in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To analyze the stated choice data, mixed logit models capturing users’ taste heterogeneity as well as flexible substitution patterns among choice alternatives have been estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The model estimation results indicate that delivery attributes including fee, time, and time slot size are significant determinants of the delivery option choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Associations between users’ preferences and socio-demographic characteristics, such as age, gender, teleworking frequency and the presence of a delivery box, were also suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Moreover, we analyzed two willingness-to-pay measures for delivery, namely, the value of delivery time savings (VODT) and the value of time slot shortening (VOTS), and applied a non-semiparametric approach with polynomial approximation to estimate their distributions in a data-oriented manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Although VODT has a large heterogeneity among the respondents, the estimated median VODT is 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='6 JPY per day, implying that more than half of the respondents would wait an additional day if the delivery fee were increased by only 26 JPY, that is, they do not necessarily need a fast delivery option but often request it when cheap or almost free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Moreover, VOTS was found to be low, distributed with the median of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='0 JPY per hour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' that is, users do not highly value the reduction in time slot size in monetary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' These findings on e-commerce users’ preferences can help in designing levels of service for last-mile delivery to significantly improve its efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Keywords: E-commerce, next day delivery, last mile delivery, delivery option choice behavior, stated preference, willingness-to-pay 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Background, Motivation, and Objectives E-commerce has experienced rapid growth over the past two decades, significantly changing consumer shopping behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Today, many people enjoy online shopping that frees them from travel costs and provides the access to an extensive product range along with the option of shopping at home 24/7 (Farag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Hsiao, 2009), and the COVID-19 pandemic has boosted and rather necessitated this transition (UNCTAD, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' This digital transformation associated with e-commerce has dramatically increased the demand for last-mile parcel delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Experienced e-commerce users further desire fast, cost-effective, and punctual (within a narrow time window) delivery, and e-retailers face the challenge of balancing the needs of such users with logistics preferences for flexible and efficient delivery (Gawor and Hoberg, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' However, ∗Corresponding author Email address: oyama@shibaura-it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='jp (Yuki Oyama) Preprint January 3, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='00666v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='GN] 30 Dec 2022 under the existing circumstances, e-marketplaces often provide user-oriented services, with many of them offering their users next- or same-day delivery options (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Klapp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Many users are unwilling to pay extra for fast delivery and are often not asked to pay for it (Savelsbergh and Van Woensel, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Naturally, users choose a fast delivery option even though they may not necessarily need it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The increasing needs of e-commerce users for delivery lead to strict time constraints, imposing an excessive burden on city logistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Hua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Based on an experiment conducted in London, Nockold (2001) reports that scheduled delivery with time windows costs three times as much as normal delivery, where items can be delivered at any time of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' As a result, the workload of delivery service providers has been intensifying and is recognized today as one of the major social issues faced by many countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' A solution to this challenge is the appropriate management of demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Although optimizing logistics networks and costs of operators have been the primary focus of the literature, the behav- ior of e-commerce users can be one of the dominant factors in determining logistics strategies, and understanding users’ behavior and preferences is one of the core challenges faced while designing urban logistics policies (Neslin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Holguín-Veras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' E-retailers and delivery providers may also want to level over-concentration of delivery demand, thereby, increasing efficiency without compromising on users’ satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For delivery demand manage- ment, allocation and pricing of delivery options/time slots have been considered representative techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Agatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2011, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Yang and Strauss, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The former deals with which options/time slots to offer, while the latter considers determining the delivery fees to offer the options (Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Better understanding of users’ preference regarding delivery options and willingness to pay for fast/preferred-time delivery enhances allocation and pricing of delivery options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In other words, to appropriately design such demand management techniques, it is necessary to understand users’ preferences for delivery option choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Modeling users’ choices also allows for demand prediction in reaction to changes in policy design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' However, despite its importance, the literature investigating how different e-commerce users make choices among available delivery options with different attributes is limited (Garver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Some studies have investigated the willingness-to-pay (WTP) measure for the delivery time, which may be useful in designing a pricing strategy (Dinlersoz and Li, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Hsiao, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Gawor and Hoberg, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Yet, to the best of our knowledge, no research has analyzed the WTP for delivery in the context of an e-retailer/marketplace offering multiple delivery options, with users making a choice among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The complex trade-off between delivery time, charge, and slot attributes in delivery option choice utilities still needs to be investigated to appropriately determine the level-of-service and the fee of delivery options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' There is also a need to understand in detail the variation in the preference for delivery options among users, because they are generally heterogenous in terms of WTP, schedule preferences, and flexibility, and therefore, differentiation/personalization may have a significant impact on delivery demand management (Agatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Furthermore, the analyst may not be able to capture taste heterogeneity based only on observed variables, and has to take into account unobserved user heterogeneity to analyze WTP measures (Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2005, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The objective of this study is to advance the understanding of e-commerce users’ preferences for delivery options and the potential impact of level-of-service design on e-commerce delivery demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To this end, we perform a choice-based analysis of users’ behavior in the context that they are faced with a choice task among different delivery options and time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To explore detailed users’ taste heterogeneity, we design and implement a large-scale stated preference (SP) survey in a realistic option choice context, particularly focusing on two proposed WTP measures for delivery option choice: the value of delivery time savings (VODT) and the value of delivery time slot shortening (VOTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The former indicates how much additional delivery fee users are willing to pay to save their waiting time for their orders by one day, and the latter indicates 2 how much they are willing to pay to shorten a delivery time slot by one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Since users do not know exactly when the ordered item is delivered within the time slot, the uncertainty of the delivery timing increases with a wide time slot range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Although fast delivery and narrow time slots are preferred by users, in the opposite sense, they impose strict constraints on last-mile delivery on the logistic service provider side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The concepts of VODT and VOTS enable to quantitatively analyze the trade-off between levels-of-service and prices and thus may provide useful information to appropriately design pricing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Moreover, we incorporate both observed and unobserved heterogeneity into the analysis, by employing a mixed logit (MXL) model (Train, 2009), which is an extension of the standard multinomial logit (MNL) model to flexibly capture both taste heterogeneity and substitution patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We further introduce a flexible semi-nonparametric approach by Fosgerau and Mabit (2013) that does not require a distributional assumption because no prior knowledge regarding the WTPs of our interest is available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Related Literature An extensive body of literature has investigated e-commerce users’ reactions to delivery, in terms of satisfaction, retention, and loyalty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Ramanathan, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Koufteros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2014), highlighting the importance of delivery fee (Lewis, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2006), delivery time (Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2017), time slot convenience (Agatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Goebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2012), for e-commerce users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We refer the reader to Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2018) for a systematic review of the literature on e-commerce users’ behavior and order fulfillment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' However, the literature on the analysis of e-commerce users’ choice behavior is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' A choice-based analysis, particularly, the discrete choice analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Ben-Akiva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 1985) provides insights on how users change their behavior in reaction to service design, thereby, evaluating various behavioral indicators, which are the key to demand management for parcel de- livery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Focusing on the benefits, some studies have proposed choice-based delivery management frameworks (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Yang and Strauss, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Mackert, 2019), however, the choice models in their studies are not yet sufficiently realistic due to a lack of prior understanding of users’ preferences for the choice of delivery options and time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Many studies on e-commerce users’ choice behavior have focused on the choice between online or physical in-store shopping (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Farag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Kollmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Hsiao (2009) analyzed such choice behavior by estimating a binary logit model based on stated choice data in the book purchase context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The author investigated VODT and found that it was approximately $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='53 per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' However, the value was calculated as the trade-off between delivery time and travel cost for physical in-store shopping, not between delivery time and fee, and thus this measure cannot be used for designing a pricing scheme for delivery options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Similarly, Gawor and Hoberg (2019) analyzed VODT by conducting a choice-based conjoint analysis in the context of choice among e-retailers in an electronic marketplace offering the same product with different prices (including delivery fees) with different delivery options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' As a result, they found that VODT was $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='61 per day at an aggregated average, which was much higher than that reported by Dinlersoz and Li (2006) and Hsiao (2009) in the book purchase context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The authors discussed that this difference comes from a difference in marketplaces since an electronic market generally offers customers more expensive products than those offered in a book market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' However, because they calculated VODT with respect to the sum of the item price and delivery fee, it is not possible to separate the effect of the change in delivery fee from the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To the best of our knowledge, no research has analyzed VODT in the context of an e-marketplace offering multiple delivery options with different levels-of-service and users making a choice among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Further, VODT in the literature is a mean estimate, meaning that users’ preferences are implicitly assumed to be homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 3 Moreover, the literature on the delivery option choice behavior of e-commerce users is scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Agatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2021) recently explored the impact of green labels on time slot choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' By estimating an MNL model, they analyzed stated choice data among non-overlapping time slots for delivery, some of which had green labels and/or price incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The authors found that green labels work as an incentive and are particularly effective for people who are eco-conscious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2016) and Yang and Strauss (2017) also estimated MNL models of time slot choice in more general settings, but included only the sensitivity to slot price and slot dummy effects in the utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The effects of delivery attributes on delivery option choice behavior have not yet been sufficiently investigated (Garver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Most relevant to our study are the studies by Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2019) and Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2019), who investigated the delivery option choice behavior of e-commerce users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2019) conducted a choice-based conjoint analysis, focusing on the delivery attributes of fee, time, reception, and return possibility, as well as the attitude of users to sustainable options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The authors found that although e-commerce users prefer free and fast delivery to their home during regular office hours, they are willing to accept more sustainable options, such as collecting their orders themselves or waiting longer, when delivery and returns are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2019) also performed a choice-based conjoint analysis to investigate how users value delivery attributes when selecting a delivery option for their online purchases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' They found that the most important attribute for users was the delivery fee, followed by non-price attributes, which was consistent with the result of Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The authors also discussed a significant difference in users’ preferences among gender and income groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Several important aspects are still missing in the previous analysis of e-commerce users’ choice behavior toward appropriate demand management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' First, users’ taste heterogeneity in de- livery option preference has to be further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Although the effects of basic demographic characteristics such as age and gender (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Hsiao, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2019), different user segments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Gawor and Hoberg, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Hjort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2013), or specific attitudinal characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Agatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2021) were analyzed in the literature, attributes associated with lifestyle and household can also have a significant impact on choice behavior (Goebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2012) and need to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Furthermore, because the analyst may not be able to capture users’ taste heterogeneity only with observed attributes, unobserved user heterogeneity should also be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Second, a detailed analysis of the WTP measures with respect to delivery attributes is missing but can be very useful in delivery demand management, particu- larly in designing a pricing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Although some studies have investigated WTP for delivery (Dinlersoz and Li, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Hsiao, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Gawor and Hoberg, 2019), no research has analyzed it in the context of delivery option choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Third, to analyze the taste heterogeneity and WTP measures in detail, advanced econometric modeling needs to be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The existing literature on e-commerce users’ behavior or delivery demand management mostly relies on the standard MNL model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Yang and Strauss, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Agatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2021), which does not capture users’ taste heterogeneity or the correlation among utilities of choice alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Contributions and Structure of the Paper To advance the understanding of e-commerce users’ delivery option preferences, we perform a discrete choice analysis in the context of users facing a choice problem among different delivery options and time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We enumerate our contributions to the literature below: Delivery option choice analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' This study is the first to analyze the integrated choice of a delivery option, date, and time slot offered by an e-marketplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Available delivery options include next-day, normal, and scheduled delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Next-day and normal delivery options do not allow users to select a time slot, while scheduled delivery does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' This 4 choice problem is a realistic situation that e-commerce users often face when shopping online, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', in Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The alternatives may exhibit complex substitution patterns and thus construct a cross-nested choice structure, which we incorporate into the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Stated choice survey design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To perform the choice analysis, we implement a web- based stated preference (SP) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' It also includes a revealed preference (RP) survey on respondents’ most recent experience shopping online so that they can readily imagine a realistic situation when conducting the stated choice tasks, and the item category and price are not limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In addition to basic demographic information, lifestyle characteristics such as teleworking frequency during the COVID-19 pandemic were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The survey was conducted in collaboration with Yamato Holdings, which has the largest market share of parcel delivery in Japan, and obtained a valid sample of 4,062 respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To the best of our knowledge, this is the largest SP survey conducted on e-commerce users’ delivery option choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' WTP measures for delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In the analysis, we focus on three delivery attributes: deliv- ery fee (option-specific price and additional charges for holiday and night slot delivery), expected delivery time, and time slot size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Based on these attributes, we calculate novel WTP measures for delivery, namely VODT and VOTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The former indicates the amount of additional delivery fee that users are willing to pay to save their order waiting time by one day, and the latter indicates the amount that they are willing to pay to shorten the time slot size by one hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Both are calculated purely with respect to the delivery fee, not including item price (Gawor and Hoberg, 2019) or travel cost (Hsiao, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Users’ taste heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The rich stated choice data allows us to incorporate detailed users’ taste heterogeneity into the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We explore the interactions of delivery option preferences with various user information such as demographic and household charac- teristics, lifestyle attributes, and their membership in some e-marketplaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Analyzing such interactions gives an insight into users’ observed heterogeneity in their preferences of options and time slots, which may be the key to differentiation in demand management (Agatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We also consider the nonlinear interactions of the WTP measures with the price of the purchased item and the frequency of online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Advanced discrete choice analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For analysis of the stated choice data, we estimate an MXL model (Train, 2009), which is an extended version of the MNL model to flexibly capture unobserved taste heterogeneity and substitution patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We treat the WTP mea- sures for delivery as random parameters and analyze their distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Because we do not have prior knowledge of their distributional forms, we apply a flexible semi-nonparametric approach (Fosgerau and Mabit, 2013) that does not require a distributional assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We further incorporate error components capturing the underlying correlations among the utilities of the alternatives, focusing on a cross-nested structure of delivery options and time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Section 2 describes the details of the design of the stated choice experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Section 3 reports the data collection and sample statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Section 4 introduces the MXL model of delivery option choice behavior to analyze the stated choice data, and Section 5 presents the estimation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In Section 6, we conclude the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Experimental design We designed a web-based SP survey for e-commerce delivery option choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The survey consists of three parts: (1) questions on the most recent online shopping experience, (2) stated choice 5 tasks on delivery options, and (3) questions on socio-demographic characteristics, and household and lifestyle attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Note that the RP survey in the first part has mainly been conducted to establish a realistic situation for the respondents during the stated choice tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Stated choice task For the stated choice experiment, we asked respondents to assume that they order the same item at the same price as those ordered in their most recent online shopping transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The date they responded to the survey was assumed to be the order date for the stated choice tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Figure 1 shows an example of the stated choice task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For each task, the respondent first chooses a delivery option among “next-day”, “scheduled” and “normal” delivery (Q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Respondents who chose the scheduled delivery option are asked to jointly select the date and time slot for delivery (Q2 and Q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The available dates for scheduled delivery range from 2 to 8 days after the order date (notated as “2+” ∼ “8+”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For Q2, a calendar pops up on the click, and the respondent specifies the order date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' While the entire period of delivery hours in a day is fixed 9:00–21:00 (12 hours in total), the size of a time slot varies from 2 to 4 hours by choice scenarios, resulting in from six to three available slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Figure 1 shows a case where the time slot size is 3 hours, and therefore, the number of available slots is four (Q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For the next-day delivery, the ordered item is delivered on the following day, but the respondent is not allowed to select a time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' When respondents choose the normal delivery option, they can neither select the date nor the time slot for delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Note that this integrated choice task of delivery option, date, and time slot is a more realistic choice problem that e-commerce users often face when shopping online, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', in Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Nev- ertheless, existing literature has focused only on a single aspect, such as option (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2019) or time slot choice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Agatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Attributes The delivery attributes on which this study mainly focuses are fee, time, and time slot size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To characterize the delivery options, the eight attributes summarized in Table 1 were controlled in the stated choice tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The delivery fees are differentiated by the delivery options, ranging from 300 to 600 JPY (Attributes 1–3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For scheduled delivery, an additional charge of 100 JPY can be imposed to deliver on holidays and in the latest time slot (Attributes 4–5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The expected delivery time for normal delivery is based on the earliest possible date (Attribute 7) and the range of delivery dates (Attribute 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The delivery time for next-day delivery is always one day, and that for scheduled delivery directly depends on the choice of the respondents, ranging from 2 to 8 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' As mentioned in the previous subsection, the size of a time slot is an attribute for scheduled delivery and varies from 2 to 4 hours (Attribute 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The entire delivery period in a day ranges from 9:00 to 21:00, with the delivery time slots defined based on the slot size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The number of available slots, therefore, can be three, four, or six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' When the size of a slot is three hours, for example, the available slots are [9–12, 12–15, 15–18, 18–21] (as in Q3 in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In total, the experiment has 3 four-level, 2 three-level, and 3 two-level attributes, implying 43 × 32 × 23 profiles for a full factorial design, which is very hard to manage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Therefore, we created a fractional factorial design consisting of 400 profiles where the main effects are orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' From this set of choice tasks, we randomly displayed five tasks to each respondent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We did not consider optimal designs like D-efficient designs (Kuhfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 1994) because prior values of the parameters for the delivery option choice model are not empirically known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Moreover, Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2018) recently presented that random design, despite its simplicity, performs as well as any design in terms of robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 6 (1) Next-day Delivery Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Which delivery option would you choose?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' If you chose “(2) Scheduled Delivery,” please select your preferred delivery date from the calendar below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' If you chose “(2) Scheduled Delivery,” please select your preferred delivery time from the list below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (1) Next-day Delivery (2) Scheduled Delivery (from 2 to 8 days after) (3) Normal Delivery (specified day or time are not available) Note: please consider today the order date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Calendar pops up on click ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Delivery Fee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='600 JPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Next day ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Specified date ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3 ~ 5 days after (not specified) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Holiday: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='+ 100 JPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Night ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=': ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='+ 100 JPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='500 JPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='300 JPY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Additional Charge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Delivery Date ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Specified Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Not available ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Not available ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Select from 4 slots of 3 hours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='(3) Normal Delivery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='(Specified day or time are not available) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='(2) Scheduled Delivery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='(From 2 to 8 days after) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9:00 - 12:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='12:00 - 15:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='15:00 - 18:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='18:00 - 21:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='Figure 1: Example of the stated choice task: Items in red indicate task attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' This is an English translation of the survey (see Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='7 for the original).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 7 Table 1: Attributes and attribute levels Attribute Alternative No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' of levels Values 1 Delivery fee (JPY) Next-day 4 [300, 400, 500, 600] 2 Delivery fee (JPY) Normal 4 [300, 400, 500, 600] 3 Delivery fee (JPY) Scheduled 4 [300, 400, 500, 600] 4 Additional charge for holiday (JPY) Scheduled 2 [±0, +100] 5 Additional charge for the latest slot (JPY) Scheduled 2 [±0, +100] 6 Size of a slot (hrs) Scheduled 3 [2, 3, 4] 7 Earliest possible date (d+) Normal 3 [2+, 3+, 4+] 8 Range of delivery dates (days) Normal 2 [2, 3] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Stated choice data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Data collection We conducted the stated choice survey from April 30 to May 14, 2021, during the COVID-19 pandemic, collaborating with Yamato Holdings Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' that has the largest market share of parcel delivery in Japan (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3 % in the Financial Year 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The respondents were among the “Kuroneko Members” (more than 50 million in Japan), registered users of the Yamato Holdings’ delivery service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In this survey, we randomly selected users who live within the three major metropolitan areas (Greater Tokyo, Osaka, and Nagoya) in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The population of the three areas comprise of 52 million, accounting for 41% of Japan’s total population1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The survey invitation was sent to 100,000 Kuroneko Members, of whom 4,872 completed it (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', the response rate was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='87%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Since five choice tasks were given to each respondent, the original sample size observed was 24,360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Data cleaning We cleaned the data as follows: First, we removed 635 observations where the scheduled delivery option is chosen for the next day or after more than 8 days 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' As mentioned earlier, we accept the scheduled delivery only for 2-8 days after the order date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (24,360 → 23,725) Then, we removed 745 observations of 151 respondents who have never experienced shopping online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To analyze the preferences for delivery options in e-commerce, this study focuses on experienced users 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (23,725 → 22,980) Finally, we removed 3120 observations of 655 respondents who chose an alternative dominated by other options once or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Such alternatives include the normal delivery option with higher delivery fees than the next-day or scheduled delivery options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We eliminate the potential bias of responses with a lack of seriousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (22,980 → 19,860) After data cleaning, we finally obtained 19,860 observations of 4,062 unique respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 1Source: 2015 Population Census (Statistics Bureau, Ministry of Internal Affairs and Communications) https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='jp/english/data/kokusei/2015/summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='html 2We could not restrict the available order dates for each participant due to system specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Alternatively, we carefully explained that the available dates for the scheduled delivery were 2 to 8 days after the order date (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', survey date).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 3This processing is also done in Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2019), who also focused on respondents who have at least some experience shopping online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Sample statistics Table 2 lists the descriptive statistics of the demographic characteristics, household information, and e-commerce attributes of the respondents, as well as their recent online shopping transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Because of the COVID-19 pandemic, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='7 % of the respondents telework once a week or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In addition, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3 % of the respondents have a delivery box at their places, which is a locker- type device where delivery persons can leave parcels even when the recipient is not at home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Regarding online shopping experiences, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='6 % of the respondents usually buy an item online once a week or more, and 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1 % of the respondents have registered for a membership with a free delivery privilege.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Note that it was clearly explained in the stated choice experiment that all respondents were required to pay the delivery fee even if they have such a membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We observed a wide range of variety in category and price of items purchased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The information regarding the item price has been used in analyzing the WTPs, which is important, given that the literature targeting different marketplaces reported significantly different values of WTP for delivery (Dinlersoz and Li, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Hsiao, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Gawor and Hoberg, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Aggregation Out of 19,860 observations, the next-day, normal and scheduled delivery options were chosen 8,429 (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4 %), 4,265 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5 %), and 7,166 times (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1 %), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Of scheduled delivery, weekdays and holidays were chosen 1,327 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5 %) and 5,839 times (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5 %), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' and the earliest, the latest and the other time slots were chosen 3,262 (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5 %), 2,082 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1 %) and 1,822 times (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4 %), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Although next-day delivery is the most popular alternative, the other options were also chosen a considerably large number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To see the trade-off between delivery time and fee, we plot the aggregate choice results focusing on the two attributes in Figure 2, which includes the results of next-day (the delivery time is one day) and scheduled delivery (two to eight days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Since a delivery date is not fixed in the normal delivery option, this aggregation does not include the observations choosing that option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Figure 2(a) shows the choice results based on the delivery time and fee that each option offers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Most of the respondents seem to prefer deliveries within three days after the order date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' There is no major change in the number of choices from the fourth to the eighth day after the order date, which may be because, for deliveries later than three days after, users care more about their schedule than the delivery time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Delivery within three days seems to be popular with a wide range of delivery fees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' even up to JPY600, which is the highest possible fee for next-day delivery, one to three days after the order date were chosen relatively high numbers of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Note that, due to the SP task design, JPY of 500 and 600 are not always expensive among the available options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' This can be seen in Figure 2(b), which focuses on the delivery fee difference to the minimum among the fees of the offered alternatives, instead of the pure delivery fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' It shows that the next-day delivery option was chosen 6,426 times (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4% of the total) when it was the least expensive among the available options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' That is to say, e-commerce users consider the trade-off between delivery fee and time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' although some still prefer their parcels to be delivered on the next day, many of them may change their preference when the fee is not the cheapest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Delivery option choice model To analyze the stated choice data, we perform a discrete choice analysis of e-commerce delivery option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The alternatives for the choice problem are defined based on delivery options (next-day, scheduled, and normal), dates, and time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' As mentioned earlier, the number of available time slots varies by choice scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To analyze all samples in a unified manner, we categorize the time slots into three categories: “morning” (the earliest), “night” (the latest), and “other” 9 Table 2: Sample statistics (N = 4062) Category Characteristic Value Sample frequency Socio-demographic Age ≤ 29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='8% 30–49 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9% 50–69 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='8% ≥ 70 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5% Gender Female 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3% Male 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='7% Occupation Employee 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% Self-employed 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2% Public officer 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='6% Professional (incld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' healthcare worker) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4% Part-time job 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5% Student 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='0% No job 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='0% Other 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2% Telework frequency Everyday 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='8% 3–4 days per week 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2% 1–2 days per week 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='7% 2–3 days per month 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% 1 day per month or less 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% Never 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% Household Residence Tokyo 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='6% Other 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4% Household composition Live alone 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% Couple with double income 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='0% Couple with single income 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9% Live with children 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9% Live with family (w/o children) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4% Live with friends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5% Other 4.' metadata={'source': 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+page_content='7% E-commerce attributes E-shopping frequency 3–4 times per week 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='0% 1–2 times per week 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='6% 2–3 times per month 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3% Once per month or less 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% E-commerce membership Member w/ free-delivery privilege 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% Member w/o privilege 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='8% Not a member 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% Recent e-shopping experience Category of item Books 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9% Software 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='0% Computing devices 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='7% Grocery 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9% Clothing 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='0% CD / DVD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9% Gift 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='8% Daily necessities 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='6% Toys 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2% Electric devices 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5% Healthcare goods 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% Cosmetics 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2% Office supplies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5% Interior items 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% Other 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='6% Price of item (JPY) ≤ 999 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9% 1,000–2,999 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='7% 3,000–4,999 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3% 5,000–7,499 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4% 7,500–9,999 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2% 10,000–14,999 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4% 15,000–19,999 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1% 20,000–29,999 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9% 30,000–49,999 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='9% ≥ 50,000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2% 10 (a) (b) Figure 2: Choice results with the focus on the delivery time (horizontal axis) and delivery fee (vertical axis): (a) takes a pure delivery fee, and (b) focuses on the fee difference among the offered alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The numbers in the grids indicate the number of times chosen, with deeper colors representing higher numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Note that this aggregation includes only the observations that chose the next-day or scheduled delivery options, and therefore the total number does not correspond to 19,860, which is the number of the total observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Note that the additional charge is consistent with this definition (Attribute 5 in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Thus, we have 21 (7 × 3) schedules for the scheduled delivery, resulting in 23 alternatives in the choice set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Model formulation For the analysis, we estimate a MXL model that captures unobserved inter-individual taste heterogeneity and flexible substitution patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Based on the random utility (RUM) theory, in choice task t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' , Tn}, individual n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' , N} is assumed to choose the alternative that maximizes the utility Untj as defined below: Untj = V(Xntj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' βn) + εntj, (1) where V is the systematic utility, which is a function of observed explanatory variables Xntj, and εntj is the unobserved utility following extreme value distribution Type I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', εntj ∼ Gumbel(0, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We assume that the scale µ is standardized as one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The preference parameters βn of individual n are assumed as realizations from distribution f (β|Ω) where Ω is a vector of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' When βn is given, the probability that choice ynt of individual n in choice scenario t is j is given by the MNL model (also called the logit kernel): Pn(ynt = j|βn) = exp{V(Xntj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' βn)} ∑j′∈Cnt exp{V(Xntj′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' βn)}, (2) where Cnt is the choice set for individual n at choice tasik t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Therefore, the joint probability of the sequence of choices {jt}Tn t=1 for the given βn is Pn � ynt = {jt}Tn t=1|βn � = Tn ∏ t=1 Pn(ynt = jt|βn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (3) 4This study assumes that Cnt is always the same for all choice tasks of all individuals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Cnt = C, and consists of 23 alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 11 The MXL model marginalizes this probability over the parameter distribution f (β|Ω), obtaining the unconditional probability Pn � ynt = {jt}Tn t=1|Ω � = � β Pn � ynt = {jt}Tn t=1|β � f (β|Ω)dβ, (4) and the log-likelihood function is given by LL(Ω) = N ∑ n=1 ln Pn � ynt = {jt}Tn t=1|Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (5) Since the integral in (4) is hard to compute, we estimate the MXL model by maximizing the following simulated log-likelihood function SLL(Ω) = N ∑ n=1 ln � 1 R R ∑ r=1 Pn(ynt = {jt}Tn t=1|βnr) � ≈ LL(Ω) (6) where βnr is the rth realization (r = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' , R}) of n’s preference parameters drawn from distribution f (β|Ω) and R is the total number of random draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Model specification 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Utility function This study investigates the distributions of two WTP measures for delivery, VODT wd and VOTS wh, as well as the effects of socio-economic characteristics s on delivery option choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Therefore, instead of taking the ratio of marginal utilities of the target attributes and the delivery fee (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', modeling in the preference space), we formulate the model in the WTP space (Train and Weeks, 2005), leading to a straightforward estimation of the WTP distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' With this formulation, the WTP distribution is directly estimated in a stable manner without any restriction on the distribution of the delivery fee coefficient βc (Scarpa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The utility Untj for alternative j of individual n in choice situation t is: Untj = βc,n(DFntj + wd,nDTntj + wh,nSSntj) + α′snj + δjξ + εntj (7) where DFntj, DTntj, and SSntj are the delivery fee (JPY), expected delivery time (days), and time slot size (hours), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' DTntj equals 1 for next-day delivery, 2–8 for scheduled delivery, and is defined as EDntj + RDntj/2 for normal delivery, where EDntj is the days to the earliest possible date and RDntj is the range of delivery dates (see Table 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' SSntj is fixed to be zero for the next-day and normal delivery options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The parameter α is a vector of the coefficients for the socioeconomic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The error component ε is assumed to be independent and identically distributed (iid) extreme value across observations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', ε ∼ Gumbel(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The normal error components ξ = {ξm ∼ N (0, σ2 m)}m∈M capture the heteroskedasticity and correlations among utilities of different delivery options, with a nesting structure M (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4 for the detail), and δjm equals one if alternative j belongs to nest m and zero otherwise (Train, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We also assume that βc, wd and wh are randomly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Observed heterogeneity We structurize the WTPs, by incorporating interactions with users’ socio-economic characteris- tics, capturing the nonlinear observed taste heterogeneity across individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' By following the continuous interaction specification taken in the estimation of the value of travel time savings 12 (Axhausen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2008), the delivery fee coefficient βc,n is structured by the price of item PIn that user n ordered5: βc,n = ˆβc,n � PIn PIref �γPI , (8) and VODT wd is structured by the frequency of online shopping FRn, wd,n = ˆwd,n � FRn FRref �γFR , (9) where ˆβc,n and ˆwd,n are realizations from the distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' PIref and FRref are the sample averages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' and γPI and γFR are the parameters to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Unobserved heterogeneity: distributional specification We assume that βc follows a log-uniform distribution, which has a shorter tail than a log-normal distribution and has been tested for the value of travel time savings in studies by Fosgerau (2006) and Hess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' If y = log(x) is uniformly distributed, x is defined to be log-uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Therefore, the fee coefficient is then given by ˆβc = − exp(a + bu), (10) where u ∼ Uniform(0, 1), and a and b are the lower bound and the spread, which are the parameters to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Unlike the value of travel time savings in the transportation research field, we do not have sufficient empirical evidence for the WTPs of interest, therefore, assuming the parametric distri- butional form a priori may bias the understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' After extensive testing, for the distributional specification of the WTPs wd, wh, we apply a semi-nonparametric approach by Fosgerau and Mabit (2013) that does not require a distributional assumption and that can flexibly describe the shape of distribution in a data-oriented manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We transform a draw u from a distribution using a power series f (u|η) = K ∑ k=1 ηkuk, (11) where K is the dimension of the polynomial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We then compute random draws of WTPs: ˆwd = f (ud|ηd), (12) ˆwh = f (uh|ηh), (13) where ηd = (ηd,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' , ηd,Kd) and ηh = (ηh,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' , ηh,Kh) are the vectors of the parameters to be estimated and define the distributional forms of the WTPs of interest, and Kd and Kh are the dimensions of the power serieses for VODT and VOTS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Error components We have 23 alternatives in the choice set, some of which may introduce correlations violating the independence of irrelevant alternatives (IIA) assumption of the MNL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To capture the heteroscedasticity and cross-correlated structures among the utilities of delivery options, we introduce the error components ξ in the MXL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Figure 3 shows the cross-nested structure 5This is to capture the difference in sensitivity to the delivery fee depending on the item price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Gawor and Hoberg (2019) reported such a difference between electronic and book marketplaces by comparing their results with the those of Dinlersoz and Li (2006) and Hsiao (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' However, how the item price actually affects the WTPs has never been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 13 of the model, where we introduce eight error components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', |M| = 8, for the delivery options (next-day, normal and scheduled), as well as date (holiday and non-holiday) and time slot nests (morning, night and other);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' for example, the utility of scheduled delivery two days after the order is placed and in the morning slot is defined as: Unt,(2+,morning) = · · · + ξscheduled + ξholiday + ξmorning + εnt,(2+,morning) where ξm ∼ N (0, σ2 m), ∀m ∈ M, and the standard deviation σm is to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Note that whether “2+” is a holiday or not depends on the order date, therefore, Unt,(2+,morning) does not necessarily contain the error component ξholiday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Because we rely on panel data for model estimation, all scale parameters are identified (Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Next-day delivery Normal delivery Scheduled delivery Morning Night Other Holiday Options Time slots Dates Non-holiday (2+, Morning) (2+, Night) (3+, Night) (2+, Other) (8+, Other) (3+, Morning) … Figure 3: Cross-nested structure of the choice model approximated by the error components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Model estimation results This section reports the model estimation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Maximum simulated likelihood estimation was performed using the Apollo package in R (Hess and Palma, 2019a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Final model specification An extensive specification search has been conducted to reach the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Table 3 reports the comparison of the models developed during the critical stages of the modeling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' First, a standard MNL model was estimated, where no parameters were assumed to be random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Since its framework restricts substitution patterns and does not adequately account for preference heterogeneity across respondents, MXL specifications were explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The error components were introduced into the model (MXL 1), to capture heteroskedasticity and flexible substitution patterns across the alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Furthermore, three MXL specifications were explored in terms of unobserved preference heterogeneity by assuming the delivery fee coefficient and the two WTPs to be randomly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For the distributions of the WTPs, the normal and log-normal distributions were tested (MXL 2 and MXL 3), as well as the polynomial approximation of (11) (MXL 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' After a comprehensive search6, the dimensions of the polynomial approximations 6We tried two candidates for bases, namely uniform and normal distributions, and different dimensions from two to five for polynomial approximations respectively in (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Considering the resultant distributions, goodness-of-fit, and estimation stability to the settings, we finally selected the base and dimensions reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 14 were finally set at Kd = 4 and Kh = 2, and uniformly distributed random numbers were used as the base u in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The overall goodness-of-fit of the models appears to be satisfactory, and the incorporation of both error components and random WTP parameters significantly improved the model performance (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Consequently, MXL 4 obtained the best goodness-of-fit in terms of the values of log-likelihood, AIC, BIC and ¯ρ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' especially, the polynomial approximations allowed the model to better fit compared to normal and log-normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Table 3: Model comparison MNL MXL 1 MXL 2 MXL 3 MXL 4 Error component No Yes Yes Yes Yes Distribution of VODT Fixed Fixed Normal Log-normal Polynomial w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Kd = 4 Distribution of VOTS Fixed Fixed Normal Log-normal Polynomial w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Kh = 2 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' of parameters 46 54 57 57 61 Log-likelihood 35,068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2 28,359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='06 26,812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='02 26,796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='48 26,624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='67 AIC 70,228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4 56,826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='12 53,738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='05 53,706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='96 53,371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='35 BIC 70,591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='64 57,252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='53 54,188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='15 54,157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='06 53,853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='03 ¯ρ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5437 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='5715 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Estimation results Table 4 enumerates the estimation results for three models: MNL, MXL 1 (error com- ponent model) and MXL 4 (error component and random parameter model with polynomial approximations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In all models, the sensitivities to the delivery attributes were estimated with statistical sig- nificance and have the expected signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Most of the other parameters also correspond to our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Below we discuss the results in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Observed heterogeneity: effects of socio-demographic characteristics Here, we discuss observed heterogeneity for delivery option choice preferences, captured by interactions between alternative specific constants and socio-demographic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For the sake of representativeness in the effects of socio-demographic variables, the categories of some variables were merged before the model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' As such, we defined respondents who telework for 1-2 days per week or more as “teleworker”s (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='8% of sample, see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For the subgroups of choices, normal delivery (option choice), weekday delivery (date choice), and “other” slot (slot choice) are treated as references for the alternative specific constants, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Option choice The estimation result indicates that users who are 70 years or older tend to choose normal delivery compared to next-day and scheduled delivery, while users younger than 30 years old exhibit opposite tendency, suggesting a clear difference in the delivery option preference be- tween different age-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Additionally, male users prefer next-day delivery but not scheduled delivery, which implies that they tend to order items just before they need them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' People living in Tokyo, the largest and busiest city in Japan, prefer scheduled delivery that allows them to select the delivery date and slot to fit in their schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In contrast, the estimation result indicates that teleworkers do not need scheduled delivery, reflecting the higher flexibility of their schedule to receive the parcels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' During the COVID-19 pandemic many people started teleworking and last-mile delivery may have been released from complex time window constraints than before;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' however, non-teleworkers who prefer scheduled delivery and delivery providers may still have a problem due to the rapid growth in total demand of last-mile delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Moreover, users who have a privileged membership in an e-commerce marketplace particularly prefer next-day delivery to 15 normal delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Since we clearly explained that their privilege was not valid during the stated choice tasks, this result highlights the inertia of their choices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', they are used to requesting fast delivery options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Finally, the result suggests a strong relationship between the presence of a delivery box and users’ delivery option choice preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' If users have a delivery box equipped with their homes, they do not need to worry about the timing of receipt of items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' therefore, they are willing to opt for next-day delivery, thereby, not needing scheduled delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Slot choice In terms of delivery time slot preference, the analysis focused on its relationships with the characteristics of occupations and households, reflecting users’ lifestyles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' First, users living with their family do not prefer the latest slot delivery, implying that they can afford to receive their parcels during the day, instead of in the evening, when the family spends time together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The time slot choice also depends on housing types: people who live in detached houses tend to choose slots during the day, indicated by the negative signs of the coefficients of morning and night slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' During holidays, users prefer to receive parcels in the morning, but not in the evening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' A possible explanation is that they may want to leave home after receiving the parcels or likely return home late in holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' As for occupation, users who are self-employed, have part-time or no jobs clearly do not opt for the night slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Error components The standard deviations of the heteroskedastic error components are all significantly differ- ent from zero, except the one for the normal delivery option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' This suggests that there exists unobserved heterogeneity and significant substitution patterns associated with choices of deliv- ery options, dates, and time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' First, the difference in the scales of the error components of next-day, scheduled, and normal delivery indicates the heteroskedasticity among the option alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In addition, the estimation results of the error scales of the day and slot alternatives support the cross-nested structure of Figure 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' alternatives within a common nest are correlated with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Table 4: Estimation results MNL MXL 1 MXL 4 Estimate Rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Estimate Rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Estimate Rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Delivery attributes Delivery fee (JPY) Fixed param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='009∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='018∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='000 Random param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Lower bound (log) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='232∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='078 Spread (log) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='036∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='052 Observed heterogeneity Item price (JPY) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='106∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='089∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='069∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='026 Value of delivery time savings (JPY/day) Fixed param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='870∗∗∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='724 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='465∗∗∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='158 Random param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' ηd,0 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='651∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='992 ηd,1 1,034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='516∗∗∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='175 ηd,2 1,061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='363∗∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='815 ηd,3 618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='977∗∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='384 ηd,4 936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='408∗∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='250 Observed heterogeneity Online shopping frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='104∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='146∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='160∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='033 Value of time slot shortening (JPY/hr) Fixed param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='412∗∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='275 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='455∗∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='088 Random param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' ηh,0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='248∗∗∗ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='596 ηh,1 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='229∗∗∗ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='598 ηh,2 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='399∗∗ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='696 (Continued on next page) 16 Table 4 Estimation results (continued) MNL MXL 1 MXL 4 Estimate Rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Estimate Rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Estimate Rob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Alternative specific constant Option choice (ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' = Normal delivery) Next-day delivery Baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='593∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='479∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='046∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='142 Age <= 29 years old 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='479∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='742∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='297 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='254∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='463 Age >= 70 years old 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='373∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='569∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='611∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='249 Male 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='177∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='297∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='141 Resident in Tokyo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='064 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='824∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='494 Delivery box 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='494∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='078 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='091∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='160 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='284∗∗∗ 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='263 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='649∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='316 Scales of error components Option choice Next-day delivery 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='235∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='076 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='082∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='122 Scheduled delivery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='713∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='197 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='108∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='530 Normal delivery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='239∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='143 Day choice Holiday 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='953∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='144 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='588∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='343 Weekday 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='652∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='208 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='474∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='414 Slot choice Morning slot 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='250∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='161 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='241∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='218 Night slot 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='628∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='170 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='231∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='328 Other slots 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='545∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='195 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='400∗∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='259 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Willingness-to-pay for delivery Finally, we discuss the distributions of WTPs for delivery obtained from the estimated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In the final model, all the distributional parameters of VODT, VOTS, and the sensitivity to the delivery fee are estimated with statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The polynomial approximation enabled estimating their distributions in a data-oriented manner, without imposing a parametric distributional form a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The estimation results of γPI and γFR, capturing the observed heterogeneity in the willingness-to-pay distributions, are consistent with our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The negative sign of the power γPI of the item price suggests that users who order more expensive items are less sensitive to the delivery fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The positive sign of the power γFR of the user’s online shopping frequency indicates that the scale of VODT increases according to the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Value of delivery time savings The distribution of VODT obtained from the estimation of MXL 4 is shown in Figure 4, and its summary statistics are reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The VODT ranges from −47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='937 to 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='445 JPY/day7, with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='6 % of users having negative values for delivery time savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The mean and median are 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='459 and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='608 JPY/day, respectively, which are relatively smaller than the values reported in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Hsiao, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Gawor and Hoberg, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' These results imply that, in the delivery option choice context, some users do not necessarily need fast delivery, and rather prefer delivery as per their convenience, given their schedule constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In other words, not everyone wants next-day delivery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' in the current e-commerce situation, many users choose the delivery option just because it is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' They (more than 50% of users) would be willing to wait an additional day if the delivery fee were increased by only 26 JPY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Nevertheless, we note that some users would highly value saving the delivery time, as the maximum of VODT is 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='445 JPY/day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Therefore, there would be a large variety (or heterogeneity) in VODT in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' It should also be noted that in the literature VODT was analyzed only separately in different markets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='53 and $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='61 per day for books (Hsiao, 2009) and electronic items (Gawor and Hoberg, 2019), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Because our survey did not impose any restriction on the item category, we can analyze the variation of VODT across different categories in a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Based on the estimation result, we performed an ex-post segmentation analysis of VODT, and the results in Figure 5(a) show its heterogeneity across different categories of ordered items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' For software and computer-related devices, VODT indicates the highest values, followed by CD/DVDs, toys, and electric devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' A possible explanation for this is that these items are not readily available in nearby stores, but users often want them urgently, thus requesting fast delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In fact, groceries have a lower VODT value than these items, because groceries for daily use can be obtained in nearby supermarkets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Moreover, users ordering books and cosmetics have low VODT values, suggesting that such items are relatively inexpensive and are often not needed immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' This is also seen in the result of Figure 5(b), which shows that the more expensive the ordered item, the higher the VODT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Finally, Figure 5(c) clearly shows that users who frequently shop online have higher VODT values, that is, frequent users need fast delivery more than low-frequent users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Table 5: Distribution characteristics of VODT Mean Std Min 25% 50% 75% Max P(v < 0) VODT (JPY/day) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='459 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='308 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='937 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='29 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='608 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='226 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='036 7The average rate over 10 years of 2012–2021 is 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='1 JPY/$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Source: International Monetary Fund (IMF) Data https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='imf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 18 VODT (JPY/day) Relative frequency Figure 4: Distribution of VODT among respondents Books Software Computing devices Grocery Clothing CD/DVD Gifts Daily necessaties Toys Electric devices Healthcare goods Cosmetics Office supplies Interior items Others < 1 / year 1 / year 1 / 6 months 1 / 2-3 months 1 / month 2-3 / month 1-2 / week 3-4 /week max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 75% 25% min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' mean median VODT (JPY/day) VODT (JPY/day) Item Category (a) (b) (c) E-shopping Frequency Item Price (JPY) <1000 1000-2999 3000-4999 5000-7499 7500-9999 10000-15000 15000-20000 20000-30000 30000-50000 > 50000 Figure 5: VODT for different segments: focusing on (a) ordered item category, (b) ordered item price, and (c) e-shopping frequency of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Value of time slot shortening The distribution of VOTS obtained from MXL 4 is shown in Figure 6, and its statistics are reported in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The VOTS ranges from −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='827 to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='086 JPY/hour, with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='2% of the respondents having negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The mean and median are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='716 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='968 JPY/hour, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' As such, although a majority of users would be willing to pay for the shortening of the delivery time slot size, the payment would be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' This means that users do not highly value the reduction in time slot size in monetary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Since the size of a time slot (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', time window constraint) has a significant impact on last-mile delivery (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Nockold, 2001), this result may be an important finding for delivery demand management to improve logistics efficiency without a serious reduction in user satisfaction8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' VOTS (JPY/hour) Relative frequency Figure 6: Distribution of VOTS among respondents Table 6: Distribution characteristics of VOTS Mean Std Min 25% 50% 75% Max P(v < 0) VOTS (JPY/hour) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='716 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='792 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='827 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='979 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='968 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='216 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='042 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Concluding remarks This study analyzed e-commerce users’ preferences for delivery options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' To this end, we designed and implemented a stated choice survey, where users were asked to indicate which option of next, scheduled, and normal delivery they would choose to deliver the ordered item, and if they chose scheduled delivery, to jointly select the delivery date and time slot too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The stated choice data of 4,062 users living in the three major metropolitan areas of Japan were analyzed by estimating a mixed logit model, capturing users’ taste heterogeneity and substitution patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We also applied a semi-nonparametric approach by Fosgerau and Mabit (2013) to flexibly estimate the distributions of willingness-to-pay (WTP) for delivery attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The results of this study contribute to advancing the understanding of e-commerce user behavior, which plays a key role in delivery demand management as well as in designing urban logistics policies (Neslin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Holguín-Veras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Specifically, the present results 8Note that in our survey the time slot size ranged from two to four hours, and a two-hour slot might not have been sufficiently tight to increase the convenience of experienced e-commerce users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 20 suggest that delivery service attributes including fee, time, and time slot size are significant determinants of the choice of a delivery option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' We found that elderly people do not tend to choose user-oriented delivery options such as next-day and scheduled delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The frequency of teleworking and the presence of a delivery box also have a strong relationship with a low propensity to request scheduled delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' However, users who have a delivery box at home tend to choose next-day delivery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' that is, although installing a delivery box reduces the demand for scheduled delivery and the risk of delivery failure, it can lead to an increased demand for fast delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The analysis also revealed that the value of delivery time savings (VODT) is widely distributed among the respondents, and its maximum is 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='4 JPY per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Nevertheless, the median VODT is only 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='6 JPY per day, debunking the myth that everyone needs fast delivery and adding to the statement and results of Rai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Today many e-commerce marketplaces offer users fast delivery for free, imposing a strict time constraint on urban logistics, but our results suggest that more than half of users would be willing to wait an additional day if the delivery fee increased only by 26 JPY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' In terms of heterogeneity, VODT has a high value for users who frequently shop online and/or order expensive items, and varies according to the category of the item ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Additionally, the value of time slot shortening (VOTS) was found to be low and distributed with the median 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='0 JPY per hour, which means that users do not highly value the reduction in time slot size in monetary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Since time windows are important constraints for last-mile delivery, the result may suggest that lengthening the size of delivery time slots would be a way to significantly improve its efficiency without a serious reduction in user satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' The present WTP measures were calculated purely with respect to the delivery fee, not including item price (Gawor and Hoberg, 2019) or travel cost (Hsiao, 2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' therefore, the results of this study can be used for the level-of-service design for last-mile delivery, independent of the retailer/marketplace strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Given the findings of e-commerce user behavior, future work includes a study of the impact of the design of delivery attributes on the efficiency of last-mile delivery (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Agatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Since our model describes the choices of option, date and time slot for delivery, it would be possible to analyze both day-to-day and within-day dynamics of delivery demand and their impact on operational efficiency, combined with a multi-period vehicle routing problem (Archetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2015) or agent-based simulation (Sakai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Note that this study was carried out during the COVID-19 pandemic, under the declaration of a state of emergency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Although we collected detailed information on users’ lifestyles, including teleworking frequency, the results may still include the effects of unobserved variables specific during the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Therefore, the comparison of users’ preferences for delivery options before, during (and possibly after) the pandemic is considered another direction of future work that could add an important contribution to the literature (Choi, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Chowdhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Acknowledgements We are grateful to Yamato Holdings Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' for sending emails to their customers “Kuroneko Members” for invitation to our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' CRediT author statement Yuki Oyama: Conceptualization, Methodology, Software, Validation, Formal analysis, Investi- gation, Data Curation, Writing - Original Draft, Visualization, Supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Daisuke Fukuda: Methodology, Software, Validation, Formal analysis, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Naoto Imura: Investigation, Project administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Katsuhiro Nishinari: Resources, Funding acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' All authors approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 21 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Screen image of the stated choice survey ①翌日配送 ②翌々日以降の日時指定配送 ③通常配送(日時指定なし) 料 金 ○○○円 ○○○円 ○○○円 追 加 料 金 なし 土日祝日:+○○○円 夜間:+○○○円 なし 配 送 日 翌日 指定日 ○○○ 時 間 指 定 不可 ○○○ 不可 ①翌日配送 ②翌々日以降の日時指定配送 ③通常配送(日時指定なし) あなたは、下記の配送オプション①～③の中から、どれを選びますか。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 1 2 3 ②「翌々日以降の日時指定配送」を選択された方は、ご希望の ここからは、あなたが直近のオンラインショッピングで購入された ○○○ について、本日、再度購入すると仮定してお考えいただきます。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 配送の料金や配送日が異なる5つのパターンが順に表示されますので、各パターンにおいて 「翌日配送・日時指定配送・通常配送」の3つからどれを選択したいかお考えください。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 配達日を以下の中からお選 びください。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' ※『以下回答欄をクリックして現れるカレンダーから』お選びください。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' ※本日を注文日としてご回答をお願いします。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 9時～11時 11時～13時 13時～15時 15時～17時 17時～19時 ②「翌々日以降の日時指定配送」を選択された方は、ご希望の配達時間帯を以下の中から お選びください。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' ※○○○時間ごとの選択肢になっております。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 19時～21時 1 2 3 4 5 6 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='7: Screen image of a stated choice task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', original version of Figure 1 (in Japanese).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Items with three circles “◦ ◦ ◦” depend on respondent’s answers from RP survey or the attributes of the choice task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' References Agatz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Campbell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Fleischmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', Savelsbergh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' 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B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Urban freight policymaking: The role of qualitative and quantitative research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Transport Policy 56, 75–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content=' Hsiao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAyT4oBgHgl3EQfxfkk/content/2301.00666v1.pdf'} +page_content='H.' metadata={'source': 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0000000000000000000000000000000000000000..7a9959f1a3484c0a7a55fc5ad14faf838f4f73f5 --- /dev/null +++ b/gtFLT4oBgHgl3EQfZS_m/content/tmp_files/2301.12069v1.pdf.txt @@ -0,0 +1,1049 @@ +Dynamical Generation of Epsilon-Near-Zero Behaviour via Tracking and Feedback +Control +Jacob Masur,∗ Denys I. Bondar,† and Gerard McCaul‡ +Tulane University Department of Physics and Engineering Physics +(Dated: January 31, 2023) +To date, epsilon near zero (ENZ) responses, characterized by an infinite phase velocity, are pri- +marily achieved by applying a monochromatic light source to a tailored metamaterial. Here, we +derive the equations for inducing a dynamically generated broadband ENZ response in a large class +of many-body systems via tracking and feedback control. We further find that this response leads to +a current-energy relationship identical to that of an ideal inductor. Using a Fermi-Hubbard model, +we numerically confirm these results which have the potential to advance optical computation on +the nanoscale. +Introduction. +One of the principal research goals of +the twenty first century optics has been the develop- +ment and application of epsilon-near-zero (ENZ) materi- +als. These form a subclass of near zero index (NZI) ma- +terials, whose common feature is the extraordinary opti- +cal properties they possess [1–6]. Such materials exhibit +a near zero permittivity or permeability at a given fre- +quency, leading to a decoupling of the electric and mag- +netic fields [2–4, 7, 8]. Such behaviour has exceptional +potential to further our capabilities in the generation and +manipulation of nonlinear optical effects at the nanoscale +[9], and represent a vital element in many proposed quan- +tum technologies. For example, ENZ materials can be +used to control the phase and direction of emission in +antennas [10–12], while in optical circuits, the can be- +have effectively as components such as switches [13–18] +and modulators [19–25]. +To date, experimental realization of these materials is +primarily achieved through the construction of artificially +constructed metamaterials [26–34] and at the plasma fre- +quency in materials with a Drude dispersion relation [35– +43]. +Though convenient and compatible with comple- +mentary metal oxide semiconductor (CMOS) devices [1], +there are large intrinsic loses in the metals and semi- +conductors in which irradiation at the plasma frequency +induces an ENZ response [3]. This is precisely a conse- +quence of the definitional property of an ENZ, namely +its near-zero dielectric permittivity [44]. Moreover, the +bandwidth of current metamaterials is narrow, limiting +ENZ metamaterials’ usefulness in real-world applications +[45–48]. Recent work has shown however that coupling +metamaterials to circuits with negative reactance, so- +called non-Foster circuits [49], can increase their band- +width and ameliorate this issue [45, 50, 51]. +Given the current limitations of metamaterial design +and fabrication, one might ask if there is an alternative +route to achieving the sought-for optical properties ex- +hibited by an ENZ material. Since this behaviour char- +acterises the response of a material to optical driving, an +alternative to engineering the equilibrium properties of +materials might be to instead consider how the relation- +ship between driving and response can characterised and +manipulated. Specifically, we ask whether an ENZ-like +response be generated dynamically by designing a driving +field whose response functionally emulates that expected +from an ENZ, namely an infinite phase velocity. +To achieve this objective, it is necessary to import tech- +niques from the field of quantum control [52, 53]. +In +particular, tracking control [54–61] provides a method +for calculating driving fields such that the evolution of +a given observable tracks a specified trajectory. These +methods have been deployed profitably in the context of +atomic [57], molecular [55, 62, 63] and solid-state systems +[60, 61, 64, 65]. Complementary to this method is feed- +back control [66]. This has been implemented in a wide +variety of systems for applications ranging from cruise +control [67–69] to computer network protocols [70–73] +and atomic force microscopy [74, 75]. Recently the con- +cept of feedback control has been further extended into +the realm of quantum algorithm optimisation [76, 77], +but to date has not been applied to the manipulation of +quantum system’s nonlinear optical properties. +In this letter, we demonstrate that using either track- +ing or feedback control allows for the dynamical gener- +ation of an ENZ response in a many-body system. Fur- +thermore, we show that this behaviour mimics that of +an ideal inductor, raising the possibility that feedback +control mechanisms can be used to create optical ana- +logues to the fundamental components of electrical cir- +cuitry. +Significantly, the proposed method admits the +possibility of negative inductances, furnishing precisely +the type of non-Foster behaviour necessary for increas- +ing the bandwidth of metamaterials. +Tracking Control. +We begin by outlining the proce- +dure for tracking control, and detail how this is applied to +generate an ENZ response in a many-body system. We +consider a one-dimensional model throughout this letter +for the sake of simplicity, but the following analysis can +be generalized to higher dimensions. We take as a model +a general many-body fermionic system in which Ne elec- +trons are localized to lattice sites j = 1, ..., Ns with spin +σ =↑, ↓. The kinetic energy is described by the hopping +parameter t0, acting as an effective single band system +coupled to a driving laser via the dipole approximation. +arXiv:2301.12069v1 [physics.optics] 28 Jan 2023 + +2 +The effect of this is to introduce a phase exp[±iΦ(t)] onto +the hopping parameter, where Φ(t) is the Peierls phase +[78, 79]. The Hamiltonian is then described by (in atomic +units): +ˆH(t) = −t0 +� +j,σ +� +e−iΦ(t)ˆc† +j,σˆcj+1,σ + h.c. +� ++ ˆU +(1) +where ˆc† +j,σ and ˆcj,σ are, respectively, the canonical +fermionic creation and annihilation operators, and ˆU de- +scribes the potential resulting from electron-electron re- +pulsion. +Optical driving induces a current in the system which +is calculated by the expectation of the current operator. +The current operator is derived by a continuity equation +[60, 61, 64], giving: +ˆJ(t) = −iat0 +� +j,σ +� +e−iΦ(t)ˆc† +j,σˆcj+1,σ − h.c. +� +(2) +where a is the lattice constant. +The optical emission +of a driven material is determined exclusively by this +current, and forms the basis for the study of optical re- +sponse in the solid state [80–83]. This is due to the fact +that under the dipole approximation, the emitted field +is proportional to the dipole acceleration +dJ(t) +dt , where +J(t) ≡ ⟨ψ(t)| ˆJ(t)|ψ(t)⟩. +For this reason, it is via the +control of this observable that an ENZ response can be +generated. +The condition for an ENZ response can be formulated +in terms of the relationship between the driving field and +response. The infinite phase velocity implied by a near- +zero dielectric permittivity implies a zero lag between +the applied and emitted electrical fields, i.e. Eout(t) ∝ +Ein(t). This relationship can be expressed in terms of +Φ(t) and J by recalling that Ein(t) = − 1 +a +dΦ(t) +dt +[60]. Thus, +the ENZ criteria can be succinctly represented by +dJ(t) +dt += − 1 +aL +dΦ(t) +dt +, +(3) +where L is a non-zero real constant. +Integrating both +sides with respect to time, we obtain a condition for an +ENZ response JENZ, +JENZ(t) = − 1 +aLΦENZ(t) + C +(4) +which can be implemented directly in a tracking control +framework. +The laser field − 1 +adΦENZ(t)/dt that induces an ENZ +response is given by the solution to the implicit equation +1 +aLΦENZ(ψ) = 2at0R(ψ) sin [ΦENZ(ψ) − θ(ψ)] + J(0). +(5) +Here, J(0) is the current expectation of the initial state, +and R(ψ) and θ(ψ) are, respectively, the real and imag- +inary components of the nearest neighbor expectation +⟨� +j,σ ˆc† +j,σˆcj+1,σ⟩. The derivation of Eq. (5) is provided +in Sec. I of the supplemental materials, and the existence +of such a field and the conditions under which it is unique +are demonstrated in Sec. II [84]. Finally, it is worth not- +ing that the physical validity of this tracking equation in +the dt → 0 limit can be checked by comparison to the +Ehrenfest theorem for J(t) (see e.g. [61]). +Feedback Control. +An alternative route to obtaining +an ENZ response can be obtained by using the mate- +rial’s own response to control the input field. Here, we +implement a simple form of feedback control known as +proportional feedback control, a schematic for which is +given in Fig. 1. In this scenario, we begin by applying a +transform limited pulse with electric field Etl(t) which is +subsequently modified by the introduction of the control +field +u(t) = kp +� +LdJ(t) +dt +− Ein(t) +� +(6) +where kp is a positive constant representing the ampli- +fication gain. The control field enforces the ENZ con- +dition (3) by correcting the input field when the error +e = L dJ +dt − Ein is nonzero. +The combined input to the system when we correct the +transform limited field with the control field (6) is then +Ein(t) = Etl(t) + u(t). +(7) +Substituting Ein(t) as given above into Eq. (6) together +with some algebraic rearrangement allows us to express +it as +u(t) = +kp +1 + kp +� +LdJ(t) +dt +− Etl(t) +� +, +(8) +which in the limit kp → ∞, gives u(t) = L dJ(t) +dt +− Etl(t). +Then, by Eq. (7), the input field approaches +Ein(t) → EENZ(t) = LdJ(t) +dt +, +(9) +the ENZ condition. Hence, given sufficiently strong am- +plification kp, the feedback scheme Fig. 1 will drive any +system to produce ENZ response. +To computationally illustrate this, we calculate the +Peierls phase by multiplying Eq. +(7) by −a and inte- +grating over time: +Φin(t) = Φtl(t) + +kp +1 + kp +[−aL (J(t) − J(0)) − Φtl(t)] . +(10) +Physical Interpretation of L. +A natural question is +how the free parameter L should be interpreted. When +considering Eq. (3), it bears a strong resemblance to the +dynamics of an inductor [85]. For an inductor with intrin- +sic inductance L and length d subject to a homogeneous +time dependent electric field E(t), we have +dI(t) +dt += d +LE(t) +(11) + +3 +∑ +∑ +amplifier +Unshaped +(transform +limited) input +𝐸tl(𝑡) +𝐸in(𝑡) +driven system +Response of +driven system +𝔏 𝑑𝐽 +𝑑𝑡 − 𝐸in(𝑡) +𝕷 +-1 +Identical input and output +𝑑𝐽 +𝑑𝑡 +𝔏 𝑑𝐽 +𝑑𝑡 +𝑢 𝑡 + 𝐸tl(𝑡) +𝑢(𝑡) +FIG. 1. The schematic depicting the setup of ENZ feedback control. The response of the system is amplified or dampened by +some value proportional to L, from this the input field is subtracted, and the resulting field is amplified by kp before being +combined with the transform limited pulse. +Here, I(t) is the macroscopic current which varies only in +time. The total energy E stored by an inductor at time +t is +E(t) − E(0) = L +2 +� +I2(t) − I2(0) +� +. +(12) +In the scenario we consider, the dynamics take place on +the nanoscale, so we substitute the macroscopic current +for the current expectation J(x, t) = ⟨ ˆJ(x, t)⟩. +Then, +we discretise with J(x, t) → +1 +aJj(t) and +� +dx → a � +j +to obtain the relationship between energy and current in +a system defined by Eq. (1). The power stored by the +solid-state system will be +P(t) = E(t) +� +j +Jj(t) = E(t)J(t), +(13) +which after time integration yields +E(t) − E(0) = −1 +a +� t +0 +dΦ(t′) +dt′ +J(t′)dt′. +(14) +Upon inserting the specific form of Φ(t) imposed by the +ENZ condition given in Eq. (4), we obtain an identical +energy-current relationship to that of the inductor shown +in Eq. (12): +E(t) − E(0) = L +2 +� +J2(t) − J2(0) +� +, +(15) +and hence we are able to identify the value of L chosen +as an effective inductance. +Simulation results. +We now demonstrate that both +tracking and feedback control induce the desired ENZ +response via numerical calculations. Though the Hamil- +tonian in Eq. (1) and subsequent analysis of the field re- +quired to induce an ENZ response are valid for any ˆU that +will commute with electron number operators [64, 65], for +simulations we take our system to be a half-filled Fermi- +Hubbard model (Ns = Ne) [86], where N↑ = N↓ and +onsite interaction of the form +ˆU = U +N +� +j=1 +ˆnj,↑ˆnj,↓ +(16) +where U parametrises interaction energy and ˆnj,σ = +ˆc† +j,σˆcj,σ is the number operator for site j and spin σ. +Numerical simulations of both types of control are per- +formed using the QuSpin package in Python [87]. +To +avoid trivial solutions of zero field and current, after ini- +tialising the simulation in the ground state of the sys- +tem, we then excite it with 10 cycles of a transform lim- +ited pump pulse of strength F0 = 10 MV +cm and frequency +ω0 = 32.9 THz. After this pre-pump, we implement the +chosen control procedure. The system is evolved using +the DOP853 algorithm [88] implemented in SciPy [89] +for a total time equal to the duration of the pump pulse +T = 2πM +ω0 +where M is the number of cycles. At each time +step, we solve Eq. (5) for tracking control and Eq. (10) +for feedback control. +A specific example of tracking and feedback control is +shown in Fig. 2. Most importantly, when +��2a2t0NsL +�� ≤ 1 +we expect the ENZ response to be unique (see Sec. II +of the supplemental material [84]), and consequently we +find that both feedback and tracking control yield identi- +cal solutions. Note that the derived driving field exhibits +a high degree of complexity and bandwidth, which would + +4 +0 +50 +100 +150 +200 +250 +0.02 +0.00 +0.02 +(a) +J(t) +0 +50 +100 +150 +200 +250 +Time +0.02 +0.00 +0.02 +(b) +(t) +a ++ J(0) +Current +FIG. 2. +Plots of the current induced (solid blue line) and +− Φ(t) +aL +J(0) (dashed orange line) for Fermi-Hubbard material +simulated exactly with 10 sites, +U +t0 = 1, a = 4 ˚A, and L = 1. +Plot (a) shows the response induced by tracking control and +plot (b) shows the response induced by feedback control with +kp = 10000. +be challenging to reproduce with current pulse shaping +technology. There is a real prospect, however, of con- +trolling the input electric field using the materials own +response via feedback control. +0.0002 +0.0000 +0.0002 +0.0004 +0.0006 +0.0008 +J2(t) +J2(0) +0.0006 +0.0004 +0.0002 +0.0000 +0.0002 +0.0004 +(t) +(0) += +2.0 += +1.0 += +0.5 += 0.5 += 1.0 += 2.0 +FIG. 3. The relationship between the change in energy and +the change in current induced by an ENZ pulse for a Fermi- +Hubbard material simulated exactly with 10 sites, +U +t0 = 0.5, +a = 4 ˚A. The slope of each line is L +2 . +As predicted by Eq. (15) and demonstrated in Fig. 3, +the relationship between the change in energy and the +change in the square of the current is exactly linear and +the slope increases with increasing L. The relationship +holds for positive as well as negative values of L, and thus +allows us to induce a non-Foster response by specifying +L < 0. +Discussion. +In this letter, we have demonstrated the +possibility of inducing an ENZ response in a large class +of solid-state systems using either tracking or feedback +control. Tracking control gives extremely precise control +over the induced current, but faces significant hurdles to +practical implementation due to the bandwidth of driv- +ing pulses. In contrast, feedback control replaces pulse +shaping with amplification, and can therefore be imple- +mented relatively straightforwardly. In the current work, +we have assumed an instantaneous feedback, but this can +easily be adapted to a finite optical path by preparing two +identical copies of the system, and using feeding the re- +sponse of one system to the initial pulse into the input +of the second at a suitable time delay. Given the equiva- +lence in the solutions they predict, it is possible that the +most direct route to implementing many tracking control +proposals will be to adapt them to a feedback procedure. +It is also worth noting that the particular response one +obtains from tracking will depend on the initial state, +and hence the specific pump profile one applies before +tracking. This offers a route to tailoring the response ob- +tained. For example, one could generate a maximum or +minimum bandwidth for the response by performing an +optimisation over pump pulse parameters, conditioned +on the desired property. +The equations developed here allow one to tune the +scaling of the system response relative to the input field +by a factor L. Analytically, this parameter is shown to +be analogous to macroscopic inductance (15), and we +demonstrate that the predicted energy-current relation- +ship is realized in simulations in Fig. 3. There are no +restrictions on the sign of this inductance can take, mean- +ing it is possible to specify L < 0 to induce a non-Foster +reactance. Since coupling negative inductance materials +to metamaterials has been shown to increase metamate- +rials’ bandwidth, not only can we drive an ENZ response +in these materials, but optical feedback control may offer +an indirect route to realising ENZ behaviour in metama- +terials by broadening their bandwidth. +The mapping between the ENZ response and an ideal +inductor also hints at the possibility that feedback con- +trol may be employed to develop quantum optical analogs +to other basic components of classical electrical circuits, +such as resistance and capacitance. The possibility of an +all optical computation has already been demonstrated +in [90], and with the limit of Moore’s law being ap- +proached [91], the need for alternative computing plat- +forms is pressing. Optical feedback control therefore rep- +resents a potential opportunity to take advantage of the +inherent nonlinearities present in quantum optics, in or- +der to develop computing at the nanoscale. +Acknowledgement. +This work was supported by the +W. M. Keck Foundation and by Army Research Of- + +5 +fice (ARO) (grant W911NF-19-1-0377, program manager +Dr. James Joseph, and cooperative agreement W911NF- +21-2-0139). The views and conclusions contained in this +document are those of the authors and should not be +interpreted as representing the official policies, either ex- +pressed or implied, of ARO or the U.S. Government. +The U.S. Government is authorized to reproduce and dis- +tribute reprints for Government purposes notwithstand- +ing any copyright notation herein. +∗ jacobmasur1@gmail.com +† dbondar@tulane.edu +‡ gmccaul@tulane.edu +[1] J. Wu, Z. T. Xie, Y. Sha, H. Y. Fu, and Q. Li, Epsilon- +near-zero photonics: infinite potentials, Photon. Res. 9, +1616 (2021). +[2] R. W. 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Eeckhout, Is moore’s law slowing down? what’s next?, +IEEE Micro 37, 4 (2017). + +1 +Supplemental Materials for “Dynamical Generation of Epsilon-Near-Zero Behaviour +via Tracking and Feedback Control” +I. +DERIVATION OF THE ENZ FIELD DETERMINED BY TRACKING CONTROL +We begin our derivation from the condition for an ENZ-like response: +JENZ(t) = − 1 +aLΦENZ(t) + C. +(S1) +Note that as Φ(t) is proportional to the time integral of the field Ein(t), we require Φ(0) = 0, which fixes C = J(0). +In order to perform tracking control, it is necessary to invert the expectation of the current operator +ˆJ(t) = −iat0 +� +j,σ +� +e−iΦ(t)ˆc† +j,σˆcj+1,σ − h.c. +� +(S2) +in order to express the control field in terms of J(t), the expectation to be tracked. If the tracking condition is fulfilled +at a time t (which will be identically true at t = 0), at time t + dt, we require: +⟨ψ(t + dt)| ˆJ(t + dt)|ψ(t + dt)⟩ = −ΦENZ(t + dt) +aL ++ J(0). +(S3) +In order to evaluate Eq. (S3), we obtain the state of the system at time t + dt by the first order approximation to +the Schr¨odinger equation: +|ψ(t + dt)⟩ = |ψ(t)⟩ − idt ˆH(t)|ψ(t)⟩ + O(dt2). +(S4) +Substituting this into Eq. (S3) gives +− ΦENZ(t + dt) +aL ++ J(0) = ⟨ψ(t)| ˆJ(t + dt)|ψ(t)⟩ + idt⟨ψ(t)|[ ˆH(t), ˆJ(t + dt)]|ψ(t)⟩ + O(dt2). +(S5) +The commutator in the second term of the RHS can be expanded to +[ ˆH(t), ˆJ(t+dt)] = iat0 +� +e−iΦENZ(t+dt) � +t0eiΦENZ(t))[ ˆK†, ˆK] −[ ˆU, ˆK] +� ++ eiΦENZ(t+dt) � +[ ˆU, ˆK†] − t0e−iΦENZ(t)[ ˆK, ˆK†] +�� +, +(S6) +where for convenience we have defined the nearest neighbor operator ˆK as +ˆK = +� +j,σ +ˆc† +j,σˆcj+1,σ. +(S7) +Under periodic boundary conditions (which we assume throughout) [ ˆK, ˆK†] = 0, meaning that it is possible rewrite +Eq. (S3) by inserting Eq. (S6), together with the definition of the current operator given by Eq. (S2): +− ΦENZ(t + dt) +aL ++ J(0) = −iat0e−iΦENZ(t+dt) � +⟨ψ| ˆK|ψ⟩ +idt⟨ψ|[ ˆU, ˆK]|ψ⟩ +� ++ h.c. + O(dt2) +(S8) +with the definition |ψ⟩ ≡ |ψ(t)⟩. +This expression is further simplified by representing the expectation values in polar form as follows: +⟨ψ| ˆK|ψ⟩ = ⟨ψ| ˆK†|ψ⟩† = R(ψ)eiθ(ψ) +(S9) +⟨ψ|[ ˆU, ˆK]|ψ⟩ = −⟨ψ|[ ˆU, ˆK†]|ψ⟩† = P(ψ)eiλ(ψ) +(S10) +where we use the argument ψ to indicate that the parameter is a functional of the state of the system |ψ(t)⟩. +Thus, (S8) becomes +−ΦENZ(t + dt) +aL ++ J(0) = −2at0R(ψ) sin [ΦENZ(t + dt) − θ(ψ)] ++2at0P(ψ) cos [ΦENZ(t + dt) − λ(ψ)] dt + O(dt2). +(S11) + +2 +Since the system in question is finite, each term in the above equation is bounded, so we can take the limit dt → 0 to +obtain an implicit equation for ΦENZ: +1 +aLΦENZ(ψ) = 2at0R(ψ) sin [ΦENZ(ψ) − θ(ψ)] + J(0) +(S12) +where we have replaced the argument of ΦENZ with ψ as all other variables are functionals of ψ, and hence the +time dependence of ΦENZ enters solely through the state of the system |ψ(t)⟩. The solution to this implicit equation +corresponds to the the laser field − 1 +adΦENZ(t)/dt that will induce an ENZ response. +II. +EXISTENCE AND UNIQUENESS OF THE ENZ FIELD +Before using (S12) to induce an ENZ response, it is important to consider under what conditions such a field exists. +It is useful to recast the problem such that, instead of finding a solution, Φ(ψ), to (S12), we find the field for which +fψ(Φ) = sin [Φ − θ(ψ)] − +Φ +Y (ψ) + G(ψ) +(S13) +has a zero. Where, for simplicity, we have defined G(ψ) = R(ψ0) +R(ψ) sin[θ(ψ0)] and Y (ψ) = 2a2t0R(ψ)L. Note that we +treat Φ as a scalar parameter to this function, and the constants are entirely determined by the current state of the +system. +Theorem 1 - If Φ is a solution to (S13), then Φ − Y (ψ)G(ψ) lies within the interval [−|Y (ψ)|, |Y (ψ)|]. +Proof: |sin [Φ − θ]| ≤ 1, so any solution must obey +���� +Φ +Y (ψ) − G(ψ) +���� ≤ 1. +(S14) +It follows that +− |Y (ψ)| ≤ Φ − Y (ψ)G(ψ) ≤ |Y (ψ)|□. +(S15) +Theorem 2 - At least one solution to (S13) exists. +Proof: First, we evaluate the value of (S13) at the endpoints of the interval in which all possible solutions lie. Using +ξ1 = −Y (ψ) + Y (ψ)G(ψ) and ξ2 = Y (ψ) + Y (ψ)G(ψ) the value of the function (S13) at the two endpoints of the +solution interval is +fψ(ξ1) = sin [ξ1 − θ(ψ)] + 1 ≥ 0 +(S16) +fψ(ξ2) = sin [ξ2 − θ(ψ)] − 1 ≤ 0. +(S17) +If at least one of fψ(ξ1) and fψ(ξ2) is zero, then there is at least one solution at one (or both) of the endpoints. If +both are non-zero, then fψ(ξ1) is positive and fψ(ξ2) is negative, and since f is continuous, there must be at least +one zero in the interval [−|Y (ψ)| + Y (ψ)G(ψ), |Y (ψ)| + Y (ψ)G(ψ)] □. +Theorem 3 - If |Y (ψ)| ≤ 1, there exists a unique solution to (S12). +Proof: The derivative of (S13) is +f ′ +ψ(Φ) = cos [Φ − θ(ψ)] − +1 +Y (ψ) +(S18) +Hence, if |Y (ψ)| ≤ 1, f ′(Φ) is non-positive/negative and therefore f(Φ) is monotonic non-increasing/decreasing. Since +fψ(Φ) = 0 for some Φ by Theorem 2, this must be the only value for which fψ(Φ) obtains zero, and hence it is the +unique solution to (S12) □. Note that since R(ψ) ≤ Ns, the ENZ field is guaranteed to be unique regardless of the +current state when +��2a2t0NsL +�� ≤ 1. + diff --git a/gtFLT4oBgHgl3EQfZS_m/content/tmp_files/load_file.txt b/gtFLT4oBgHgl3EQfZS_m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e63a129726dbe2355233ab835c8c3d5b88f32702 --- /dev/null +++ b/gtFLT4oBgHgl3EQfZS_m/content/tmp_files/load_file.txt @@ -0,0 +1,886 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf,len=885 +page_content='Dynamical Generation of Epsilon-Near-Zero Behaviour via Tracking and Feedback Control Jacob Masur,∗ Denys I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Bondar,† and Gerard McCaul‡ Tulane University Department of Physics and Engineering Physics (Dated: January 31, 2023) To date, epsilon near zero (ENZ) responses, characterized by an infinite phase velocity, are pri- marily achieved by applying a monochromatic light source to a tailored metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Here, we derive the equations for inducing a dynamically generated broadband ENZ response in a large class of many-body systems via tracking and feedback control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' We further find that this response leads to a current-energy relationship identical to that of an ideal inductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Using a Fermi-Hubbard model, we numerically confirm these results which have the potential to advance optical computation on the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' One of the principal research goals of the twenty first century optics has been the develop- ment and application of epsilon-near-zero (ENZ) materi- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' These form a subclass of near zero index (NZI) ma- terials, whose common feature is the extraordinary opti- cal properties they possess [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Such materials exhibit a near zero permittivity or permeability at a given fre- quency, leading to a decoupling of the electric and mag- netic fields [2–4, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Such behaviour has exceptional potential to further our capabilities in the generation and manipulation of nonlinear optical effects at the nanoscale [9], and represent a vital element in many proposed quan- tum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' For example, ENZ materials can be used to control the phase and direction of emission in antennas [10–12], while in optical circuits, the can be- have effectively as components such as switches [13–18] and modulators [19–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' To date, experimental realization of these materials is primarily achieved through the construction of artificially constructed metamaterials [26–34] and at the plasma fre- quency in materials with a Drude dispersion relation [35– 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Though convenient and compatible with comple- mentary metal oxide semiconductor (CMOS) devices [1], there are large intrinsic loses in the metals and semi- conductors in which irradiation at the plasma frequency induces an ENZ response [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' This is precisely a conse- quence of the definitional property of an ENZ, namely its near-zero dielectric permittivity [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Moreover, the bandwidth of current metamaterials is narrow, limiting ENZ metamaterials’ usefulness in real-world applications [45–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Recent work has shown however that coupling metamaterials to circuits with negative reactance, so- called non-Foster circuits [49], can increase their band- width and ameliorate this issue [45, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Given the current limitations of metamaterial design and fabrication, one might ask if there is an alternative route to achieving the sought-for optical properties ex- hibited by an ENZ material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Since this behaviour char- acterises the response of a material to optical driving, an alternative to engineering the equilibrium properties of materials might be to instead consider how the relation- ship between driving and response can characterised and manipulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Specifically, we ask whether an ENZ-like response be generated dynamically by designing a driving field whose response functionally emulates that expected from an ENZ, namely an infinite phase velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' To achieve this objective, it is necessary to import tech- niques from the field of quantum control [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' In particular, tracking control [54–61] provides a method for calculating driving fields such that the evolution of a given observable tracks a specified trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' These methods have been deployed profitably in the context of atomic [57], molecular [55, 62, 63] and solid-state systems [60, 61, 64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Complementary to this method is feed- back control [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' This has been implemented in a wide variety of systems for applications ranging from cruise control [67–69] to computer network protocols [70–73] and atomic force microscopy [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Recently the con- cept of feedback control has been further extended into the realm of quantum algorithm optimisation [76, 77], but to date has not been applied to the manipulation of quantum system’s nonlinear optical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' In this letter, we demonstrate that using either track- ing or feedback control allows for the dynamical gener- ation of an ENZ response in a many-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Fur- thermore, we show that this behaviour mimics that of an ideal inductor, raising the possibility that feedback control mechanisms can be used to create optical ana- logues to the fundamental components of electrical cir- cuitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Significantly, the proposed method admits the possibility of negative inductances, furnishing precisely the type of non-Foster behaviour necessary for increas- ing the bandwidth of metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Tracking Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' We begin by outlining the proce- dure for tracking control, and detail how this is applied to generate an ENZ response in a many-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' We consider a one-dimensional model throughout this letter for the sake of simplicity, but the following analysis can be generalized to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' We take as a model a general many-body fermionic system in which Ne elec- trons are localized to lattice sites j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=', Ns with spin σ =↑, ↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The kinetic energy is described by the hopping parameter t0, acting as an effective single band system coupled to a driving laser via the dipole approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='12069v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='optics] 28 Jan 2023 2 The effect of this is to introduce a phase exp[±iΦ(t)] onto the hopping parameter, where Φ(t) is the Peierls phase [78, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The Hamiltonian is then described by (in atomic units): ˆH(t) = −t0 � j,σ � e−iΦ(t)ˆc† j,σˆcj+1,σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' � + ˆU (1) where ˆc† j,σ and ˆcj,σ are, respectively, the canonical fermionic creation and annihilation operators, and ˆU de- scribes the potential resulting from electron-electron re- pulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Optical driving induces a current in the system which is calculated by the expectation of the current operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The current operator is derived by a continuity equation [60, 61, 64], giving: ˆJ(t) = −iat0 � j,σ � e−iΦ(t)ˆc† j,σˆcj+1,σ − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' � (2) where a is the lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The optical emission of a driven material is determined exclusively by this current, and forms the basis for the study of optical re- sponse in the solid state [80–83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' This is due to the fact that under the dipole approximation, the emitted field is proportional to the dipole acceleration dJ(t) dt , where J(t) ≡ ⟨ψ(t)| ˆJ(t)|ψ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' For this reason, it is via the control of this observable that an ENZ response can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The condition for an ENZ response can be formulated in terms of the relationship between the driving field and response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The infinite phase velocity implied by a near- zero dielectric permittivity implies a zero lag between the applied and emitted electrical fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Eout(t) ∝ Ein(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' This relationship can be expressed in terms of Φ(t) and J by recalling that Ein(t) = − 1 a dΦ(t) dt [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Thus, the ENZ criteria can be succinctly represented by dJ(t) dt = − 1 aL dΦ(t) dt , (3) where L is a non-zero real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Integrating both sides with respect to time, we obtain a condition for an ENZ response JENZ, JENZ(t) = − 1 aLΦENZ(t) + C (4) which can be implemented directly in a tracking control framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The laser field − 1 adΦENZ(t)/dt that induces an ENZ response is given by the solution to the implicit equation 1 aLΦENZ(ψ) = 2at0R(ψ) sin [ΦENZ(ψ) − θ(ψ)] + J(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (5) Here, J(0) is the current expectation of the initial state, and R(ψ) and θ(ψ) are, respectively, the real and imag- inary components of the nearest neighbor expectation ⟨� j,σ ˆc† j,σˆcj+1,σ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (5) is provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' I of the supplemental materials, and the existence of such a field and the conditions under which it is unique are demonstrated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' II [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Finally, it is worth not- ing that the physical validity of this tracking equation in the dt → 0 limit can be checked by comparison to the Ehrenfest theorem for J(t) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' [61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Feedback Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' An alternative route to obtaining an ENZ response can be obtained by using the mate- rial’s own response to control the input field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Here, we implement a simple form of feedback control known as proportional feedback control, a schematic for which is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' In this scenario, we begin by applying a transform limited pulse with electric field Etl(t) which is subsequently modified by the introduction of the control field u(t) = kp � LdJ(t) dt − Ein(t) � (6) where kp is a positive constant representing the ampli- fication gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The control field enforces the ENZ con- dition (3) by correcting the input field when the error e = L dJ dt − Ein is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The combined input to the system when we correct the transform limited field with the control field (6) is then Ein(t) = Etl(t) + u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (7) Substituting Ein(t) as given above into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (6) together with some algebraic rearrangement allows us to express it as u(t) = kp 1 + kp � LdJ(t) dt − Etl(t) � , (8) which in the limit kp → ∞, gives u(t) = L dJ(t) dt − Etl(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Then, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (7), the input field approaches Ein(t) → EENZ(t) = LdJ(t) dt , (9) the ENZ condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Hence, given sufficiently strong am- plification kp, the feedback scheme Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 1 will drive any system to produce ENZ response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' To computationally illustrate this, we calculate the Peierls phase by multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (7) by −a and inte- grating over time: Φin(t) = Φtl(t) + kp 1 + kp [−aL (J(t) − J(0)) − Φtl(t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (10) Physical Interpretation of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' A natural question is how the free parameter L should be interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' When considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (3), it bears a strong resemblance to the dynamics of an inductor [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' For an inductor with intrin- sic inductance L and length d subject to a homogeneous time dependent electric field E(t), we have dI(t) dt = d LE(t) (11) 3 ∑ ∑ amplifier Unshaped (transform limited) input 𝐸tl(𝑡) 𝐸in(𝑡) driven system Response of driven system 𝔏 𝑑𝐽 𝑑𝑡 − 𝐸in(𝑡) 𝕷 1 Identical input and output 𝑑𝐽 𝑑𝑡 𝔏 𝑑𝐽 𝑑𝑡 𝑢 𝑡 + 𝐸tl(𝑡) 𝑢(𝑡) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The schematic depicting the setup of ENZ feedback control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The response of the system is amplified or dampened by some value proportional to L, from this the input field is subtracted, and the resulting field is amplified by kp before being combined with the transform limited pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Here, I(t) is the macroscopic current which varies only in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The total energy E stored by an inductor at time t is E(t) − E(0) = L 2 � I2(t) − I2(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (12) In the scenario we consider, the dynamics take place on the nanoscale, so we substitute the macroscopic current for the current expectation J(x, t) = ⟨ ˆJ(x, t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Then, we discretise with J(x, t) → 1 aJj(t) and � dx → a � j to obtain the relationship between energy and current in a system defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The power stored by the solid-state system will be P(t) = E(t) � j Jj(t) = E(t)J(t), (13) which after time integration yields E(t) − E(0) = −1 a � t 0 dΦ(t′) dt′ J(t′)dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (14) Upon inserting the specific form of Φ(t) imposed by the ENZ condition given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (4), we obtain an identical energy-current relationship to that of the inductor shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (12): E(t) − E(0) = L 2 � J2(t) − J2(0) � , (15) and hence we are able to identify the value of L chosen as an effective inductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' We now demonstrate that both tracking and feedback control induce the desired ENZ response via numerical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Though the Hamil- tonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (1) and subsequent analysis of the field re- quired to induce an ENZ response are valid for any ˆU that will commute with electron number operators [64, 65], for simulations we take our system to be a half-filled Fermi- Hubbard model (Ns = Ne) [86], where N↑ = N↓ and onsite interaction of the form ˆU = U N � j=1 ˆnj,↑ˆnj,↓ (16) where U parametrises interaction energy and ˆnj,σ = ˆc† j,σˆcj,σ is the number operator for site j and spin σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Numerical simulations of both types of control are per- formed using the QuSpin package in Python [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' To avoid trivial solutions of zero field and current, after ini- tialising the simulation in the ground state of the sys- tem, we then excite it with 10 cycles of a transform lim- ited pump pulse of strength F0 = 10 MV cm and frequency ω0 = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='9 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' After this pre-pump, we implement the chosen control procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The system is evolved using the DOP853 algorithm [88] implemented in SciPy [89] for a total time equal to the duration of the pump pulse T = 2πM ω0 where M is the number of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' At each time step, we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (5) for tracking control and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (10) for feedback control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' A specific example of tracking and feedback control is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Most importantly, when ��2a2t0NsL �� ≤ 1 we expect the ENZ response to be unique (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' II of the supplemental material [84]), and consequently we find that both feedback and tracking control yield identi- cal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Note that the derived driving field exhibits a high degree of complexity and bandwidth, which would 4 0 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='02 (a) J(t) 0 50 100 150 200 250 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='02 (b) (t) a + J(0) Current FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Plots of the current induced (solid blue line) and − Φ(t) aL +J(0) (dashed orange line) for Fermi-Hubbard material simulated exactly with 10 sites, U t0 = 1, a = 4 ˚A, and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Plot (a) shows the response induced by tracking control and plot (b) shows the response induced by feedback control with kp = 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' be challenging to reproduce with current pulse shaping technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' There is a real prospect, however, of con- trolling the input electric field using the materials own response via feedback control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0008 J2(t) J2(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0004 (t) (0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The relationship between the change in energy and the change in current induced by an ENZ pulse for a Fermi- Hubbard material simulated exactly with 10 sites, U t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='5, a = 4 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The slope of each line is L 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' As predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (15) and demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 3, the relationship between the change in energy and the change in the square of the current is exactly linear and the slope increases with increasing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The relationship holds for positive as well as negative values of L, and thus allows us to induce a non-Foster response by specifying L < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' In this letter, we have demonstrated the possibility of inducing an ENZ response in a large class of solid-state systems using either tracking or feedback control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Tracking control gives extremely precise control over the induced current, but faces significant hurdles to practical implementation due to the bandwidth of driv- ing pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' In contrast, feedback control replaces pulse shaping with amplification, and can therefore be imple- mented relatively straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' In the current work, we have assumed an instantaneous feedback, but this can easily be adapted to a finite optical path by preparing two identical copies of the system, and using feeding the re- sponse of one system to the initial pulse into the input of the second at a suitable time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Given the equiva- lence in the solutions they predict, it is possible that the most direct route to implementing many tracking control proposals will be to adapt them to a feedback procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' It is also worth noting that the particular response one obtains from tracking will depend on the initial state, and hence the specific pump profile one applies before tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' This offers a route to tailoring the response ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' For example, one could generate a maximum or minimum bandwidth for the response by performing an optimisation over pump pulse parameters, conditioned on the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The equations developed here allow one to tune the scaling of the system response relative to the input field by a factor L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Analytically, this parameter is shown to be analogous to macroscopic inductance (15), and we demonstrate that the predicted energy-current relation- ship is realized in simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' There are no restrictions on the sign of this inductance can take, mean- ing it is possible to specify L < 0 to induce a non-Foster reactance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Since coupling negative inductance materials to metamaterials has been shown to increase metamate- rials’ bandwidth, not only can we drive an ENZ response in these materials, but optical feedback control may offer an indirect route to realising ENZ behaviour in metama- terials by broadening their bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The mapping between the ENZ response and an ideal inductor also hints at the possibility that feedback con- trol may be employed to develop quantum optical analogs to other basic components of classical electrical circuits, such as resistance and capacitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The possibility of an all optical computation has already been demonstrated in [90], and with the limit of Moore’s law being ap- proached [91], the need for alternative computing plat- forms is pressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Optical feedback control therefore rep- resents a potential opportunity to take advantage of the inherent nonlinearities present in quantum optics, in or- der to develop computing at the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' This work was supported by the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Keck Foundation and by Army Research Of- 5 fice (ARO) (grant W911NF-19-1-0377, program manager Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' James Joseph, and cooperative agreement W911NF- 21-2-0139).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either ex- pressed or implied, of ARO or the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Government is authorized to reproduce and dis- tribute reprints for Government purposes notwithstand- ing any copyright notation herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' ∗ jacobmasur1@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='com † dbondar@tulane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='edu ‡ gmccaul@tulane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='edu [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' T.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Jacobs, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Bondar, Towards sin- gle atom computing via high harmonic generation, EPJ PLus (accepted) (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' [91] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Eeckhout, Is moore’s law slowing down?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' what’s next?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=', IEEE Micro 37, 4 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' 1 Supplemental Materials for “Dynamical Generation of Epsilon-Near-Zero Behaviour via Tracking and Feedback Control” I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' DERIVATION OF THE ENZ FIELD DETERMINED BY TRACKING CONTROL We begin our derivation from the condition for an ENZ-like response: JENZ(t) = − 1 aLΦENZ(t) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S1) Note that as Φ(t) is proportional to the time integral of the field Ein(t), we require Φ(0) = 0, which fixes C = J(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' In order to perform tracking control, it is necessary to invert the expectation of the current operator ˆJ(t) = −iat0 � j,σ � e−iΦ(t)ˆc† j,σˆcj+1,σ − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' � (S2) in order to express the control field in terms of J(t), the expectation to be tracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' If the tracking condition is fulfilled at a time t (which will be identically true at t = 0), at time t + dt, we require: ⟨ψ(t + dt)| ˆJ(t + dt)|ψ(t + dt)⟩ = −ΦENZ(t + dt) aL + J(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S3) In order to evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S3), we obtain the state of the system at time t + dt by the first order approximation to the Schr¨odinger equation: |ψ(t + dt)⟩ = |ψ(t)⟩ − idt ˆH(t)|ψ(t)⟩ + O(dt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S4) Substituting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S3) gives − ΦENZ(t + dt) aL + J(0) = ⟨ψ(t)| ˆJ(t + dt)|ψ(t)⟩ + idt⟨ψ(t)|[ ˆH(t), ˆJ(t + dt)]|ψ(t)⟩ + O(dt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S5) The commutator in the second term of the RHS can be expanded to [ ˆH(t), ˆJ(t+dt)] = iat0 � e−iΦENZ(t+dt) � t0eiΦENZ(t))[ ˆK†, ˆK] −[ ˆU, ˆK] � + eiΦENZ(t+dt) � [ ˆU, ˆK†] − t0e−iΦENZ(t)[ ˆK, ˆK†] �� , (S6) where for convenience we have defined the nearest neighbor operator ˆK as ˆK = � j,σ ˆc† j,σˆcj+1,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S7) Under periodic boundary conditions (which we assume throughout) [ ˆK, ˆK†] = 0, meaning that it is possible rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S3) by inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S6), together with the definition of the current operator given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S2): − ΦENZ(t + dt) aL + J(0) = −iat0e−iΦENZ(t+dt) � ⟨ψ| ˆK|ψ⟩ +idt⟨ψ|[ ˆU, ˆK]|ψ⟩ � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' + O(dt2) (S8) with the definition |ψ⟩ ≡ |ψ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' This expression is further simplified by representing the expectation values in polar form as follows: ⟨ψ| ˆK|ψ⟩ = ⟨ψ| ˆK†|ψ⟩† = R(ψ)eiθ(ψ) (S9) ⟨ψ|[ ˆU, ˆK]|ψ⟩ = −⟨ψ|[ ˆU, ˆK†]|ψ⟩† = P(ψ)eiλ(ψ) (S10) where we use the argument ψ to indicate that the parameter is a functional of the state of the system |ψ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Thus, (S8) becomes −ΦENZ(t + dt) aL + J(0) = −2at0R(ψ) sin [ΦENZ(t + dt) − θ(ψ)] +2at0P(ψ) cos [ΦENZ(t + dt) − λ(ψ)] dt + O(dt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S11) 2 Since the system in question is finite, each term in the above equation is bounded, so we can take the limit dt → 0 to obtain an implicit equation for ΦENZ: 1 aLΦENZ(ψ) = 2at0R(ψ) sin [ΦENZ(ψ) − θ(ψ)] + J(0) (S12) where we have replaced the argument of ΦENZ with ψ as all other variables are functionals of ψ, and hence the time dependence of ΦENZ enters solely through the state of the system |ψ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' The solution to this implicit equation corresponds to the the laser field − 1 adΦENZ(t)/dt that will induce an ENZ response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' EXISTENCE AND UNIQUENESS OF THE ENZ FIELD Before using (S12) to induce an ENZ response, it is important to consider under what conditions such a field exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' It is useful to recast the problem such that, instead of finding a solution, Φ(ψ), to (S12), we find the field for which fψ(Φ) = sin [Φ − θ(ψ)] − Φ Y (ψ) + G(ψ) (S13) has a zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Where, for simplicity, we have defined G(ψ) = R(ψ0) R(ψ) sin[θ(ψ0)] and Y (ψ) = 2a2t0R(ψ)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Note that we treat Φ as a scalar parameter to this function, and the constants are entirely determined by the current state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Theorem 1 - If Φ is a solution to (S13), then Φ − Y (ψ)G(ψ) lies within the interval [−|Y (ψ)|, |Y (ψ)|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Proof: |sin [Φ − θ]| ≤ 1, so any solution must obey ���� Φ Y (ψ) − G(ψ) ���� ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S14) It follows that − |Y (ψ)| ≤ Φ − Y (ψ)G(ψ) ≤ |Y (ψ)|□.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S15) Theorem 2 - At least one solution to (S13) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Proof: First, we evaluate the value of (S13) at the endpoints of the interval in which all possible solutions lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Using ξ1 = −Y (ψ) + Y (ψ)G(ψ) and ξ2 = Y (ψ) + Y (ψ)G(ψ) the value of the function (S13) at the two endpoints of the solution interval is fψ(ξ1) = sin [ξ1 − θ(ψ)] + 1 ≥ 0 (S16) fψ(ξ2) = sin [ξ2 − θ(ψ)] − 1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' (S17) If at least one of fψ(ξ1) and fψ(ξ2) is zero, then there is at least one solution at one (or both) of the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' If both are non-zero, then fψ(ξ1) is positive and fψ(ξ2) is negative, and since f is continuous, there must be at least one zero in the interval [−|Y (ψ)| + Y (ψ)G(ψ), |Y (ψ)| + Y (ψ)G(ψ)] □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Theorem 3 - If |Y (ψ)| ≤ 1, there exists a unique solution to (S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Proof: The derivative of (S13) is f ′ ψ(Φ) = cos [Φ − θ(ψ)] − 1 Y (ψ) (S18) Hence, if |Y (ψ)| ≤ 1, f ′(Φ) is non-positive/negative and therefore f(Φ) is monotonic non-increasing/decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Since fψ(Φ) = 0 for some Φ by Theorem 2, this must be the only value for which fψ(Φ) obtains zero, and hence it is the unique solution to (S12) □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} +page_content=' Note that since R(ψ) ≤ Ns, the ENZ field is guaranteed to be unique regardless of the current state when ��2a2t0NsL �� ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf'} diff --git a/hdE3T4oBgHgl3EQfIQm9/vector_store/index.faiss b/hdE3T4oBgHgl3EQfIQm9/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ae203214cb52dd970f09763330f7e965e057ca62 --- /dev/null +++ b/hdE3T4oBgHgl3EQfIQm9/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cec28c0ffc22756ad304903755c747422fc12c8577d6a93926521cdbbb667831 +size 6684717 diff --git 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b/m9FST4oBgHgl3EQfKzjA/content/tmp_files/2301.13738v1.pdf.txt @@ -0,0 +1,4087 @@ +Submitted to: +© Alexander Cowtan & Simon Burton +This work is licensed under the +Creative Commons Attribution License. +CSS code surgery as a universal construction +Alexander Cowtan +Dept. of Computer Science, University of Oxford +Wolfson Building, Parks Road, Oxford, UK +Quantinuum +Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom +akcowtan@gmail.com +Simon Burton +Quantinuum +Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom +simon.burton@quantinuum.com +We define code maps between Calderbank-Shor-Steane (CSS) codes using maps between chain com- +plexes, and describe code surgery between such codes using a specific colimit in the category of chain +complexes. As well as describing a surgery operation, this gives a general recipe for new codes. As +an application we describe how to ‘merge’ and ‘split’ along a shared X or Z operator between ar- +bitrary CSS codes in a fault-tolerant manner, so long as the participating qubits satisfy a technical +condition related to gauge fixing. We prove that such merges and splits on LDPC codes yield codes +which are themselves LDPC. +1 +Introduction +Quantum computers have become larger and more sophisticated in recent years [10, 35], but fault- +tolerance is necessary to perform practically relevant quantum algorithms. Qubit stabiliser error-correction +codes are a well-studied approach to fault-tolerant quantum computing [20] and are favourable both for +their practicality and theoretical simplicity. Such codes store logical data using entangled states of phys- +ical qubits and repeated many-body measurements, and so long as the physical errors on the qubits stay +below a certain threshold the logical data is protected. +The most well-known example of a qubit stabiliser code is the toric code, in which qubits are em- +bedded on the surface of a torus, and properties of the logical space are determined by the topology of +the surface [15, 28]. This is a basic example of a qubit Calderbank-Shor-Steane (CSS) code; there are +several equivalent ways of defining CSS codes, but for our purposes we shall describe them as codes +which are all homological in a suitable sense [3]. +This means that we can study CSS codes using the tools of homological algebra [38]. This approach +has recently seen much success, for example in the construction of so-called good low-density parity +check (LDPC) code families using a balanced product of chain complexes [33]. Such code families have +an encoding rate k/n of logical to physical qubits which is constant in the code size, while maintaining +a linear code distance d, a substantial asymptotic improvement over simpler examples such as the toric +code. The main caveat is, informally, that the connectivity between physical qubits is non-local. This +complicates the architecture of the system, and also complicates the protocols for performing logical +gates. +There have been several recent works on protocols for logical gates in CSS codes [29, 9, 6, 34, 24], +of varying generality. Here, we build on this work by defining surgery, in the abstract, using arbitrary +arXiv:2301.13738v1 [quant-ph] 31 Jan 2023 + +2 +Please define \titlerunning +CSS codes which form a categorical span, although the practical implementation of such surgery has +several important caveats. The idea is that merging two codes works by identifying a common structure +in each code and quotienting it out. CSS code surgery is particularly convenient when the CSS codes +are compatible, in the sense that they have at least one identical Z or X logical operator. In this case, the +common structure being quotiented out is the logical operator. In order to formalise this, we take a step +back and look at the category of chain complexes Ch(MatF2). +We start by giving a recap of the relevant categorical background of chain complexes, and the view +of classical linear binary codes and qubit CSS codes using chain complexes. We then define code maps +between CSS codes using morphisms between chain complexes. These are maps which send X-checks to +X-checks and Z-checks to Z-checks in a coherent way, and have a convenient presentation as phase-free +ZX diagrams, which we prove in Proposition 4.12. +We believe that code maps crop up throughout the CSS code literature. We see 3 primary use-cases +for code maps: +1. Encoders/decoders [16, 14, 22]. +2. Constructing new codes. +3. Designing fault-tolerant logical operations. +We intend to expound on code maps in future work, but presently we focus on items 2 and 3. +We define CSS code merges as a colimit – specifically, a coequaliser/pushout – in the category of +chain complexes. Not only does the construction describe a surgery operation, but it also gives a general +recipe for new codes. An application of our treatment is the description of certain classes of code surgery +whereby the codes are merged or split along a Z or X operator, closely related to the notion of ‘welding’ +in [32]. We prove that merging two LDPC codes in such a manner still yields an LDPC code. We +give a series of examples, including the specific case of lattice surgery between surface codes. Lastly, +we discuss how to apply such protocols in practice, and prove that when a technical condition related +to gauge fixing is satisfied then code surgery can be performed fault-tolerantly, allowing us to perform +logical parity measurements on codes. +1.1 +Guide to reading the paper +Section 2 gives a bird’s eye view of category theory and universal constructions, which will be useful +later on. Section 3 describes the category of chain complexes with morphisms as matrices over F2. +Category theorists may wish to skip past these sections. +We then give a rundown of CSS codes viewed as chain complexes in Section 4. Readers familiar +with basic category theory and this perspective of CSS codes can safely skip to Section 4.3, where we +introduce the notion of code maps, that is coherent transforms between codes. +We introduce surgery of codes as a colimit in Section 5. This is when the notion of ‘gluing’ codes +together comes in, and we prove several results about these codes when the colimit uses logical Z or +X-operators. Lastly, we introduce a protocol for performing logical Z ⊗Z and X ⊗X measurements +fault-tolerantly in Section 6. +2 +Universal constructions +In this section we provide a cartoon introduction to category theory and universal constructions. We +avoid any weasel phrases like “in some sense”, or even any further scare quotes. However, when we use +actual precise language, ie. jargon words, we emphasise these with italic font. + +Alexander Cowtan & Simon Burton +3 +A category is a collection of objects and morphisms. We will begin by drawing an object as a box +with a decoration, such as +. +Morphisms are arrows between objects, like this +. +The arrow notation suggests that we can compose these. +. +The product of two objects in a category is an object, together with two arrows, +. +The product decoration combines the two decorations, +. +The product also must satisfy a universal property. This states that any other object that also combines +the two decorations is already compatible with the product object in a unique way. In other words, for +all test objects there exists a unique comparison morphism: +. + +4 +Please define \titlerunning +The real product is the minimal object that projects down to the factors. Any other test object lives over +the real product. +This universal property has the immediate consequence that any other object that satisfies all these +requirements, will be isomorphic via a unique isomorphism that commutes with the other morphisms, +. +A pullback is a product with constraints: +. +The resulting square should commute: if we compose any two paths of arrows with the same source +object and the same target object then these paths should be equal. +. +As with products, we also require the pullback to satisfy a universal property. +. + +Alexander Cowtan & Simon Burton +5 +All of these statements have dual statements, which we get by reversing all the arrows. When we do +this we sometimes put a co- prefix on the terminology. For example, a coproduct, which would normally +be called a sum, looks like this +. +Once again, we require any such candidate coproduct +to satisfy a universal property +. +We think of a coproduct as a way of gluing together objects. By adding constraints we can express where +we wish to glue +The answer to this question is called a pushout: it is an object together with two morphisms, + +6 +Please define \titlerunning +that satisfies a universal property, +We have purposefully avoided describing the decorations in these diagrams: how they work, what +they mean. A more sensible introduction to category theory would describe these systematically, possibly +mentioning the category of finite sets and functions. This is a story about how the objects are sets, with +elements, and we can combine these in various ways to make other sets. Instead of telling this story, we +skip to the punchline, which is that there are no elements, or rather, what you think of as an element of +an object is really a morphism into that object: +To push this idea home, and also move toward the goals of this paper, we consider the category MatF2 +of finite dimensional matrices over the field F2. This has as objects the natural numbers and matrices over +F2 as morphisms. Composition of morphisms is matrix multiplication. We will show each object as a +box with dots. For example, here is a composition of two morphisms in this category +. +The objects have very little going on inside them, serving only as anchors for the morphisms (matri- +ces) where all the action is taking place. The vector elements of a vector space have vanished into the +morphisms: + +Alexander Cowtan & Simon Burton +7 +A coproduct (sum) in this category is an object together with two morphisms +satisfying the universal property of coproducts. Here is one candidate: +. +This coproduct will not be unique (except for some degenerate cases), but the universal property of the +coproduct guarantees it is unique up to unique isomorphism. We have reinvented the direct sum of vector +spaces. +For a pushout of vector spaces +. +we get + +8 +Please define \titlerunning +. +This is gluing of a two dimensional vector space and a three dimensional vector space along a one +dimensional vector space. +But what about products? A curious thing happens in the category MatF2; we can get the dual +universal construction by transposing matrices. For example, the above coproduct becomes the product: +and similarly with pullbacks. The transpose duality of MatF2 will follow us throughout the rest of this +paper. +Here we have been taking the objects of MatF2 to be just natural numbers. In the rest of the paper we +will use a slightly different definition for the objects: each natural number n is replaced by a basis set of +size n for an n-dimensional vector space. +3 +Chain complexes +We now recap some elementary homological algebra. All of this section is known, but we fix notation +and look explicitly at the particular category of interest. +Let MatF2 be the category which has as objects based finite-dimensional vector spaces over F2, so +each vector space V has a specified basis ˜V, and we have V ∼= F| ˜V| +2 . A morphism f : V → W in MatF2 is +a dim W ×dim V matrix valued in F2. +Each V has a dual space V ∗. As V ∼= V ∗, we may fix the duals such that V ∗ = V, and ˜V ∗ = ˜V. This +has the benefit of forcing the dual of any matrix f : V → W, which is given by f ∗ : W ∗ → V ∗, to strictly +be the transpose f ⊺ : W → V. + +Alexander Cowtan & Simon Burton +9 +Much of the following mathematics will work given any rigid Abelian category A as input, but we +only need MatF2 for our purposes in Section 4. +Let Ch(MatF2) be the category of bounded chain complexes in MatF2. We now recap some of the +basic properties of this category. A chain complex C• looks like this: +··· +Cn+1 +Cn +Cn−1 +··· +∂n +∂n−1 +where each component Ci is a based vector space and n ∈ Z is called the degree of the component in +C•. C• has F2-matrices as differentials ∂n : Cn+1 → Cn such that ∂n ◦ ∂n+1 = 0 (mod 2), ∀n ∈ Z. To +disambiguate differentials between chain complexes we will use ∂C• +n := ∂n ∈ C• when necessary. +All our chain complexes are bounded, meaning there is some k ∈ Z such that Cn>k = 0 and l ∈ Z such +that Cn◁ CG. More generally we can replace CG and C(G) with H and H∗ for any semisimple Hopf +algebra H [31, 12]. Just as there are no stabiliser generators, there are no longer Z and X-operators, but +there are ribbon operators. As special cases there are ribbon operators which correspond to actions of +only CG or C(G). The first author recently generalised lattice surgery to Kitaev models [13], albeit with +some caveats. In the same way that CSS codes generalise stabiliser codes based on cell complexes, we +imagine there could be a general class of commuting projector models using the quantum double, which +are not necessarily defined on a tessellated manifold. The details of such a class are not known to us, +and generalising the notion of ‘sites’ on a lattice seems difficult. We speculate that the notion of ‘gluing’ +along, say, a CG operator could work for such commuting projector models. + +38 +Please define \titlerunning +8 +Acknowledgements +AC thanks Aleks Kissinger for helpful discussions about Lemma 4.4, and both Aleks Kissinger and John +van de Wetering for helpful discussions about Proposition 4.12. AC also thanks the Wolfson Harrison +UK Research Council Quantum Foundation Scholarship for making this work possible. +References +[1] B. Audoux and A. Couvreur, On tensor products of CSS Codes, Ann. Inst. Henri Poincar´e Comb. Phys. +Interact. 6 (2019), no. 2, pp. 239–287 +[2] N. P. 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Babbush et al., Quantum supremacy using a programmable superconducting processor. +Nature 574, 505–510 (2019) +[36] J. van de Wetering, ZX-calculus for the working quantum computer scientist, arXiv:2012.13966 [quant-ph] +[37] C. Vuillot, L. Lao, B. Criger, C. G. Almud´ever, K. Bertels and B. M. Terhal, Code deformation and lattice +surgery are gauge fixing, New J. Phys. 21 033028 (2019) +[38] C. A. Weibel, An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics), +Cambridge University Press, doi:10.1017/CBO9781139644136 (1994) +A +Limits and colimits in Ch(MatF2) +Lemma A.1. Ch(MatF2) has all kernels and cokernels. +Proof. Recall that MatF2 has all kernels and cokernels, i.e. subspaces and quotient spaces. Then given a + +40 +Please define \titlerunning +chain map f : C• → D• we define ker(f) with +··· +Kn+1 +Kn +··· +··· +Cn+1 +Cn +··· +··· +Dn+1 +Dn +··· +∂ K• +n +ker(fn+1) +ker( fn) +∂C• +n +fn+1 +fn +∂ D• +n +where ∂ K• +n +always exists and is uniquely defined, because +fn ◦∂C• +n ◦ker(fn+1) = ∂ D• +n ◦ fn+1 ◦ker(fn+1) = 0 +and so by the universal property of ker(fn) there is a unique matrix ∂ K• +n +: Kn+1 → Kn. These satisfy +∂ K• +n ◦∂ K• +n+1 = 0 as +ker(fn)◦∂ K• +n ◦∂ K• +n+1 = ∂C• +n ◦∂C• +n+1 ◦ker(fn+2) = 0 +and then kernels are monic. Kn = {v ∈ Cn | fn(v) = 0} by the definition of kernels in MatF2. Given the +correct choice of basis, ∂ K• +n +is thus just ∂C• +n ◦ ker(fn+1) as a matrix but without the all-zero rows which +map into Cn/Kn. +That ker(f) is a genuine kernel in Ch(MatF2) is straightforward to check but we do not give further +details. +The reversed argument applies for cokernels, giving quotient complexes D•/im(f) with components +Dn/im(fn) etc. +Remark A.2. As Ch(MatF2) is additive, equalisers and coequalisers can be seen as special cases of +kernels and cokernels by defining eq(f,g) = ker(f −g) and coeq(f,g) = coker(f −g), for f,g :C• → D•. +For the chain complex part E• of an equaliser we have components En = {c | f(c) = g(c)} ⊆ Cn. For the +chain complex part F• of a coequaliser, we have components Fn = Dn/ f(c) ∼ g(c), for c ∈ Cn. +We now sketch a proof of Lemma 5.5. +Proof. Recall that an Abelian category is an additive category such that: +1. Every morphism has a kernel and cokernel. +2. Every monomorphism is the kernel of its cokernel. +3. Every epimorphism is the cokernel of its kernel. +The first is just Lemma A.1, and the other two follow using the fact that they hold degree-wise in MatF2. +We will now spell out pullbacks. While they can be defined using equalisers and products we con- +struct them explicitly, as it is easy to do so. + +Alexander Cowtan & Simon Burton +41 +Definition A.3. The pullback of chain maps f : X• → Z• and g : Y• → Z• gives the chain complex W•, +where each component is the pullback Wn of fn and gn. The differentials ∂W• +n +are given by the unique map +from each component’s pullback. Specifically, if we have the pullback +W• +Y• +X• +Z• +w +v +g +f +then for degrees n,n+1 we have +Wn +Yn +Wn+1 +Yn+1 +Xn+1 +Zn+1 +Xn +Zn +wn +vn +gn +∂W• +n +wn+1 +vn+1 +∂Y• +n +gn+1 +∂ X• +n +fn+1 +∂ Z• +n +fn +where +Wn = {(x,y) | fn(x) = gn(y)} ⊆ Xn ⊕Yn; +vn(x,y) = x ∈ Xn; +wn(x,y) = y ∈ Yn. +As fn ◦∂ X• +n ◦vn+1 = gn ◦∂Y• +n ◦wn+1 and the outer square is a pullback, there is a unique matrix ∂W• +n . One +can check by diagram chasing that the differentials ∂W• +n ◦∂W• +n+1 = 0, and then that this is indeed a pullback +in Ch(MatF2). +B +Tensor structure of Ch(MatF2) +Definition B.1. [38, Sec. 2.7] Let C•,D• be chain complexes in Ch(MatF2). Define (C⊗D)• with com- +ponents +(C⊗D)n = +� +i+j=n +Ci ⊗D j +where the latter tensor product is the normal tensor product in MatF2. Differentials between components +are given by +∂ (C⊗D)• +n += +i+j=n +� +idCi ⊗∂ D• +j |∂C• +i +⊗idDj +� +where the horizontal line +indicates that all these matrices are stacked vertically, which we illustrate +in an example below. One can check that ∂ (C⊗D)• +n +◦∂ (C⊗D)• +n+1 += 0 (mod 2), as desired. +Also define the object 1• ∈ Ch(MatF2) as +1• = ··· +0 +10 +0 +··· +where 10 = 1, and all other 1i are 0. + +42 +Please define \titlerunning +One can check that (C⊗D)• is a F2-linear monoidal product ⊗ in Ch(MatF2), which follows from +associativity and distributivity of ⊕ and ⊗ in MatF2. For the unit, observe that +(C⊗1)n = Cn ⊗1 = Cn; +∂ (C⊗1)• +n += +� +idCn ⊗∂ 1• +0 |∂C• +n ⊗id10 +� += ∂C• +n . +Example B.2. Consider two chain complexes of length 1: +C• = ··· +0 +C1 +C0 +0 +··· +∂C• +0 +D• = ··· +0 +D1 +D0 +0 +··· +∂ D• +0 +In this case we have +(C⊗D)0 = C0 ⊗D0; +(C⊗D)1 = (C1 ⊗D0)⊕(C0 ⊗D1); +(C⊗D)2 = C1 ⊗D1 +for nonzero components, and +∂ (C⊗D)• +0 += (idC0 ⊗∂ D• +0 |∂C• +0 ⊗idD0); +∂ (C⊗D)• +1 += +� +∂C• +0 ⊗idD1 +idC1 ⊗∂ D• +0 +� +for nonzero differentials. Then +∂ (C⊗D)• +0 +◦∂ (C⊗D)• +1 += ∂C• +0 ⊗∂ D• +0 +∂C• +0 ⊗∂ D• +0 += 0 +(mod 2) +as the matrix partitions factor upon multiplication. +This example illustrates an interesting property of ⊗ in Ch(MatF2): bothC•,D• have only one nonzero +differential, but (C⊗D)• has two. It is easy to see that given two complexes of lengths s,t the tensor +product will have length s+t. +Lemma B.3. [38] +Hn((C⊗D)•) ∼= +� +i+j=n +Hi(C•)⊗Hj(D•) +That is, the homology subspaces factor through the tensor product conveniently. This is also called +the K¨unneth formula. The manner in which the homology factors through does not make Hn(−) a +monoidal functor with respect to the tensor product. +The tensor product is used to build codes from other CSS codes [1]. +C +Graphs and cell complexes +In this appendix we give some categorical background on abstract cell complexes. This is not necessary +to define CSS code surgery, but codes obtained from cell complexes are an important motivating example, +as they include surface codes, toric codes [28], hyperbolic codes [7] and the balanced product codes from +[33]. In general, if a CSS code comes from tessellating a manifold, it is likely to use cell complexes. +Cell complexes are important in the study of topological spaces, and many of the constructions of CSS +codes, such as balanced/lifted products, can also be phrased in the language of topology, but we stick to +cell complexes for brevity. As a warm-up, we describe certain categories of graphs, and then move on to +a specific kind of cell complex. + +Alexander Cowtan & Simon Burton +43 +Let Γ be a finite simple undirected graph. Recall that as a simple graph, Γ has at most one edge +between any two vertices and no self-loops on vertices. Γ can be defined as a pair of sets, V(Γ) and +E(Γ), with E(Γ) ⊆ 2V(Γ), the powerset of vertices, where each e ∈ E(Γ) has 2 elements i.e. it can be +expressed as e = {v1,v2}. An example of a graph is Cn, the cycle graph with n vertices and edges. We +will also use Pn, the path graph with n edges and n+1 vertices. +Definition C.1. Let Grph be the category of finite simple undirected graphs. A morphism Γ → ∆ in Grph +is a function f : V(Γ) → V(∆) such that {v1,v2} ∈ E(Γ) =⇒ { f(v1), f(v2)} ∈ E(∆), i.e. the function +respects the incidence of edges. +Grph has several different products and other categorical features. We are particularly interested +in colimits. Grph has a coproduct Γ + ∆ being the disjoint union, with V(Γ + ∆) = V(Γ) ⊔V(∆) and +E(Γ+∆) = E(Γ)⊔E(∆). It also has an initial object I given by the empty graph. However, Grph is not +cocomplete, as it does not have all pushouts. +Example C.2. As a counterexample [30], given the diagram +no cocone exists, as the graphs are not allowed self-loops. Therefore, no pushout exists. +One can easily see that there are diagrams for which pushouts do exist, though. +More than just graphs, we would like to allow for open graphs, i.e. graphs which may have edges +which connect to only one vertex, but are not self-loops. For example, +We call G3 the 3rd open path graph, where the nth open path graph Gn has n edges in a line with n − 1 +vertices between them. We now give a particular formalisation of open graphs. +Definition C.3. Let Γ be a finite simple undirected graph with two disjoint vertex sets V(Γ) and B(Γ), +where E(Γ) ⊆ 2V(Γ)∪B(Γ). We then say that Γ is an open graph. We call V(Γ) the internal vertices and +B(Γ) the boundary vertices. +So in the picture of G3 above there are vertices at either end of the open wires, but they are considered +‘invisible’, i.e. they belong to B(Γ). +Definition C.4. Let OGrph be the category of open graphs. A morphism Γ → ∆ in OGrph is a func- +tion f : V(Γ) ∪ B(Γ) → V(∆) ∪ B(∆) such that f(x) ∈ B(∆) =⇒ x ∈ B(Γ) and {v1,v2} ∈ E(Γ) =⇒ +{ f(v1), f(v2)} ∈ E(∆). +This restriction prevents internal vertices from being ‘deleted’ by a graph morphism by converting +them to boundary vertices, although we do not prevent the reverse from occurring. OGrph has very similar +properties to Grph. Its initial object is the empty open graph. OGrph has a coproduct, where V(Γ+∆) = +V(Γ) ⊔V(∆) and B(Γ + ∆) = B(Γ) ⊔ B(∆). Like Grph, OGrph is not cocomplete, as Example C.2 also +works in the setting of open graphs. It is obvious that Grph is a subcategory of OGrph. +We now move on to cell complexes, in particular abstract cubical complexes. These are abstract cell +complexes which are ‘square’, unlike their ‘triangular’ relatives simplicial complexes. + +44 +Please define \titlerunning +Definition C.5. The abstract d-cube is the set {0,1}d, with the 0-cube {0,1}0 := {0}. A face of the +abstract d-cube is a product A1 ×···×Ad, where each Ai is a nonempty subset of {0,1}. +Definition C.6. [17] Let S be a finite set and let Ω be a collection of nonempty subsets of S such that: +• Ω covers S. +• For X,Y ∈ Ω, X ∩Y ∈ Ω or X ∩Y = /0. +• For each X ∈ Ω, there is a bijection from X to the abstract d-cube for some choice of d, such that +any Y ⊂ X is in Ω iff it is mapped to a face of the d-cube. +Then Ω is an abstract cubical complex. +Abstract cubical complexes are combinatorial versions of cubical complexes, meaning they are +stripped of their associated geometry. The elements in Ω are still called faces. We can consider Ω +to be a graded poset, with subset inclusion as the partial order, and the grading dim(X) = log2 |X|. We +also call this grading the dimension d of X, and we call X a d-face. The set of d-faces in Ω is called Ωd. +There is a relation Ωd → Ωd−1 taking a d-face to its (d −1)-face subsets. +We call the vertex set V(Ω) = S = Ω0, and also define the dimension of a cubical complex +dim(Ω) = max +X∈Ω dim(X) +The d-skeleton of Ω is the maximal subcomplex ϒ ⊆ Ω such that dim(ϒ) = d. The 1-skeleton of an ab- +stract cubical complex is a finite simple undirected graph. The 2-skeleton of an abstract cubical complex +is ‘like’ a square lattice, in that it has 2-faces which each have 4 0-faces as subsets and 4 1-faces. +Definition C.7. Let ACC be the category of abstract cubical complexes. A morphism f : Ω → ϒ in ACC +is a function f : V(Ω) → V(ϒ), such that {x,··· ,y} ∈ Ωd =⇒ {f(x),··· , f(y)} ∈ ϒd, i.e. incidence is +preserved at each dimension. +Similar to Grph, ACC has coproduct given by (Ω+ϒ)i = Ωi ⊔ϒi and an initial object I = /0, and does +not generally have pushouts, where we can reuse the same counterexample as Grph. Another categorical +property we highlight here is that ACC has a monoidal product called the box product. +Definition C.8. Let ϒ 2 Ω be the box product of abstract cubical complexes. Then +(ϒ 2 Ω)n = ∑ +i+j=n +ϒi ×Ωj. +We now check that ϒ 2 Ω is indeed an abstract cubical complex. +Proof. First, it has a vertex set V(ϒ 2 Ω) = V(ϒ) ×V(Ω), and thus trivially covers ϒ0 × Ω0. Second, +let X ×Y ∈ ϒi × Ω j and T ×U ∈ ϒk × Ωl. This has (X ×Y) ∩ (T ×U) = (X ∩ T) × (Y ∩U) which is +either in ϒm × Ωn for some m ≤ i,m ≤ k and n ≤ j,n ≤ l, and thus (X ∩ T) × (Y ∩U) ∈ ϒ 2 Ω, or +(X ∩T)×(Y ∩U) = /0. Third, if X and Y each have a bijection to an i-cube and j-cube respectively, then +X ×Y has a bijection to an (i + j)-cube. Any W ⊂ X ×Y can be expressed as T ×U, for T ⊂ X and +U ⊂ Y. Then W is in Ω 2 ϒ iff T is mapped to a face of the i-cube and U to a face of the j-cube, thus W +to a face of the (i+ j)-cube. +Let us compile this into a more digestible form for the case when ϒ and Ω are both graphs. Given +vertices (u,u′) and (v,v′) in V(ϒ 2 Ω), the 1-face {(u,u′),(v,v′)} ∈ (ϒ 2 Ω)1 iff (u = v & (u′,v′) ∈ Ω) +or ((u,v) ∈ ϒ & u′ = v′). Then (ϒ 2 Ω)2 ∼= E(ϒ) × E(Ω). The 1-skeleton of ϒ 2 Ω is just the normal +box product of graphs [25]. + +Alexander Cowtan & Simon Burton +45 +Example C.9. Let Cm and Cn be cycle graphs with m and n vertices respectively, considered as abstract +cubical complexes. Then T = Cm 2 Cn admits an embedding as a square lattice on the torus, and has +dim(Cm 2 Cn) = 2. Setting m = n = 3 we have +where the grey dots indicate periodic boundary conditions and the white circles specify 2-faces. This +example will come up in the form of the toric code in Section 4. +Obviously, Grph is a subcategory of ACC. +We are also interested in open abstract cubical complexes. +Definition C.10. Let ϒ be an open abstract cubical complex. ϒ is an abstract cubical complex where ϒ0 +is divided into two disjoint vertex sets V(ϒ) and B(ϒ). +The 1-skeleton of an open abstract cubical complex is an open graph. +Definition C.11. Let OACC be the category of open abstract cubical complexes. A morphism f : Ω → +ϒ in OACC is a function f : V(Ω) ∪ B(Ω) → V(ϒ) ∪ B(ϒ) such that f(x) ∈ B(ϒ) =⇒ x ∈ B(Ω) and +{x,··· ,y} ∈ Ωd =⇒ { f(x),··· , f(y)} ∈ ϒd. +As in our previous examples, OACC has the obvious coproduct and initial object, and does not have +pushouts in general. +Example C.12. Let ϒ be a ‘patch’, a square lattice with two rough and two smooth boundaries: +This patch has 6 2-faces, 13 1-faces and 6 0-faces. +Example C.13. We can perform the pushout of two smaller open abstract cubical complexes to acquire + +46 +Please define \titlerunning +a patch: +where the apex is P1, the blue edge indicates where the apex is mapped to, and the bottom right open +abstract cubical complex is the object of the pushout. +Example C.14. Let G3 be the open path graph, and let Ω be a patch. Then we have a pushout +This example comes up in the context of lattice surgery on surface codes. Evidently, both OGrph and +ACC are subcategories of OACC, and one can define a box product for OACC in the same way as we did for +ACC in Definition C.8. +One can define quantum codes using abstract cell complexes more generally, but abstract cubical +complexes are the specific type which we make use of in examples in Section 4 and onwards. We now +relate the above cell complexes to chain complexes by way of functors. +Definition C.15. Given an abstract cubical complex Ω we can define the incidence chain complex C• in +Ch(MatF2), where each nonzero component has a basis ˜Cn−1 = Ωn, say, and each nonzero differential +∂C• +n−1 takes an n+1-face to its n-dimensional subsets. 7 The differential is thus a matrix with a 1 where an +n-face is contained within an (n+1)-face, and 0 elsewhere. It is an elementary fact that every (d −2)- +face in a d-face is the intersection of exactly 2 (d − 1)-faces, thus ∂C• +n−1 ◦ ∂C• +n += 0 mod 2. Clearly, the +7We choose to send n-faces to (n−1)-components, rather than n-components. + +Alexander Cowtan & Simon Burton +47 +incidence chain complex of a dimension 1 abstract cubical complex is just the incidence matrix of a +simple undirected graph. +We can do essentially the same thing given an open abstract cubical complex ϒ. In this case, each +nonzero component has a basis ˜Cn−1 = {X ∈ Ωn | X ̸⊆ B(Ω)}, that is we ignore all faces which are made +up only of boundary vertices, and differentials are the same matrices as above, with a 1 where an n-face +which is not a subset of B(Ω) (and therefore would be ‘invisible’) is contained in an (n + 1)-face. It is +easy to see that we still have ∂C• +n−1 ◦∂C• +n = 0 mod 2. The incidence chain complex of a dimension 1 open +abstract cubical complex is the incidence matrix of an open graph. +Definition C.16. Let C• and D• be the incidence chain complexes of two abstract cubical complexes Ω +and ϒ with a morphism f : Ω → ϒ, and set ˜C−1, ˜D−1 as V(Ω),V(ϒ) respectively. This induces a chain +map g• : C• → D•, with the matrix g−1 given by f, and all matrices on higher components generated +inductively. Degrees i < −1 are assumed to be zero. +As a consequence, we can define a functor ϕ : ACC → Ch(MatF2), sending each abstract cell complex +to its free chain complex as described in Definition C.15. ϕ(f) ∈ Hom(ϕ(Ω),ϕ(ϒ)) for any morphism +f : Ω → ϒ between abstract cubical complexes, as the function on vertices is already F2-linear and the +matrices at higher degrees are uniquely determined. ϕ is faithful but not full, as there exist morphisms, +such as the zero morphism, which are not in the image of ϕ. +Definition C.17. There is also a functor ϑ : OACC → Ch(MatF2). On objects, this again follows Defini- +tion C.15. On morphisms this is the same as ϕ except it must obviously ignore maps between boundary +vertices everywhere. Thus ϑ is not faithful. +Example C.18. Let Ω and ϒ be two abstract cubical complexes. Then ϕ(Ω+ϒ) = ϕ(Ω)⊕ϕ(ϒ), which +is easy to check. Similarly, ϕ(/0) = 0•. The same is true of ϑ, except that ϑ(Ξ) = 0• for any Ξ with +V(Ξ) = /0. +Lemma C.19. The functors ϕ and ϑ are cocontinuous i.e. they preserve colimits. +Proof. We give a proof sketch here. We know already that ϕ preserves coproducts so it is sufficient to +check that it preserves pushouts. Let +Ξ +ϒ +Ω +χ +g +f +l +k +be a pushout in ACC. Then χ0 = Ω0 ⊔ ϒ0/ f ∼ g, and we have elements in χn of the form ([x],··· ,[y]), +which can be seen as pushouts at each dimension. Also, (x,··· ,y) ∈ Ωn =⇒ ([x],··· ,[y]) ∈ χn, and the +same for ϒn. Then +˜ +ϕ(χ)−1 = χ0, recalling that we chose to send d-faces to (d −1)-components. We then +have basis elements of the form ([x],··· ,[y]) ∈ +˜ +ϕ(χ)n−1, and differentials have their obvious form. If we +take the diagram in Ch(MatF2): +ϕ(Ξ) +ϕ(ϒ) +ϕ(Ω) +ϕ(g) +ϕ( f) +Then we have Q• as the pushout. Again, this has ˜Q−1 = χ0. Basis elements in Qn are then also of the +form ([x],··· ,[y]) for [x],[y] ∈ χ0. The differentials also match up correctly, and so Q• = ϕ(χ). + +48 +Please define \titlerunning +The same checks apply if we take ϑ : OACC → Ch(MatF2) instead. Observe that in this case f and g +may have images only in B(Ω) and B(ϒ), in which case Ξ must have empty V(Ξ). Then the pushout in +Ch(MatF2) will just be a direct sum, i.e. the pushout with ϑ(Ξ) = 0• as the apex. +Recall that ACC and OACC do not themselves have all pushouts, and therefore all colimits, but ϕ and +ϑ preserve those which they do have. +Remark C.20. For any chain complex C• we have also the pth translation C[p]•, where all indices +are shifted down by p, i.e. C[p]n = Cn+p and ∂C[p]• +n += ∂C• +n+p. This extends to an invertible endofunctor +p : Ch(MatF2) → Ch(MatF2) in the obvious way. +Lemma C.21. Let ϒ and Ω be two open abstract cubical complexes. Recalling the functor ϑ : OACC → +Ch(MatF2) from Definition C.16, we almost have ϑ(ϒ 2 Ω) = ϑ(ϒ)⊗ϑ(Ω). We do not quite have this, +as we chose to shift the indices down for convenience, so ϑ is not a monoidal functor, but for moral +purposes it is. Instead we have ϑ(ϒ 2 Ω)[−1] = ϑ(ϒ) ⊗ ϑ(Ω), using the index shifting endofunctor +−1 : Ch(MatF2) → Ch(MatF2). +D +Pushouts and properties of codes +Here we describe a few problems with using general pushouts to construct new quantum codes. First, +in a certain sense the pushout of LDPC codes is not necessarily LDPC. To illustrate this, consider the +following pushout of graphs: +As ϑ is cocontinuous this pushout exists also in Ch(MatF2). There, it represents a merge of two binary +classical codes, although we can consider a binary linear code to just be a CSS code without any Z +measurements. As a consequence, we have two initial codes with PX having maximal weights 1 each, +and the merged code has maximal weight 4. Evidently, one can scale this with the size of the input +graphs: here, the input graphs each have 3 edges, but if there are graphs with m edges each (and weight +1) and the apex with m vertices (and weight 0) then the pushout graph will have maximal weight m+1. +As a consequence the family of pushout graphs as m scales is not bounded above by a constant, and so +the corresponding family of codes is not LDPC. +Conjecture D.1. Let +A• +D• +C• +g• +f• + +Alexander Cowtan & Simon Burton +49 +be a monic span in Ch(MatF2), and let Q• be the pushout chain complex of this monic span. Further, let +the monic span be a representative of a family of monic spans which are parameterised by some n ∈ N, +and let A•, C• and D• be the Z-type complexes of quantum LDPC codes. Then (Q•,Q∗ +•) is also LDPC. +Formulating this conjecture properly requires specifying what it means for a monic span to be pa- +rameterised. The above conjecture is clearly not an if and only if, as balanced products are not pushouts +of monic spans. +Lastly, taking pushouts evidently preserves neither homologies nor code distances, as easy exam- +ples with lattice surgery demonstrate. Moreover, we do not know of a way of giving bounds on these +quantities for general pushouts, although again we suspect it should be easier for monic spans. +E +Octagonal surface code patch +Consider the following patch of surface code: +where the bristled edges are rough boundaries, and the diagonal edges are smooth boundaries. We have +abstracted away from the actual cell complex as the tessellation is not important. Z-type logical operators +take the form of strings extending from one rough boundary to another, e.g. +Two strings belong to the same equivalence class iff they are isotopic on the surface, allowing for the +endpoints to slide up and down a rough boundary. There are exactly 3 nontrivial such classes out of +which all other strings can be composed. As a consequence, this patch of surface code has logical space +V with dimV = 23 = 8. 8 We can choose a basis for this logical space, which has logical Z operators +8More generally, a patch with 2m edges, alternating rough and smooth, has dimV = 2m−1, i.e. the number of edges in a +minimal spanning tree on the complete graph with m vertices. + +50 +Please define \titlerunning +with representatives: +Z1 = +Z2 = +Z3 = +where the middle operator can be smoothly deformed to a vertical line from top to bottom if desired. +Recall that on the surface code an X operator anticommutes with a Z operator iff the strings cross an odd +number of times. Thus, given the basis above, the duality pairing of Lemma 4.4 forces a similar basis of +X operators, with representatives: +X1 = +X2 = +X3 = +We see that Z1 is contained entirely within Z2 on physical qubits. Thus it is possible to construct a Z +merge which is not separated, in the parlance of Definition 5.12. If we choose a different representative, +by deforming Z2 to be a vertical line, then we can also perform a separated Z merge. +F +Fault tolerant Z-merge with the Shor code +In this appendix we work through an example explicitly, using the techniques of Section 6 to perform a +distance 3 fault-tolerant Z ⊗Z measurement between two copies of the Shor code, for which see Exam- +ple 4.7. +Let us say the two copies are labelled (C•,C∗ +•) and (D•,D∗ +•), with +C• = D• = F6 +2 +F9 +2 +F2 +2 +∂0 +∂−1 +and +∂C• +0 = ∂ D• +0 += +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +; +∂C• +−1 = ∂ D• +−1 = +�1 +1 +1 +1 +1 +1 +0 +0 +0 +1 +1 +1 +0 +0 +0 +1 +1 +1 +� +. + +Alexander Cowtan & Simon Burton +51 +We will use the Z operator Z1 ⊗Z4 ⊗Z7, denoted u = +� +1 +0 +0 +1 +0 +0 +1 +0 +0 +�⊺, with u ∈ C0 and +u ∈ D0, to glue along. +The logical operator subcomplex V• is then +V• = F3 +2 +F2 +2 +∂V• +−1 +with ∂V• +−1 = +�1 +1 +0 +1 +0 +1 +� +and all other components of V• being 0. +We now make the tensor product chain complex W• = (P⊗V)• from Definition 6.3, where P• = +P1 +P0 +� +�1 +1 +� +� +. We have +W• = F3 +2 +F8 +2 +F4 +2 +∂W• +0 +∂W• +−1 +with +∂W• +0 += +� +� +� +� +� +� +� +� +� +� +� +� +1 +0 +0 +0 +1 +0 +0 +0 +1 +1 +0 +0 +0 +1 +0 +0 +0 +1 +1 +1 +0 +1 +0 +1 +� +� +� +� +� +� +� +� +� +� +� +� +; +∂W• +−1 = +� +� +� +� +1 +1 +0 +0 +0 +0 +1 +0 +1 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +1 +1 +0 +1 +0 +0 +0 +0 +1 +0 +1 +0 +1 +� +� +� +� +For T• we take the two pushouts from Definition 6.5. First, we have +V• +C• +W• +R• +g• +f• +q• +p• +Giving +R• = F9 +2 +F14 +2 +F4 +2 +∂ R• +0 +∂ R• +−1 +with R1 =W1 ⊕C1, as V1 = 0. The other components of R• require taking quotients, identifying elements + +52 +Please define \titlerunning +of W0 and C0, and the same for W−1 and C−1. One can then use Definition 5.1 to show that +∂ R• +0 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +0 +0 +0 +0 +0 +0 +1 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +; +∂ R• +−1 = +� +� +� +� +0 +0 +0 +1 +0 +1 +1 +1 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +1 +0 +0 +0 +1 +1 +1 +1 +1 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +� +� +� +�. +For the second pushout, that is +V• +W• +R• +D• +T• +we then have +T• = F15 +2 +F20 +2 +F4 +2 +∂ T• +0 +∂ T• +−1 +. +The differentials are somewhat unwieldy, but we include them for completeness: +∂ T• +0 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� + +Alexander Cowtan & Simon Burton +53 +∂ T• +−1 = +� +� +� +� +1 +0 +1 +1 +1 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +1 +0 +0 +0 +1 +1 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +1 +1 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +0 +0 +0 +1 +1 +1 +� +� +� +� +One can check the various properties of this code. For example, rank∂ T• +−1 = 4 and rank∂ T• +0 = 15. Thus +dimH0(T•) = dimT0 −4−15 = 1, and so the code (T•,T ∗ +• ) encodes one logical qubit. +We can compare this with (C ⊕D)• from before the merge: +(C ⊕D)• = F12 +2 +F18 +2 +F4 +2 +∂ (C⊕D)• +0 +∂ (C⊕D)• +−1 +where the differentials are easy to see from those of C• and D•, with ∂ (C⊕D)• +0 += ∂C• +0 ⊕ ∂ D• +0 +etc. As +expected, there are 2 new qubits, and 3 new Z-measurements. Each of the 2 new qubits participates in 2 +of the new Z-measurements (and no other Z-measurements). Evidently, ((C ⊕D)•,(C ⊕D)∗ +•) encodes 2 +logical qubits. +For the fault-tolerant Z ⊗Z measurement, we therefore start with the code ((C ⊕ D)•,(C ⊕ D)∗ +•). +Recall that this has d = 3. We then initialise the 2 new qubits in the |+⟩ state and measure 3 rounds +of the stabilisers specified by ∂ T• +0 +and ∂ T• +−1. As the 2 new qubits each participate in 2 of the new Z- +measurements, the product of the outcomes is insensitive to initialisation errors. We apply the gauge- +fixing operators from Example 6.2 to correct for measurements of the 3 new Z-measurements which +output the -1 measurement outcome. We end up with the code (T•,T ∗ +• ). + diff --git a/m9FST4oBgHgl3EQfKzjA/content/tmp_files/load_file.txt b/m9FST4oBgHgl3EQfKzjA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c796cb8899b7284ad2c2b6725726a0349cb278bb --- /dev/null +++ b/m9FST4oBgHgl3EQfKzjA/content/tmp_files/load_file.txt @@ -0,0 +1,2114 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf,len=2113 +page_content='Submitted to: © Alexander Cowtan & Simon Burton This work is licensed under the Creative Commons Attribution License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' CSS code surgery as a universal construction Alexander Cowtan Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' of Computer Science, University of Oxford Wolfson Building, Parks Road, Oxford, UK Quantinuum Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom akcowtan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content='com Simon Burton Quantinuum Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content='burton@quantinuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content='com We define code maps between Calderbank-Shor-Steane (CSS) codes using maps between chain com- plexes, and describe code surgery between such codes using a specific colimit in the category of chain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' As well as describing a surgery operation, this gives a general recipe for new codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' As an application we describe how to ‘merge’ and ‘split’ along a shared X or Z operator between ar- bitrary CSS codes in a fault-tolerant manner, so long as the participating qubits satisfy a technical condition related to gauge fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We prove that such merges and splits on LDPC codes yield codes which are themselves LDPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' 1 Introduction Quantum computers have become larger and more sophisticated in recent years [10, 35], but fault- tolerance is necessary to perform practically relevant quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Qubit stabiliser error-correction codes are a well-studied approach to fault-tolerant quantum computing [20] and are favourable both for their practicality and theoretical simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Such codes store logical data using entangled states of phys- ical qubits and repeated many-body measurements, and so long as the physical errors on the qubits stay below a certain threshold the logical data is protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The most well-known example of a qubit stabiliser code is the toric code, in which qubits are em- bedded on the surface of a torus, and properties of the logical space are determined by the topology of the surface [15, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This is a basic example of a qubit Calderbank-Shor-Steane (CSS) code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' there are several equivalent ways of defining CSS codes, but for our purposes we shall describe them as codes which are all homological in a suitable sense [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This means that we can study CSS codes using the tools of homological algebra [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This approach has recently seen much success, for example in the construction of so-called good low-density parity check (LDPC) code families using a balanced product of chain complexes [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Such code families have an encoding rate k/n of logical to physical qubits which is constant in the code size, while maintaining a linear code distance d, a substantial asymptotic improvement over simpler examples such as the toric code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The main caveat is, informally, that the connectivity between physical qubits is non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This complicates the architecture of the system, and also complicates the protocols for performing logical gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' There have been several recent works on protocols for logical gates in CSS codes [29, 9, 6, 34, 24], of varying generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Here, we build on this work by defining surgery, in the abstract, using arbitrary arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content='13738v1 [quant-ph] 31 Jan 2023 2 Please define \\titlerunning CSS codes which form a categorical span, although the practical implementation of such surgery has several important caveats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The idea is that merging two codes works by identifying a common structure in each code and quotienting it out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' CSS code surgery is particularly convenient when the CSS codes are compatible, in the sense that they have at least one identical Z or X logical operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' In this case, the common structure being quotiented out is the logical operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' In order to formalise this, we take a step back and look at the category of chain complexes Ch(MatF2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We start by giving a recap of the relevant categorical background of chain complexes, and the view of classical linear binary codes and qubit CSS codes using chain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We then define code maps between CSS codes using morphisms between chain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' These are maps which send X-checks to X-checks and Z-checks to Z-checks in a coherent way, and have a convenient presentation as phase-free ZX diagrams, which we prove in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We believe that code maps crop up throughout the CSS code literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We see 3 primary use-cases for code maps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Encoders/decoders [16, 14, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Constructing new codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Designing fault-tolerant logical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We intend to expound on code maps in future work, but presently we focus on items 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We define CSS code merges as a colimit – specifically, a coequaliser/pushout – in the category of chain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Not only does the construction describe a surgery operation, but it also gives a general recipe for new codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' An application of our treatment is the description of certain classes of code surgery whereby the codes are merged or split along a Z or X operator, closely related to the notion of ‘welding’ in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We prove that merging two LDPC codes in such a manner still yields an LDPC code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We give a series of examples, including the specific case of lattice surgery between surface codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Lastly, we discuss how to apply such protocols in practice, and prove that when a technical condition related to gauge fixing is satisfied then code surgery can be performed fault-tolerantly, allowing us to perform logical parity measurements on codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content='1 Guide to reading the paper Section 2 gives a bird’s eye view of category theory and universal constructions, which will be useful later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Section 3 describes the category of chain complexes with morphisms as matrices over F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Category theorists may wish to skip past these sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We then give a rundown of CSS codes viewed as chain complexes in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Readers familiar with basic category theory and this perspective of CSS codes can safely skip to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content='3, where we introduce the notion of code maps, that is coherent transforms between codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We introduce surgery of codes as a colimit in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This is when the notion of ‘gluing’ codes together comes in, and we prove several results about these codes when the colimit uses logical Z or X-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Lastly, we introduce a protocol for performing logical Z ⊗Z and X ⊗X measurements fault-tolerantly in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' 2 Universal constructions In this section we provide a cartoon introduction to category theory and universal constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We avoid any weasel phrases like “in some sense”, or even any further scare quotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' However, when we use actual precise language, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' jargon words, we emphasise these with italic font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Alexander Cowtan & Simon Burton 3 A category is a collection of objects and morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We will begin by drawing an object as a box with a decoration, such as .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Morphisms are arrows between objects, like this .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The arrow notation suggests that we can compose these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The product of two objects in a category is an object, together with two arrows, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The product decoration combines the two decorations, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The product also must satisfy a universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This states that any other object that also combines the two decorations is already compatible with the product object in a unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' In other words, for all test objects there exists a unique comparison morphism: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' 4 Please define \\titlerunning The real product is the minimal object that projects down to the factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Any other test object lives over the real product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This universal property has the immediate consequence that any other object that satisfies all these requirements, will be isomorphic via a unique isomorphism that commutes with the other morphisms, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' A pullback is a product with constraints: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The resulting square should commute: if we compose any two paths of arrows with the same source object and the same target object then these paths should be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' As with products, we also require the pullback to satisfy a universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Alexander Cowtan & Simon Burton 5 All of these statements have dual statements, which we get by reversing all the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' When we do this we sometimes put a co- prefix on the terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' For example, a coproduct, which would normally be called a sum, looks like this .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Once again, we require any such candidate coproduct to satisfy a universal property .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We think of a coproduct as a way of gluing together objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' By adding constraints we can express where we wish to glue The answer to this question is called a pushout: it is an object together with two morphisms, 6 Please define \\titlerunning that satisfies a universal property, We have purposefully avoided describing the decorations in these diagrams: how they work, what they mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' A more sensible introduction to category theory would describe these systematically, possibly mentioning the category of finite sets and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This is a story about how the objects are sets, with elements, and we can combine these in various ways to make other sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Instead of telling this story, we skip to the punchline, which is that there are no elements, or rather, what you think of as an element of an object is really a morphism into that object: To push this idea home, and also move toward the goals of this paper, we consider the category MatF2 of finite dimensional matrices over the field F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This has as objects the natural numbers and matrices over F2 as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Composition of morphisms is matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We will show each object as a box with dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' For example, here is a composition of two morphisms in this category .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The objects have very little going on inside them, serving only as anchors for the morphisms (matri- ces) where all the action is taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The vector elements of a vector space have vanished into the morphisms: Alexander Cowtan & Simon Burton 7 A coproduct (sum) in this category is an object together with two morphisms satisfying the universal property of coproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Here is one candidate: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This coproduct will not be unique (except for some degenerate cases), but the universal property of the coproduct guarantees it is unique up to unique isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We have reinvented the direct sum of vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' For a pushout of vector spaces .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' we get 8 Please define \\titlerunning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This is gluing of a two dimensional vector space and a three dimensional vector space along a one dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' But what about products?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' A curious thing happens in the category MatF2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' we can get the dual universal construction by transposing matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' For example, the above coproduct becomes the product: and similarly with pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' The transpose duality of MatF2 will follow us throughout the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Here we have been taking the objects of MatF2 to be just natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' In the rest of the paper we will use a slightly different definition for the objects: each natural number n is replaced by a basis set of size n for an n-dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' 3 Chain complexes We now recap some elementary homological algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' All of this section is known, but we fix notation and look explicitly at the particular category of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Let MatF2 be the category which has as objects based finite-dimensional vector spaces over F2, so each vector space V has a specified basis ˜V, and we have V ∼= F| ˜V| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' A morphism f : V → W in MatF2 is a dim W ×dim V matrix valued in F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Each V has a dual space V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' As V ∼= V ∗, we may fix the duals such that V ∗ = V, and ˜V ∗ = ˜V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' This has the benefit of forcing the dual of any matrix f : V → W, which is given by f ∗ : W ∗ → V ∗, to strictly be the transpose f ⊺ : W → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Alexander Cowtan & Simon Burton 9 Much of the following mathematics will work given any rigid Abelian category A as input, but we only need MatF2 for our purposes in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' Let Ch(MatF2) be the category of bounded chain complexes in MatF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' We now recap some of the basic properties of this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' A chain complex C• looks like this: ··· Cn+1 Cn Cn−1 ··· ∂n ∂n−1 where each component Ci is a based vector space and n ∈ Z is called the degree of the component in C•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' C• has F2-matrices as differentials ∂n : Cn+1 → Cn such that ∂n ◦ ∂n+1 = 0 (mod 2), ∀n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' To disambiguate differentials between chain complexes we will use ∂C• n := ∂n ∈ C• when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FST4oBgHgl3EQfKzjA/content/2301.13738v1.pdf'} +page_content=' All our chain complexes are bounded, meaning there is some k ∈ Z such that Cn>k = 0 and l ∈ Z such that Cn 0} ⊆ Ω for some r0 > 0. +Date: January 30, 2023. +1 + +2 +L. ABATANGELO AND R. OGNIBENE +Given a relatively open set Σ ⊆ B′ +r0 with Lipschitz boundary and ε ∈ (0, 1), we consider the thin +tube of section εΣ and fixed height equal to 1 attached at the origin. If we denote +Tε := εΣ × (−1, 0], +our perturbed domain will be +(1.3) +Ωε := Ω ∪ εΣ ∪ Tε +and Ω ∩ Tε = εΣ. We then consider the eigenvalue problem for the Dirichlet-Laplacian on the +perturbed domain Ωε +(1.4) +�−∆ϕ = λεϕ, +in Ωε, +ϕ = 0, +on ∂Ωε. +Again by classical spectral theory, for ε ∈ (0, 1) this problem admits a sequence of eigenvalues +tending to +∞, which will be denoted as +0 < λε +1 < λε +2 ≤ · · · ≤ λε +n ≤ · · · → +∞, +whereas {ϕε +n}n≥1 will denote a corresponding sequence eigenfunctions, assumed to be orthonormal +in L2(Ωε). +The main goal of the present paper is understanding the behavior of the perturbed eigenvalues +λε +n as the tube radius ε tends to zero. Literature on this problem is very rich. We refer to [16, +Introduction] for a presentation of the established results. Here we just mention that such a large +interest for the problem is due to physical and engineering motivations. Spectral behavior of the +Laplacian on thin branching domains appears in the theory of quantum graphs, which models +propagation of waves in quasi one-dimensional systems (quantum wires and waveguides, photonic +crystals, blood vessels and so on), as well as in the theory of elasticity and multistructure problems. +First of all, as a consequence of the convergence in the sense of Mosco of the sequence of domains +{Ωε}ε to the limit domain Ω (see the discussion in Subsection 2.2), classical results (see e.g. [12]) +ensure stability of the spectrum, in the sense that for any N ∈ N \ {0} +λε +N → λN +as ε → 0. +The analysis of the present paper originates from the main result in [16], which in turn generalizes +the papers [2] and [19]. +In [16] the authors study the sharp asymptotic behavior of Dirichlet +eigenvalues in a domain perturbed as described above (with some additional geometric assumption +on the section of the tube) using Almgren-type monotonicity formulas, Courant-Fischer min-max +characterization for eigenvalues and blow-up analysis for scaled eigenfunctions. Specifically, they +restrict to the case in which the perturbed eigenvalues are converging to a simple eigenvalue of +the limit problem. Due to the local nature of the singular perturbation taken into account, the +eigenfunctions’ local behavior at 0 ∈ ∂Ω (namely the point where the thin branch is attached) +plays a crucial role. Such behavior can be described as follows (see [7] or [13, Theorem 1.3]): if +ϕN is an eigenfunction of (1.1), there exists k ∈ N \ {0} such that +(1.5) +ϕN(rx) +rk +→ ψk(x), +in C1,α(B+ +1 ) as r → 0, for some ψk ∈ Pk +odd, +where Pk +odd denotes the space of harmonic homogeneous polynomials of degree k, odd with respect +to the last variable xd. We point out that the polynomials in the class Pk +odd, restricted to the (d−1)- +dimensional unit sphere, are spherical harmonics (i.e. eigenfunctions of the spherical Laplacian) +vanishing on {xd = 0}. If we denote +(1.6) +Π := Rd ++ ∪ Σ ∪ T, +where T := Σ × (−∞, 0) +and by D1,2(Π) the completion of C∞ +c (Π) with respect to the L2 norm of the gradient, the main +result [16, Theorem 1.1] establishes that, if Σ is starshaped with respect to the origin and if λN is +a simple eigenvalue of (1.1), then +(1.7) +λǫ +N = λN − Ck,Σ εd−2+2k + o(εd−2+2k), +as ε → 0, + +3 +where +(1.8) +Ck,Σ = −2 +inf +u∈D1,2(Π) +�1 +2 +� +Π +|∇u|2 dx − +� +Σ +u∂ψk +∂xd +dx′ +� +> 0. +Let us now briefly comment on this result. From (1.7), one can see that the local nature of the +perturbation mainly emerges in the exponent k of the radius of the tube’s section ε: in this sense, +the vanishing order k of the unique limit eigenfunction (up to multiplicative constants) at the +junction point determines the rate of convergence of the perturbed eigenvalue. The second factor +which influences the asymptotic expansion (1.7) is the positive constant Ck,Σ, whose variational +characterization (1.8) sheds some light on its nature. As already noticed for the first time in [2], +this coefficient has to do with the ability of a membrane to respond to a vertical force acting +on it. The main drawback of [16] is the geometric assumptions on the tube’s section Σ and the +hypothesis of simplicity of the limit eigenvalue λN. +In this paper, we mainly address these two open questions. Concerning nontrivial multiplicity +of eigenvalues, less is available in literature as compared to the simple case (we refer to [16] for the +state of the art in this last instance). In this regard, we mention the work by Taylor [23] which +provides an estimate for the eigenvalue variation in a similar context. The distance between the +perturbed and the limit eigenvalue is estimated by Cεa, where the rate a is independent of any +eigenvalue and the constant C depends only on the distance between the limit eigenvalue to the +nearby ones. The same author provides a similar result in collaboration with Collins (see [10]) for +problems with mixed Dirichlet–Neumann boundary conditions if the tube is attached at a point +where Dirichlet condition is imposed. Apart from these, no sharper result on eigenbranches is +available in literature, for which eigenbranches could be distinguished each other by their different +asymptotic behavior as ε → 0. Splitting of eigenbranches is proved in [19, Chapter 3] but only in +dimension 2. To the best of our knowledge, no other result is available on this issue. Nevertheless, +we strongly believe that this is a relevant topic of investigation: multiple eigenvalues appear in +domains with symmetries and this is often the case in applications. +Before stating our main results, we give the fundamental definition of the paper together with +some remarks on it. For f ∈ C1 � +B+ +r0ε +� +we introduce the functional +(1.9) +JΩε +εΣ,f(u) = Jε +f(u) := 1 +2 +� +Ωε +|∇u|2 dx − +� +εΣ +u ∂f +∂xd +dx′, +defined for u ∈ H1 +0(Ωε)1. +Definition 1.1. For any f ∈ C1 � +B+ +r0ε +� +we call the thin f-torsional rigidity of εΣ relative to Ωε +the following quantity +TΩε(εΣ, f) : = −2 +inf +u∈H1 +0 (Ωε) Jε +f(u), += +sup +u∈H1 +0 (Ωε) +� +2 +� +εΣ +u ∂f +∂xd +dx′ − +� +Ωε +|∇u|2 dx +� +. +If +∂f +∂xd = 1 on εΣ, we denote TΩε(εΣ) := TΩε(εΣ, f) and we call it the thin torsional rigidity of εΣ +relative to Ωε. +If we consider εΣ as a variable and Ωε and f as parameters, TΩε(εΣ, f) is a set-function and +will play a crucial role in our analysis. It will be the right quantity to face the perturbation theory +of eigenvalues in this framework as soon as f is a suitable relative eigenfunction. Realizing this +is one of the main novelty of this work. Broadly speaking, this notion of torsional rigidity of the +perturbing set plays a similar role as that of the capacity of a set in [3] (see Section 1.1 for a more +detailed explanation). In Section 2 we report some basic results concerning this quantity, such as +existence and uniqueness of a minimizer for Jε +f, equivalent formulations, monotonicity properties +1we note that Jε +f can be defined for less regular functions f: it may be sufficient that +∂f +∂xd ∈ +� +H1/2 +00 +�′ +provided +the integral � +εΣ u ∂f +∂xd dx′ is meant as a duality product. + +4 +L. ABATANGELO AND R. OGNIBENE +and asymptotic behavior as ε → 0. Here we just mention that there exists a unique function +U Ωε +εΣ,f = U ε +f ∈ H1 +0(Ωε) (depending also on Σ) achieving TΩε(εΣ, f). Moreover, it weakly satisfies + + + + + + + + + +−∆U ε +f = 0, +in Ωε \ εΣ +U ε +f = 0, +on ∂Ωε, +∂U ε +f|Ω +∂xd +− +∂U ε +f|Tε +∂xd += − ∂f +∂xd +, +on εΣ, +in the sense that +(1.10) +� +Ωε +∇U ε +f · ∇ϕ dx = +� +εΣ +ϕ ∂f +∂xd +dx′, +for all ϕ ∈ H1 +0(Ωε). +In addition, one can observe that the map +f �→ U ε +f +is linear. When +∂f +∂xd = 1 on εΣ, we drop the index f and we denote by U ε the unique function +in H1 +0(Ωε) achieving TΩε(εΣ). We call U ε +f and U ε the thin f-torsion function and thin torsion +function of εΣ (relative to Ωε), respectively. +Coming back to our problem, let λN be an eigenvalue of (1.1) with multiplicity m ≥ 1 and let +us denote by E(λN) ⊆ H1 +0(Ω) the associated m-dimensional eigenspace. As already mentioned in +the introduction, for i ∈ {1, . . . , m}, +λε +N+i−1 → λN, +as ε → 0. +Therefore we have exactly m eigenvalue branches departing from the multiple limit eigenvalue λN +and some of them may a priori coincide. We investigate the asymptotic behaviors of λε +N+i−1 as +ε → 0. In order to find good approximations for perturbed eigenvalues we use a slight modification +of a lemma by G. Courtois [11], itself based on the work by Y. Colin de Verdi´ere [9]. It establishes +essentially the possibility to approximate small eigenvalues of a quadratic form with eigenvalues +of the same form restricted to a finite dimensional subspace of the form domain. One pays an +error which can be estimated in terms of spectral projections. For completeness, we report its +statement in Proposition 3.3 as it appears in [3, Proposition 3.1]. Its proof is given in [3, Appendix +B]. Good approximations for perturbed eigenvalues rely on good approximations for perturbed +eigenfunctions. These will be suitable modifications of the limit eigenfunctions corresponding to +the limit eigenvalue λN = · · · = λN+m−1. Best approximations will be +Πεϕ = Φε +ϕ := ϕ + U ε +ϕ, +with ϕ ∈ E(λN), +where U ε +ϕ is the thin ϕ-torsion function, with ϕ an eigenfunction relative to λN. In view of (1.10), +the function Φε +ϕ ∈ H1 +0(Ωε) weakly solves +�−∆Φε +ϕ = λNϕ, +in Ωε, +Φε +ϕ = 0, +on ∂Ωε. +This means that Φε +ϕ is in the domain of the perturbed operator, but −∆Φε +ϕ acts as −∆ϕ in the +sense of distributions on H1 +0(Ω). For further discussion see Remark 2.2. +If we apply Proposition 3.3 in a suitable way (see Section 3.1 for the details) we obtain our first +main result. We state it in the following +Theorem 1.2. For i ∈ {1, . . . , m}, +(1.11) +λε +N+i−1 = λN − µε +i + o(χ2 +ε) as ε → 0, +where +χ2 +ε := sup{TΩε(εΣ, u) : u ∈ E(λN) and ∥u∥L2(Ω) = 1} +and {µε +i}m +i=1 are the eigenvalues (taken in non-increasing order) of the quadratic form rε, defined +for u, v ∈ E(λN) as +rε(u, v) := +� +Ωε +∇U ε +u · ∇U ε +v dx + λN +� +Ωε +U ε +u U ε +v dx, + +5 +where U ε +u (U ε +v) is the thin u-torsion function of εΣ (the thin v-torsion function of εΣ, respectively). +Remark 1.3. In fact, one can obviously even consider m = 1, for this is allowed in Proposition +3.3. In this case, Theorem 1.2 readily implies the main result in [16] (see (1.7) and (1.8)), if +supplied with the blow-up analysis for TΩε(εΣ, f) given in Theorem 2.8. +Although Theorem 1.2 provides a good approximation for perturbed eigenvalues when the limit +one is simple, it is not exhaustive for multiple ones. For instance, for some i it may be µε +i = o(χ2 +ε): +in this case (1.11) reduces to λε +N+i−1 − λN = o(χ2 +ε), providing no sharp asymptotic behavior but +just an estimate. Nevertheless, we expect that eigenbranches’ asymptotic rates will depend on +local behaviors at 0 of properly chosen limit eigenfunctions. In order to improve Theorem 1.2 +in this way, we introduce the following proposition, which is originally given in [3, Proposition +1.10] in a slight different context. It sheds light on the proper choice of the limit eigenbasis: the +limit eigenspace can be uniquely split in several subspaces where eigenfunctions share the same +vanishing order at 0. It is called the order decomposition of E(λN). +Proposition 1.4. ([3, Proposition 1.10]) There exists an integer p ≥ 1, a decomposition of E(λN) +into a sum of orthogonal subspaces +E(λN) = E1 ⊕ · · · ⊕ Ep +and an associated finite increasing sequence of integers +0 < k1 < · · · < kp +such that, for all 1 ≤ j ≤ p, a function ϕ ∈ Ej \ {0} has vanishing order kj at 0, that is +ϕ(rx) +rkj +→ ψkj(x) +in C1,α(B+ +1 ), as ε → 0, +for some harmonic polynomial ψkj, homogeneous of degree kj and odd with respect to xd. +In +addition, such a decomposition is unique. +Secondly, we will need a blow-up analysis for torsion functions. Let us describe in a few words +our procedure. By the change of variables x �→ εx, we zoom in closely to the origin: the perturbed +domain Ωε is transformed into +1 +εΩε = 1 +εΩ ∪ Σ ∪ 1 +εTε, +which is intuitively “converging” to Π (1.6) as ε → 0. +Given the order decomposition as in +Proposition 1.4 and fixed j ∈ {1, . . . , p}, we then consider the map +Bj : Ej → Pkj +odd, +where Bjϕ is the harmonic homogeneous polynomial of degree kj which describes the local be- +havior of ϕ ∈ Ej near the origin, namely +ϕ(rx) +rkj +→ (Bjϕ) (x) +in C1,α(B+ +1 ), as r → 0 +and Pkj +odd denotes the space of harmonic polynomials homogeneous of degree kj odd with respect +to the xd variable. We observe that for ϕ ∈ Ej a simple change of variables gives +TΩε(εΣ, ϕ) = −2 +�1 +2 +� +Ωε +��∇U ε +ϕ +��2 dx − +� +εΣ +U ε +ϕ +∂ϕ +∂xd +dx′ +� += −2εd−2+2kj +� +1 +2 +� +1 +ε Ωε +|∇ ˆU ε +ϕ|2 dx − +� +Σ +ˆU ε +ϕ +∂ ˆϕε +∂xd +dx′ +� +, +where +ˆU ε +ϕ(x) := U ε +ϕ(εx) +εkj +and +ˆϕε(x) := ϕ(εx) +εkj +. + +6 +L. ABATANGELO AND R. OGNIBENE +In view of this, it is reasonable to investigate the behavior of ˆU ε +ϕ as ε → 0 and to expect that +ε−d+2−2kjTΩε(εΣ, ϕ) admits a nontrivial, finite, limit. This motivates the following quantity: +(1.12) +TΠ(Σ, Ψ) := −2 inf +�1 +2 +� +Π +|∇u|2 dx − +� +Σ +u ∂Ψ +∂xd +dx′ : u ∈ D1,2(Π) +� +, +which we define for Ψ ∈ C1(B+ +r0)2. If +∂Ψ +∂xd = 1 on Σ, we denote TΠ(Σ) := TΠ(Σ, Ψ). Therefore, +Theorem 2.8 will establish +ε−d+2−kjTΩε(εΣ, ϕ) → TΠ(Σ, Bjϕ) +as ε → 0, for all ϕ ∈ Ej, +for all j = 1, . . . , p. Hereafter we denote by U Π +Σ,ϕ the function achieving TΠ(Σ, Bjϕ), for ϕ ∈ Ej. +Taking into account Proposition 1.4 and this blow-up analysis, we are able to prove the following +result about the sharp asymptotic behavior of perturbed eigenvalues. Before stating it, we need +the following notation. For 1 ≤ j ≤ p, we let Ej as in Proposition 1.4 and we write +mj := dim (Ej), +so that +m = m1 + · · · + mj + · · · + mp, +whereas we denote by +µj,1 ≥ · · · ≥ µj,ℓ ≥ · · · ≥ µj,mj > 0 +Tj(u, v) = +� +Π +∇Uu · ∇Uv dx +defined for u, v ∈ Ej ⊆ E(λN), +where Uu = U Π +Σ,u and Uv = U Π +Σ,v achieve TΠ(Σ, Bju) and TΠ(Σ, Bjv), respectively. We are now +ready to state the main result of our paper. +Theorem 1.5. For any i ∈ {1, . . ., m}, there holds +(1.13) +λε +N+i−1 = λN − µj,ℓ εd−2+2kj + o(εd−2+2kj), +as ε → 0, +where +(j, ℓ) = + + + + + + + + + + + +(1, i) +if 1 ≤ i ≤ m1 +(2, i − m1) +if m1 + 1 ≤ i ≤ m1 + m2 +... +... +(p, i − (m − mp)) +if m − mp + 1 ≤ i ≤ m +This result concludes our analysis on this class of problems: it provides sharp asymptotics for +any perturbed eigenvalue. +Let us now analyze more in depth the particular case when p = 1 and k1 = 1. This is relevant +since the gradient of limit eigenfunctions vanish at most on a subset of ∂Ω ∩ {xd = 0} which +has zero (d − 1)-dimensional measure (see e.g. [20]). Thus, the limit eigenfunctions vanish with +order 1 for Ld−1-almost every point in ∂Ω ∩ {xd = 0}. Broadly speaking, vanishing order 1 occurs +generically with respect to the points in ∂Ω ∩ {xd = 0}. We also observe that if d = 2 p = 1 can +only uccur if m = 1 (see Remark 4.2). If u ∈ E(λN), we have that +B1u = ∂u +∂xd +(0) xd, +that is +u(rx) +r +→ ∂u +∂xd +(0) xd, +in C1,α(B+ +1 ), as ε → 0. +Hence, by linearity there holds +U Π +Σ,u = ∂u +∂xd +(0) U Π +Σ,xd. +2We point out that a little abuse of notation has been made, since this minimization is made within D1,2(Π), +which is strictly larger than H1 +0(Π). + +7 +Therefore, we can write down the quadratic form +T1(u, v) = ∂u +∂xd +(0) ∂v +∂xd +(0) +� +Π +��∇U Π +Σ,xd +��2 dx = ∂u +∂xd +(0) ∂v +∂xd +(0) TΠ(Σ). +In particular, it is possible to choose an eigenbasis {ϕN, . . . , ϕN+m−1} for E(λN) which diagonal- +izes T1, in such a way that +µ1,i = +�∂ϕN+i−1 +∂xd +(0) +�2 +TΠ(Σ), +for i = 1, . . . , m, +thus leading to the following. +Corollary 1.6. Let us assume Proposition 1.4 holds with p = 1 and k1 = 1. Then there exists a +basis {ϕN, . . . , ϕN+m−1} of E(λN), orthonormal in L2(Ω) and such that, for all i ∈ {1, . . ., m}, +there holds +λε +N+i−1 = λN − +�∂ϕN+i−1 +∂xd +(0) +�2 +TΠ(Σ) εd + o(εd), +as ε → 0. +We observe that this result recovers, in the case m = 1, what the authors obtained in [2]. +Finally, we would like to emphasize another particular instance, which concerns the behavior of +the first eigenvalue, being one of the most widely studied set functions. In this case, it is known +that the limit eigenvalue is simple (i.e. m = 1) and that the corresponding eigenfunctions have +nonzero gradient on any regular boundary point, being them positive in Ω (i.e. p = 1 and k1 = 1). +Hence, we have the following. +Corollary 1.7. Let ϕ1 ∈ H1 +0(Ω) be a normalized eigenfunction corresponding to λ1. Then there +holds +λε +1 = λ1 − +�∂ϕ1 +∂xd +(0) +�2 +TΠ(Σ) εd + o(εd), +as ε → 0. +It is worth noticing that, in these two last results, the coefficients of the first term in the +asymptotic expansion of the eigenvalue variation split as a product of two factors: one of them +only depending on the behavior of the limit eigenfunctions at the origin and the other one only +depending on the geometry of the set Σ. +The paper is organized as follows. Section 2 is devoted to present basic properties of the thin f- +torsional rigidity and contains preliminary result in view of the main theorems. Section 3 contains +the proof of Theorem 1.2, whereas Section 4 contains the proof of Theorem 1.5. +1.1. Mixed Dirichlet–Neumann boundary conditions. The preceeding arguments apply also +when we perturb problem (1.1) by prescribing that eigenfunctions satisfy homogeneous Neumann +boundary conditions on εΣ, in place of attaching a thin tube with section εΣ to the fixed domain +Ω. +The problem has been already studied in dimension 2 in [18], achieving a full asymptotic +expansion of perturbed eigenvalues (see also [1] for related results) and in any dimension by [15] +but only for simple eigenvalues. This last paper is our starting point. +Let us consider the weak form of the eigenvalue problem +(1.14) + + + + + + + + + +−∆ξ = ˜λǫξ, +in Ω +ξ = 0, +on ∂Ω \ εΣ +∂ξ +∂ν = 0 +on εΣ. +To this aim, we set the functional framework as it appears in [15], by introducing the space +H1 +0,∂Ω\εΣ(Ω), defined as the closure in H1(Ω) of C∞ +c (Ω∪εΣ). We say that ˜λε ∈ R is an eigenvalue +of (1.14) if there exists ξε ∈ H1 +0,∂Ω\εΣ(Ω) \ {0} (named eigenfunction) such that +� +Ω +∇ξε · ∇v dx = ˜λε +� +Ω +ξεv dx +for all v ∈ H1 +0,∂Ω\εΣ(Ω). + +8 +L. ABATANGELO AND R. OGNIBENE +For any ε ∈ (0, 1) there exists a non-decreasing sequence of positive eigenvalues +0 < ˜λε +1 < ˜λε +2 ≤ · · · ≤ ˜λε +n ≤ · · · → +∞. +When ε → 0 the Neumann region disappears and the Dirichlet region covers the entire boundary. +One then expects the eigenelements of the mixed Dirichlet-Neumann problem (1.14) to converge +to the ones of the limit Dirichlet problem (1.1). This problem revealed to be more involved than +its counterpart, in which Neumann boundary condition are prescribed on a large part of ∂Ω and +Dirichlet boundary conditions on a vanishing portion of it. A capacitary approach (such as the one +developed in [14] for the case of disappearing Dirichlet region) turns out to be particularly effective +when functions are required to vanish in “small” sets; this is basically related to the known fact +that Sobolev spaces are not affected if their functions are prescribed to vanish on zero capacity +sets. So, the case introduced in this subsection falls outside a capacitary context. Furthermore, +to the best of our knowledge, in literature there is no analogue to the capacity, which can play a +similar role in the converse case (1.14). Because of this, in [15] the authors undertake the problem +through another approach, based on Almgren-type monotonicity formula. In this work they prove +the convergence of the perturbed spectrum, as ε → 0, to the spectrum of the Dirichlet-Laplaciann +(see [15, Proposition 2.3]), that is +(1.15) +˜λǫ +n → λn +as ε → 0 for any n ∈ N \ {0}. +Moreover, in case of simple limit eigenvalues, they provide an explicit asymptotic expansion of the +perturbed eigenvalues, which is sharp only when the set Σ fulfills certain geometric assumptions. +In particular, they proved that, if Σ is strictly starshaped with respect to the origin, if λN is simple +and if (1.5) holds, then +(1.16) +˜λε +N = λN − ˜Ck,Σ εd−2+2k + o(εd−2+2k), +as ε → 0, +where +˜Ck,Σ := −2 +inf +u∈D1,2(Rd ++∪Σ) +� +1 +2 +� +Rd ++ +|∇u|2 dx + +� +Σ +u∂ψk +∂ν dx′ +� +and D1,2(Rd ++ ∪ Σ) denotes the completion of C∞ +c (Rd ++ ∪ Σ) with respect to the L2 norm of the +gradient. We also observe that here ν = (0, . . . , 0, −1) and ∂ψk +∂ν = − ∂ψk +∂xd . +With the very same method used to prove the results in the previous subsection, we are able to +remove both the simplicity assumption on the limit eigenvalue λN and the geometric assumption +on Σ and to prove a sharp asymptotic expansion for ˜λε +N in the general case. In the same spirit of +the previous case, here we are able to detect the proper quantity which measures the magnitude +of the perturbation and, consequently, the stability of eigenvalues. It is still a notion of torsional +rigidity and plays the role of perfect counterpart of the capacity, in the present framework. Let +us first introduce the functional +˜JΩ +εΣ,f(u) = ˜Jε +f(u) := 1 +2 +� +Ω +|∇u|2 dx + +� +εΣ +u ∂f +∂ν dx′, +defined for u ∈ H1 +0,∂Ω\εΣ(Ω), where f ∈ C1(B+ +r0). In the same lines as in the previous subsection, +we introduce the notion of relative torsional rigidity of a set which is suitable for our problem. +Definition 1.8. For any f ∈ C1(B+ +r0) we call the boundary f-torsional rigidity of εΣ relative to +Ω the following quantity +TΩ(εΣ, f) : = −2 +inf +u∈H1 +0,∂Ω\εΣ(Ω) +˜Jε +f(u) += −2 +inf +u∈H1 +0,∂Ω\εΣ(Ω) +�1 +2 +� +Ω +|∇u|2 dx + +� +εΣ +u ∂f +∂ν dx′ +� +If ∂f +∂ν = −1 on εΣ, we denote TΩ(εΣ) := TΩ(εΣ, f) and we call it the boundary torsional rigidity +of εΣ relative to Ω. + +9 +We point out that Definition 1.1 and Definition 1.8 are completely matching each other. As in +the previous case, by standard minimization methods there exists ˜U ε +f = ˜U Ω +εΣ,f ∈ H1 +0,∂Ω\εΣ(Ω)\{0} +achieving TΩ(εΣ, f). We call ˜U ε +f the boundary f-torsion function of εΣ, relative to Ω. In particular, +˜U ε +f satisfies + + + + + + + + + +−∆ ˜U ε +f = 0, +in Ω, +˜U ε +f = 0, +on ∂Ω \ εΣ, +∂ ˜U ε +f +∂ν = − ∂f +∂ν , +on εΣ +in a weak sense, that is ˜U ε +f ∈ H1 +0,∂Ω\εΣ(Ω) and +(1.17) +� +Ω +∇ ˜U ε +f · ∇ϕ dx = − +� +εΣ +ϕ∂f +∂ν dx′ +for all ϕ ∈ H1 +0,∂Ω\εΣ(Ω). +Let λN be an eigenvalue to the problem (1.14) with multiplicity m and E(λN) be the associated +eigenspace. Our first result on this problem is the analogue of Theorem 1.2. +Theorem 1.9. For i ∈ {1, . . . , m}, +˜λε +N+i−1 = λN − ˜µε +i + o(˜χ2 +ε) as ε → 0, +where +˜χ2 +ε := sup{TΩ(εΣ, u) : u ∈ E(λN) and ∥u∥L2(Ω) = 1} +according to Definition 1.8 and {˜µε +i}m +i=1 are the eigenvalues (taken in non-increasing order) of the +quadratic form ˜rε, defined for u, v ∈ E(λN) as +˜rε(u, v) := +� +Ω +∇ ˜U ε +u · ∇ ˜U ε +v + λN +� +Ω +˜U ε +u ˜U ε +v, +where ˜U ε +u ( ˜U ε +v) is the boundary u-torsion function of εΣ (the boundary v-torsion function of εΣ, +respectively). +Given the order decomposition of E(λN), as in Proposition 1.4, and taking ϕ ∈ Ej (for j ∈ +{1, . . . , p}), it is possible to perform a blow-up analysis for a suitable rescaling of the ϕ-boundary +torsion function ˜U ε +ϕ. Namely, following the steps as in Section 2.2 (outlined in the previous section), +one can prove that +ε−d+2−2kj ˜U ε +ϕ(εx) +as well as +ε−d+2−2kkTΩ(εΣ, ϕ) +admit nontrivial, finite limits as ε → 0. It is then natural to introduce the following quantity, +defined for Ψ ∈ C1(B+ +1 ), +(1.18) +TRd ++(Σ, Ψ) := −2 +� +1 +2 +� +Rd ++ +|∇u|2 dx + +� +Σ +u∂Ψ +∂ν dx′ : u ∈ D1,2(Rd ++ ∪ Σ) +� +. +If ∂Ψ +∂ν = −1 on Σ we denote TRd ++(Σ) := TRd ++(Σ, Ψ). +Hence, we are able to prove the following (which is the analogous of Theorem 2.8). +Lemma 1.10. Let j ∈ {1, . . . , p} and let ϕ ∈ Ej. Then +TΩ(εΣ, ϕ) = εd−2+2kTRd ++(Σ, Bjϕ) + o(εd−2+2kj), +as ε → 0. +In addition +ε−d+2−kj ˜U ε +ϕ(εx) → ˜U +Rd ++ +Σ,ϕ, +in D1,2(Rd ++ ∪ Σ), as ε → 0, +where ˜U +Rd ++ +Σ,ϕ ∈ D1,2(Rd ++ ∪ Σ) denotes the function achieving TRd ++(Σ, Bjϕ). +Remark 1.11. Of course, even in this case one can consider m = 1, for this is allowed in +Proposition 3.3. In this way, Theorem 1.9 provides immediately the main result as stated in [15] +if supplied with Lemma 1.10. + +10 +L. ABATANGELO AND R. OGNIBENE +Taking into account the order decomposition Proposition 1.4 for E(λN), and the blow-up anal- +ysis for scaled torsion functions, we are able to improve the result of Theorem 1.9. Before stating +the main theorem, we need the following notation. For 1 ≤ j ≤ p, we let Ej as in Proposition 1.4 +and we recall that mj = dim (Ej) so that +m = m1 + · · · + mj + · · · + mp. +Moreover, we denote by +˜µj,1 ≥ · · · ≥ ˜µj,ℓ ≥ · · · ≥ ˜µℓ,mj > 0 +the eigenvalues of the quadratic form +˜Tj(u, v) = +� +Rd ++ +∇ ˜Uu · ∇ ˜Uv dx +defined for u, v ∈ Ej ⊆ E(λN), +where ˜Uu = ˜U +Rd ++ +Σ,u and ˜Uv = ˜U +Rd ++ +Σ,v achieve, respectively, TRd ++(Σ, Bju) and TRd ++(Σ, Bjv). We are now +ready to state the main result of this section, which is the analogue of Theorem 1.5. +Theorem 1.12. For any i ∈ {1, . . . , m}, there holds +(1.19) +˜λε +N+i−1 = λN − ˜µj,ℓ εd−2+2kj + o(εd−2+2kj), +as ε → 0, +where +(j, ℓ) = + + + + + + + + + + + +(1, i) +if 1 ≤ i ≤ m1 +(2, i − m1) +if m1 + 1 ≤ i ≤ m1 + m2 +... +... +(p, i − (m − mp)) +if m − mp + 1 ≤ i ≤ m +For simplicity of exposition, we do not present here the proofs of Theorem 1.9 and 1.12, since +they follow step by step the proofs of the case of domains with handles attached. +It will be +sufficient to set all the arguments in the appropriate functional setting, as described above. +Also for this kind of perturbation, we think it is interesting to understand what happens in +some particular cases, similarly to what we described for the attachment of a thin tube in the +previous section. More precisely, reasoning in a completely analogous way, if p = 1 and k1 = 1 +one can see that it is possible to find an eigenbasis {ϕN, . . . , ϕN+m−1} ⊆ E(λN), orthonormal in +L2(Ω), in such a way that +˜µ1,i = +�∂ϕN+i−1 +∂ν +(0) +�2 +TRd ++(Σ) +for i = 1, . . . , m. +Before stating the result, we would like to observe that, in view of the characterization of the +half-laplacian (−∆Rd−1) +1 +2 on Rd−1 = ∂Rd ++ as a Dirichlet-to-Neumann map on Rd ++, on can easily +see that the quantity TRd ++(Σ) coincides with 1 +2-fractional torsional rigidity of Σ in Rd−1. Namely +TRd ++(Σ) = T +1 +2 +Rd−1(Σ), where +T +1 +2 +Rd−1(Σ) := −2 inf +�1 +2 ∥u∥2 +D +1 +2 ,2(Rd−1) − +� +Σ +u: u ∈ D +1 +2 ,2 +0 +(Σ) +� +and D +1 +2 ,2 +0 +(Σ) denotes the completion of C∞ +c (Σ) with respect to the norm +∥u∥D +1 +2 ,2(Rd−1) := +� +1 +(2π) +d−1 +2 +� +Rd−1 |ζ| |ˆu(ζ)|2 dζ +� 1 +2 +. +Here by ˆu we denote the (normalized) Fourier transform of u in Rd−1. We thus have the following. +Corollary 1.13. Let us assume Proposition 1.4 holds with p = 1 and k1 = 1. Then there exists a +basis {ϕN, . . . , ϕN+m−1} of E(λN), orthonormal in L2(Ω) and such that, for all i ∈ {1, . . ., m}, +there holds +˜λε +N+i−1 = λN − +�∂ϕN+i−1 +∂ν +(0) +�2 +T +1 +2 +Rd−1(Σ) εd + o(εd), +as ε → 0. + +11 +We can also investigate, as a remarkable instance, the perturbation of the first eigenvalue and +obtain the following. +Corollary 1.14. Let ϕ1 ∈ H1 +0(Ω) be a normalized eigenfunction corresponding to λ1. Then there +holds +λε +1 = λ1 − +�∂ϕ1 +∂xd +(0) +�2 +T +1 +2 +Rd−1(Σ) εd + o(εd), +as ε → 0. +2. Facts about TΩε(εΣ, f) +In this section we collect some basic facts regarding the notion of thin f-torsional rigidity of +εΣ introduced before. +2.1. Basics. Firstly, we briefly mention the variational framework for TΩε(εΣ, f). +As already +mentioned, by standard minimization methods, it can be proved that, for any f ∈ C1(B+ +r0ε), there +exists a unique U Ωε +εΣ,f = U ε +f ∈ H1 +0(Ωε) \ {0} such that +Jf +ε (U ε +f ) = +inf +u∈H1 +0 (Ωε) Jf +ε (u), +where Jf +ε is as in (1.9). In particular, U ε +f satisfies +(2.1) +0 = dJf +ε (U ε +f )[ϕ] = +� +Ωε +∇U ε +f · ∇ϕ dx − +� +εΣ +ϕ ∂f +∂xd +dx′ +for all ϕ ∈ H1 +0(Ωε). +Letting ϕ = U ε +f in the previous equation we get +� +Ωε +��∇U ε +f +��2 dx = +� +εΣ +U ε +f +∂f +∂xd +dx′, +hence obtaining +(2.2) +TΩε(εΣ, f) = +� +Ωε +��∇U ε +f +��2 dx = +� +εΣ +U ε +f +∂f +∂xd +dx′. +The first property deals with equivalent definitions for the thin f-torsional rigidity of εΣ. +Lemma 2.1. Definition (1.1) is equivalent to the following +(2.3) +TΩε(εΣ, f) = +sup +u∈H1 +0 (Ωε)\{0} +�� +εΣ +u ∂f +∂xd +dx′ +�2 +� +Ωε +|∇u|2 dx +. +Proof. By definition, +TΩε(εΣ, f) = +sup +u∈H1 +0 (Ωε)\{0} +sup +t>0 +� +2t +� +εΣ +u ∂f +∂xd +dx′ − t2 +� +Ωε +|∇u|2 dx +� +and the inner supremum is actually attained at +t = +� +εΣ +u ∂f +∂xd +dx′ +� +Ωε +|∇u|2 dx +. +Substituting this value into the previous equality leads to (2.3). +□ +Remark 2.2. We find useful to note that if ϕ ∈ E(λN) and uε is a perturbed eigenfunction then +by (1.1) and (1.4) we have +� +εΣ +∂ϕ +∂xd +uε = (λN − λε) +� +Ω +ϕuε. + +12 +L. ABATANGELO AND R. OGNIBENE +From the latter equality and Lemma 2.1 it follows the meaningful estimate +λεTΩε(εΣ, ϕ) ≥ +� +(λN − λε) +� +Ω +ϕuε +�2 +. +On the other hand, if we denote the bounded linear +F : H1 +0(Ωε) → H−1(Ωε) +u �→ − +� +εΣ +∂ϕ +∂xd +u, +then by (2.3) +� +TΩε(εΣ, ϕ) = ∥F∥∗, +as the thin ϕ- torsion function U ε +ϕ is the least energy element in F−1(∥F∥∗) ⊆ H1 +0(Ωε). +The next properties deal with its behavior as ε → 0. +Lemma 2.3. If ε1 > ε2 then for any f ∈ C1(B+ +r0ε1) we have +TΩε1 (ε1Σ, f) ≥ TΩε2 (ε2Σ, f). +Proof. The statement is obvious thanks to the inclusion H1 +0(Ωε2) ⊆ H1 +0(Ωε1). +□ +Lemma 2.4. For any f ∈ E(λN) we have that +TΩε(εΣ, f) → 0 +as ε → 0. +Proof. Taking into account (2.3), by Cauchy-Schwarz Inequality, the trace embedding H1(B+ +r0) ֒→ +L2(εΣ) and regularity of eigenfunctions we have that +TΩ(εΣ, f) ≤ +sup +u∈H1 +0 (Ωε)\{0} +� +εΣ +u2 dS +� +εΣ +� ∂f +∂xd +�2 +dx′ +� +Ω +|∇u|2 dx += +� +εΣ +� ∂f +∂xd +�2 +dx′ +sup +u∈H1 +0 (Ωε)\{0} +� +εΣ +u2 dx′ +� +Ω +|∇u|2 dx +(2.4) +≤ Cd,Ω,r0 +���� +∂f +∂xd +���� +2 +H1(B+ +r0) +sup +u∈H1 +0 (Ωε)\{0} +� +εΣ u2 dx′ +� +Ωε |∇u|2 dx +. +(2.5) +By scaling we have +sup +u∈H1 +0 (Ωε)\{0} +� +εΣ u2 dx′ +� +Ωε |∇u|2 dx +≤ +sup +u∈H1 +0 (Ωε)\{0} +� +εΣ u2 dx′ +� +B+ +ε ∪Tε |∇u|2 dx +≤ +sup +u∈H1 +0,∂(B+ +1 ∪T1)\S+ +1 +(B+ +1 ) +ε +� +Σ u2 dx′ +� +B+ +1 ∪T1 |∇u|2 dx += CΣ ε +(2.6) +where +CΣ = +sup +u∈H1 +0,∂(B+ +1 ∪T1)\S+ +1 +(B+ +1 ) +� +Σ u2 dx′ +� +B+ +1 ∪T1 |∇u|2 dx +> 0 +and, for any compact set K ⊆ B+ +1 , the space H1 +0,K(B+ +1 ) is defined as the closure of C∞ +c (B+ +1 \ K) +with respect to the H1 norm. Actually, for regular K there holds H1 +0,K(B+ +1 ) = {u ∈ H1(B+ +1 ): u = +0 on K}. Invoking (2.5) and (2.6) we conclude the proof. +□ + +13 +2.2. Blow-up analysis for the thin f-torsion function. As already mentioned in the intro- +duction, spectral stability for this problem is ensured by the results in [12]. It is a consequence of +the uniform convergence of the resolvents. Nevertheless, in order to perform a blow-up analysis +as ε → 0 for the thin torsion function, we need a fundamental notion of convergence of sets (or +functional spaces): it is the so-called convergence in the sense of Mosco. In our setting of scaling +handles it is established in [12, Section 7]. We report here the definition for future reference. +Definition 2.5. Let Hε, H0 and H be Hilbert spaces such that Hε, H0 ⊆ H for all ε ∈ (0, 1). We +say that Hε converges to H0 in the sense of Mosco in H if the following hold: +(M1) if vε ∈ Hε for all ε ∈ (0, 1) and vε ⇀ v weakly in H, as ε → 0, then v ∈ H0; +(M2) for any v ∈ H0 there exists a sequence {vε}ε∈(0,1) such that vε ∈ Hε for all ε ∈ (0, 1) and +vε → v strongly in H. +We start this last subsection giving an important lemma for the forthcoming analysis. +Lemma 2.6. Let f ∈ E(λN) and let U ε +f ∈ H1 +0(Ωε) be the thin f-torsion function of εΣ. Then +� +Ωε +|U ε +f|2 dx = o(TΩε(εΣ, f)), +as ε → 0. +Proof. Let us assume by contradiction that there exists a sequence εn → 0 and a constant C > 0 +such that +� +Ωε +|U εn +f |2 dx ≥ 1 +C TΩε(Σεn, f). +We set +Wn := +U εn +f +∥U εn +f ∥L2(Ωε) +. +We have +∥Wn∥L2(Ωε) = 1 +and recalling (2.2) +∥∇Wn∥2 +L2(Ωε) = +1 +∥U εn +f ∥2 +L2(Ωε) +TΩε(Σεn, f) ≤ C. +By the weak compactness of the unit ball of H1 +0(Ωε0), the compactness of the inclusion H1 +0(Ωε0) ⊂ +L2(Ωε0) and thanks to the convergence of the perturbed domains in sense of Mosco (see Definition +2.5), there exists an increasing sequence of integers (nk)k≥1 and a function W ∈ H1 +0(Ω) such that +(Wnk)k≥1 converges to W when k goes to +∞, weakly in H1 +0(Ωε0) and strongly in L2(Ωε0). We +have that at the same time ∥W∥L2(Ω) = 1 and +� +Ω ∇W · ∇ϕ = 0 for any ϕ ∈ H1 +0(Ω), therefore W +is identically 0. We have reached a contradiction and proved the lemma. +□ +We now turn to the very aim of the subsection. In order to give the blow-up result on the thin +f-torsion function we start with an estimate on its energy. +Lemma 2.7. Let f ∈ E(λN) be such that +f(εx) +εk +→ ψk(x) +in C1,α(B+ +1 ) as ε → 0, +for some integer k ≥ 1 and some harmonic polynomial ψk, homogeneous of degree k. Then +TΩε(εΣ, f) = O(εd+2k−2), +as ε → 0. +Proof. We start from (2.4). Moreover, by assumption there holds +(2.7) +� +εΣ +� ∂f +∂xd +�2 +dx′ = εd+2k−3 +� +Σ +� ∂ +∂xd +�f(εx′) +εk +��2 +dx′ = O(εd+2k−3), +as ε → 0. The conclusion follows from (2.4), (2.6), and (2.7). +□ + +14 +L. ABATANGELO AND R. OGNIBENE +Theorem 2.8. Let U ε +f ∈ H1 +0(Ωε) be the thin f-torsion function of εΣ. Under the same assump- +tions as in Lemma 2.7 there holds +ˆUε(x) := +U ε +f(εx) +εk +→ U Π +Σ,ψk(x) +in D1,2(Π) as ε → 0, +where U Π +Σ,ψk achieves TΠ(Σ, ψk) as defined in (1.18). Moreover, +TΩε(εΣ, f) = εd−2+2kTΠ(Σ, ψk) + o(εd−2+2k), +as ε → 0. +Proof. From Lemma 2.7 we deduce that +(2.8) +� +1 +ε Ω∪T +|∇ ˆUε|2 dx ≤ C, +for some C > 0 independent from ε, thus implying that { ˆUε}ε is bounded in D1,2(Π), if ˆUε is +meant to be trivially extended in Π \ ( 1 +εΩ ∪ T ). Then there exist a subsequence (still denoted by +{ ˆUε}ε) and a function W ∈ D1,2(Π) such that +ˆUε ⇀ W +in D1,2(Π), +(2.9) +ˆUε → W +in L2(Σ), +(2.10) +as ε → 0 by compactness of trace embedding. Now, from (2.9) and (1.5) and the equation satisfied +by ˆUε, one can easily derive the equation satisfied by W, which is +� +Π +∇W · ∇ϕ dx = +� +Σ +ϕ∂ψk +∂xd +for all ϕ ∈ D1,2(Π). +By the uniqueness of the minimizer of TΠ(Σ, ψk) (see also [16, Proposition 2.2] and [17, Lemma +2.4]) we have that W = U Π +Σ,ψk. Finally +ε−d−2k+2TΩε(εΣ, f) = +� +1 +ε Ω∪T +|∇ ˆUε|2 dx = +� +Σ +ˆUε +∂ +∂xd +�f(εx) +εk +� +dx′ +→ +� +Σ +U Π +Σ,ψk +∂ψk +∂xd +dx′ = +� +Π +|∇U Π +Σ,ψk|2 dx = TΠ(Σ, ψk) +as ε → 0, and the proof is concluded. +□ +3. Perturbation of eigenvalues +Our subsequent analysis is close to [11, Proof of Theorem 1.2] and [3, Section 3], except that we +replace the standard capacity or the u-capacity with the thin f-torsional rigidity defined above. +We introduce the quantity χε: +(3.1) +χ2 +ε := sup{TΩε(εΣ, u) : u ∈ E(λN) and ∥u∥L2(Ω) = 1} +Lemma 3.1. There holds +χε → 0, +as ε → 0. +Proof. Let us pick u ∈ E(λN) such that ∥u∥L2(Ω) = 1. +If {uN+i−1}m +i=1 ⊆ E(λN) denotes a +L2(Ω)-orthonormal basis, we write u = �m +i=1 ciuN+i−1, with �m +i=1 c2 +i = 1. Then by linearity, + +15 +Cauchy-Schwarz inequality, the trivial inequality (�m +i=1 ai)2 ≤ m �m +i=1 a2 +i and (2.2) +TΩε(εΣ, u) = +������ +� +1≤i,j≤m +cicj +� +Ωε +∇U ε +uN+i−1 · ∇U ε +uN+j−1 dx +������ +≤ +� +1≤i,j≤m +|ci||cj| +�� +Ωε +|∇U ε +uN+i−1|2 dx +� 1 +2 �� +Ω +|∇U ε +uN+j−1|2 dx +� 1 +2 += +� m +� +i=1 +|ci| +�� +Ωε +|∇U ε +uN+i−1|2 dx +� 1 +2 �2 +≤ m +� +max +1≤i≤m +� +Ωε +|∇U ε +uN+i−1|2 dx +� m +� +i=1 +c2 +i = m max +1≤i≤m TΩε(εΣ, uN+i−1). +By Lemma 2.4 TΩε(εΣ, uN+i−1) → 0 for all 1 ≤ i ≤ m: the proof is complete. +□ +For ε > 0, we denote by Πε the linear mapping +Πε : +E(λN) +→ +H1 +0(Ωε) +u +�→ +u + U ε +u, +where E(λN) and H1 +0(Ωε) are considered to be endowed, respectively, with the L2(Ω) and L2(Ωε) +norms. +Lemma 3.2. If Mε := ∥I − Πε∥L(E(λN ),H1 +0 (Ωε)), there holds +Mε = o(χε), +as ε → 0. +Proof. Let v ∈ E(λN) such that ∥v∥L2(Ω) = 1 and let us write v = �m +i=1 ciuN+i−1, for some +{ci}m +i=1 such that �m +i=1 c2 +i = 1, being {uN+i−1}m +i=1 a basis of E(λN) orthonormal in L2(Ω). By +definition, we have (Πε − I)v = U ε +v. Hence, by linearity and Cauchy-Schwarz inequality we find +that +∥(Πε − I)v∥L(E(λN ),L2(Ωε)) = ∥U ε +v∥L2(Ωε) +≤ +m +� +i=1 +|ci|∥U ε +uN+i−1∥L2(Ωε) ≤ +� m +� +i=1 +c2 +i +� 1 +2 � m +� +i=1 +∥U ε +uN+i−1∥2 +L2(Ωε) +� 1 +2 += +� m +� +i=1 +TΩε(εΣ, uN+i−1) +∥U ε +uN+i−1∥2 +L2(Ωε) +TΩε(εΣ, uN+i−1) +� 1 +2 +≤ √m χε max +1≤i≤m +∥U ε +N+i−1∥L2(Ωε) +TΩε(εΣ, uN+i−1)1/2 . +According to Lemma 2.6, the last term is o(χε), as ε → 0, and this concludes the proof. +□ +We observe that, in particular, Lemma 3.2 that Mε < 1, meaning that Πε is injective, for ε +small enough. We will always assume this to be the case in the rest of this section. +3.1. Application of the abstract lemma. We here recall the abstract result needed in order +to find good approximation of perturbed eigenvalues. +Proposition 3.3 ([3], Proposition 3.1). Let (H, ∥ · ∥) be a Hilbert space and q be a quadratic +form, semi-bounded from below (not necessarily positive), with domain D dense in H and with +discrete spectrum {νi}i≥1. Let {gi}i≥1 be an orthonormal basis of eigenvectors of q. Let N and +m be positive integers, F an m-dimensional subspace of D and {ξF +i }m +i=1 the eigenvalues of the +restriction of q to F. +Assume that there exist positive constants γ and δ such that +(H1) 0 < δ < γ/ +√ +2; +(H2) for all i ∈ {1, . . . , m}, |νN+i−1| ≤ γ, νN+m ≥ γ and, if N ≥ 2, νN−1 ≤ −γ; + +16 +L. ABATANGELO AND R. OGNIBENE +(H3) |q(ϕ, g)| ≤ δ ∥ϕ∥ ∥g∥ for all g ∈ D and ϕ ∈ F. +Then we have +(i) +��νN+i−1 − ξF +i +�� ≤ 4 +γ δ2 for all i = 1, . . . , m; +(ii) ∥ΠN − I∥L(F,H) ≤ +√ +2δ/γ, where ΠN is the projection onto the subspace of D spanned by +{gN, . . . , gN+m−1}. +We are going to apply Proposition 3.3 in the following way. Here we follow the outline of [3]. +For ε > 0 small enough, we introduce the following set of definitions (3.2)–(3.5): +Hε := L2(Ωε) and ∥·∥ := ∥·∥L2(Ωε) ; +(3.2) +Dε := H1 +0(Ωε); +(3.3) +qε(u) := +� +Ωε +|∇u|2 dx − λN +� +Ωε +u2 dx, +for all u ∈ Dε; +(3.4) +Fε := Πε(E(λN)). +(3.5) +By construction, the eigenvalues of qε are {λε +i − λN}i≥1. We use the notation νε +i := λε +i − λN. +If ε is small enough, Lemma 3.2 implies that Πε is injective, so that Πε is bijective from E(λN) +onto Fε and Fε is proved to be m-dimensional. Since λε +i → λi for all i ∈ N \ {0} and since λN is +of multiplicity m, the assumption (H2) in Proposition 3.3 is fulfilled for ε > 0 small enough if we +take, for instance, +γ := 1 +2 min{λN − λN−1, λN+m − λN+m−1} +when N ≥ 2 and, when N = 1 (in which case m = 1), +γ := 1 +2 (λ2 − λ1) . +It remains to check whether condition (H3) in Proposition 3.3 is satisfied. Let us choose u ∈ Fε +and w ∈ Dε. Since Πε is injective, as a consequence of Lemma 3.2, there exists a unique v ∈ E(λN) +such that u = Πεv. Hence, we have +qε(u, w) = +� +Ωε +∇(v + U ε +v) · ∇w dx − λN +� +Ωε +(v + U ε +v)w dx += +� +Ω +∇v · ∇w dx + +� +Ωε +∇U ε +v · ∇w dx − λN +� +Ω +vw dx − λN +� +Ωε +U ε +vw dx += +� +εΣ +∂v +∂νΩ +w dx′ + +� +Ωε +∇U ε +v · ∇w dx − λN +� +Ωε +U ε +vw dx += − +� +εΣ +∂v +∂xd +w dx′ + +� +εΣ +∂v +∂xd +w dx′ − λN +� +Ωε +U ε +vw dx += −λN +� +Ωε +U ε +vw dx +where we have used the facts that v is an eigenfunction relative to λN, the exterior normal vector +to Ω on εΣ is νΩ = (0, . . . , 0, −1) and U ε +v satisfies (2.1). We then obtain +|qε(u, w)| = |qε(Πεv, w)| ≤ λN∥U ε +v∥∥w∥ ≤ λNMε∥v∥∥w∥ ≤ λN +Mε +1 − Mε +∥u∥∥w∥ +because ∥U ε +v∥L2(Ωε) = ∥v + U ε +v − v∥L2(Ωε) = ∥(Πε − I)v∥L2(Ωε) and ∥v∥L2(Ωε) = ∥v − Πεv + +Πεv∥L2(Ωε) ≤ Mε∥v∥L2(Ωε) + ∥u∥L2(Ωε), so that +∥v∥L2(Ωε) ≤ +∥u∥ +1 − Mε +. +Lemma 3.2 then implies +|qε(u, w)| ≤ δε∥u∥∥w∥, + +17 +for some δε > 0 such that δε = o(χε) as ε → 0. We can now apply Proposition 3.3 with δ = δε, +which implies that for 1 ≤ i ≤ m +(3.6) +λε +N+i−1 = λN + ξε +i + o(χ2 +ε), +where {ξε +i }m +i=1 are the eigenvalues of the restriction of qε to Fε. +3.2. Analysis of the restricted quadratic form. We now need to study {ξε +i }m +i=1. To do so, we +are now going to investigate how the quadratic form qε acts when restricted to the m-dimensional +subspace Fε = Πε(E(λN)), still endowed with the L2(Ωε)-norm. Let us introduce the following +bilinear form +rε(u, v) := +� +Ωε +∇U ε +u · ∇U ε +v dx + λN +� +Ωε +U ε +u U ε +v dx, +defined for u, v ∈ E(λN). +Lemma 3.4. For all ϕi, ϕj ∈ E(λN), +qε (Πεϕi, Πεϕj) = −rε(ϕi, ϕj). +Proof. For simplicity, in the sequel we write U ε +i in place of U ε +ϕi. We have +qε (Πεϕi, Πεϕj) = +� +Ωε +∇(ϕi + U ε +i ) · ∇(ϕj + U ε +j ) dx − λN +� +Ωε +(ϕi + U ε +i ) (ϕj + U ε +j ) dx += +� +Ωε +∇ϕi · ∇U ε +j dx + +� +Ωε +∇ϕj · ∇U ε +i dx + +� +Ωε +∇U ε +i · ∇U ε +j dx +− λN +� +Ωε +ϕi U ε +j dx − λN +� +Ωε +ϕj U ε +i dx − λN +� +Ωε +U ε +i U ε +j dx, +where we have used the fact that ϕi, ϕj are both eigenfunctions relative to λN. Note that the +integral involving ϕi or ϕj taken over Ωε are the same if taken over Ω. Integrating by parts we +obtain +� +Ω +∇ϕi · ∇U ε +j dx = λN +� +Ωε +ϕi U ε +j dx + +� +εΣ +∂ϕi +∂νΩ +U ε +j dx′, +where νΩ = (0, . . . , 0, −1). We can go ahead obtaining +qε (Πεϕi, Πεϕj) = − +� +εΣ +∂ϕi +∂xd +U ε +j dx′ − +� +εΣ +∂ϕj +∂xd +U ε +i dx′ + +� +Ωε +∇U ε +i · ∇U ε +j dx′ − λN +� +Ωε +U ε +i U ε +j dx. +Taking into account (2.1) with U ε +j and U ε +i as test functions we obtain +� +Ωε +∇U ε +i · ∇U ε +j dx = +� +εΣ +∂ϕi +∂xd +U ε +j dx′ = +� +εΣ +∂ϕj +∂xd +U ε +i dx′, +respectively. In this way, +qε (Πεϕi, Πεϕj) = − +� +Ωε +∇U ε +i · ∇U ε +j dx − λN +� +Ωε +U ε +i U ε +j dx′, +and the proof is concluded. +□ +Remark 3.5. As detailed in [3, Appendix C], Lemma 3.4 implies that +ξε +j = µε +j + o(χε2) +as ε → 0, +where µε +j denote the eigenvalues of the form rε(·, ·) defined on E(λN). +Therefore, in view of (3.6) and Remark 3.5, the proof of Theorem 1.2 is complete. + +18 +L. ABATANGELO AND R. OGNIBENE +4. Ramification of eigenvalues +The aim of the present section is to prove Theorem 1.5. In order to investigate the occurrence +of ramification of multiple eigenvalues, we need to study µε +j, i.e. the eigenvalues of the form rε(·, ·) +defined on E(λN). As already mentioned in the introduction, we expect that as ε > 0 multiple +eigenvalues split according to the order of vanishing of suitably chosen limit eigenfunctions at the +origin. Hence, we first introduce the aforementioned order decomposition of E(λN), which drives +us towards the choice of the proper limit eigenbasis. Secondly, we iteratively apply the abstract +result Proposition 3.3, by choosing smaller and smaller approximating spaces F. +4.1. Order decomposition of the eigenspace. For clarity of exposition, we report here the +statement of Proposition 1.4, which consists of a slight variation of [3, Proposition A.1]. +Proposition 4.1. There exists a decomposition of E(λN) into a sum of orthogonal subspaces +E(λN) = E1 ⊕ · · · ⊕ Ep, +for some integer p ≥ 1, and an associated finite increasing sequence of integers +0 < k1 < · · · < kp +such that, for all 1 ≤ j ≤ p, a function in Ej \ {0} has the order of vanishing kj at 0, that is +ϕ(rx) +rkj +→ ψkj(x) +in C1,α(B+ +1 ), as ε → 0, +for some harmonic polynomial ψkj, homogeneous of degree kj and odd with respect to xd. +In +addition, such a decomposition is unique. +Proof. Given any u ∈ E(λN), let us consider its restriction to B+ +R = {x ∈ BR : xd > 0}, for R < r0 +sufficiently small so that B+ +R ⊆ Ω (with r0 as in (1.2)), let us extend it to BR oddly with respect +to xd, and let us call it ˜u. Then +−∆˜u = λN ˜u, +in BR. +Hence, in view of classical regularity results, ˜u is analytic at 0 and any truncation of its Taylor +expansion at 0 is odd with respect to xd. For any k ∈ N, let us define the mapping Πk : E(λN) → +Rodd +k +[X1, . . . , Xd] that associates to a function its (upper) Taylor expansion at 0, truncated to +order k. Here Rodd +k +[X1, . . . , Xd] is the the space of polynomials odd with respect to xd of degree +at most k. The proof can then proceed as in [3, Appendix A]. +□ +Remark 4.2. Let E(λN) = E1 ⊕ · · · ⊕ Ep, be the order decomposition of Proposition 4.1. Then +the dimension of Ej is at most the dimension of the space of spherical harmonics in d variables of +degree kj (see, e.g., [6, pp. 159–165]) vanishing on {xd = 0}. Explicitly, +dim(Ej) ≤ +�kj + d − 2 +kj +� ++ +�kj + d − 3 +kj − 1 +� +− 1. +As a consequence, in the case d = 2 dim(Ej) ≤ 1 for all 1 ≤ j ≤ p. +4.2. Eigenvalues µε +j. Let us denote by +E(λN) = E1 ⊕ · · · ⊕ Ep +the order decomposition of the eigenspace E(λN) (see Proposition 4.1), with +0 < k1 < · · · < kp +the associated finite sequence of vanishing orders. +Up to a change of basis, we can therefore assume, in the course of the proof, that the orthonormal +basis {uN+i−1}m +i=1 has a form which is convenient for our computations. The final result will not +depend on this choice of basis. +More precisely, we introduce the bilinear forms +Tℓ(u, v) = +� +Π +∇U Π +Σ,u · ∇U Π +Σ,v +for u, v ∈ Eℓ ⊆ E(λN) + +19 +where U Π +Σ,u and U Π +Σ,v achieve +TΠ(Σ, Bℓu) = −2 +inf +w∈D1,2(Π) +�1 +2 +� +Π +|∇w|2 dx − +� +Σ +w∂Bℓu +∂xd +dx′ +� +and +TΠ(Σ, Bℓv) = −2 +inf +w∈D1,2(Π) +�1 +2 +� +Π +|∇w|2 dx − +� +Σ +w∂Bℓv +∂xd +dx′ +� +respectively. We can assume that the orthonormal basis {uN+i−1}m +i=1 agrees with the order de- +composition and diagonalizes each of the quadratic forms Tℓ. Explicitly, this means that, for all +ℓ ∈ {1, . . . , p}, +Eℓ = span{uN+m1+···+mℓ−1, . . . , uN+m1+···+mℓ−1+mℓ−1} +and, for all 1 ≤ s < t ≤ mℓ, +Tℓ(uN+m1+···+mℓ−1+s−1, uN+m1+···+mℓ−1+t−1) = 0. +It follows that, for all 1 ≤ s ≤ mℓ, +Tℓ(uN+m1+···+mℓ−1+s−1, uN+m1+···+mℓ−1+s−1) = µℓ,s. +According to Remark 3.5, we start from the lowest rate of convergence k1 and we look for the +m1 largest eigenvalues (as ε → 0) of the matrix of the quadratic form rε in the basis {uN+i−1}m +i=1, +namely Aε. It follows from Lemma 2.6 and Theorem 2.8 that +Aε = + + + + + + + + + +0 +0 +µ1,1 εd−2+2k1 +0 +0 +... +0 +µ1,m1 εd−2+2k1 + + + + + + + + + ++ o +� +εd−2+2k1� +. +Using the min-max characterization of eigenvalues, we conclude that, for 1 ≤ i ≤ m1, +µε +i = µ1,i εd−2+2k1 + o +� +εd−2+2k1� +and, for m1 + 1 ≤ i ≤ m, +µε +i = o +� +εd−2+2k1� +. +Proposition 3.3, Remark 3.5 and the fact that χ2 +ε and εd−2+2k1 are of the same order, tell us that +the same estimates hold for the difference λε +N−1+i − λN: for 1 ≤ i ≤ m1 +νε +N−1+i = −µ1,iεd−2+2k1 + o +� +εd−2+2k1� +and, for m1 + 1 ≤ i ≤ m, +νε +N−1+i = o +� +εd−2+2k1� +. +The rest of the proof consists of a step-by-step procedure, in which we rescale the quadratic form +qε and apply the same arguments in order to identify successive groups of eigenvalues converging +to λN with the same rate. Let us sketch the next step. We set, for u, v ∈ Dε, +qε +2(u, v) ≡ +1 +εd−2+2k1 qε(u, v), +and we define the subspace +F ε +2 = Πε(E2 ⊕ · · · ⊕ Ep). +The eigenvalues of qε +2 are +� +1 +εd−2+2k1 νε +i +� +i≥1 = +1 +εd−2+2k1 {λε +i − λN}i≥1. We know from the first +step that, for 1 ≤ i ≤ m1, +lim +ε→0 +1 +εd−2+2k1 νε +N−1+i = −µ1,i < 0. +It follows immediately that there exists γ > 0 such that, for ε > 0 small enough, +���� +1 +εd−2+2k1 νε +N−1+i +���� ≤ γ for m1 + 1 ≤ i ≤ m +and +1 +εd−2+2k1 νε +N+m ≥ 2γ; + +20 +L. ABATANGELO AND R. OGNIBENE +whereas in case N ≥ 2 even +1 +εd−2+2k1 νε +N−1 ≤ −2γ. +Repeating the arguments of Section 3.1, we can show that for all v ∈ F ε +2 and w ∈ Dε, +|qε +2(v, w)| ≤ o +��εd−2+2k2 +εd−2+2k1 +�1/2� +∥v∥∥w∥ = o +� +εk2−k1� +∥v∥∥w∥. +Using the arguments in the proof of Theorem 1.2 and in the first step, we conclude that, for +1 + m1 ≤ i ≤ m1 + m2, +1 +εd−2+2k1 νε +N−1+i = −µ2,i−m1 ε2k2−2k1 + o +� +ε2k2−2k1� +and, for m1 + m2 + 1 ≤ i ≤ m, +1 +εd−2+2k1 νε +N−1+i = o +�εd−2+2k2 +εd−2+2k1 +� +. +This gives us finally, for m1 + 1 ≤ i ≤ m1 + m2, +νε +N−1+i = −µ2,i−m1 εd−2+2k2 + o +� +εd−2+2k2� +and, for m1 + m2 + 1 ≤ i ≤ m, +νε +N−1+i = o +� +εd−2+2k2� +. +Carrying on the procedure for ℓ from 3 to m, we reach the conclusion. +Remark 4.3. By 4.2, in dimension d = 2 the eigenfunctions associated to a multiple eigenvalue +have necessarily different vanishing order at 0. We then recover the result proved in [19, Section +11]. +Appendix A. Remarks on mixed Dirichlet–Neumann boundary conditions +The case of mixed Dirichlet–Neumann boundary conditions possesses additional features worthy +to come to light. Firstly, let us recall the following definition, already introduced in Section 1.1. +Definition A.1. Let Γ ⊆ ∂Ω be a relatively open set. We call boundary torsional rigidity of Γ +relative to Ω the quantity +TΩ(Γ) := −2 inf +�1 +2 +� +Ω +|∇u|2 dx − +� +Γ +u dS : u ∈ H1 +0,∂Ω\Γ(Ω) +� +which coincides to the energy of the unique weak solution of the problem +(A.1) + + + + + + + +−∆UΓ = 0, +in Ω +∂UΓ +∂ν = 1, +on Γ +UΓ = 0, +on ∂Ω \ Γ. +We also recall that H1 +0,∂Ω\Γ(Ω) denotes the closure of C∞ +c (Ω ∪ Γ) in H1(Ω). We would like to +stress that Problem (A.1) has to do with the so-called boundary torsional rigidity of Ω as it is +introduced in [8], there denotated by T (Ω, δ). As explained in [8, Section 2], T (Ω; δ) is modeled on +the trace Sobolev embedding W 1,2(Ω) ֒→ L1(∂Ω) and it is closely related to the Steklov eigenvalue +problem. In this case it is worthwhile to mention that, equivalently +(A.2) +TΩ(Γ) = +sup +ϕ∈H1 +0,∂Ω\Γ(Ω)\{0} +�� +Γ +ϕ dS +�2 +� +Ω +|∇ϕ|2 dx +, + +21 +for it is related to the best constant for the Sobolev embedding H1 +0,∂Ω\Γ(Ω) ֒→ L1(Γ). Moreover, +TΩ(Γ) is related to the lowest of the so-called Dirichlet–Steklov eigenvalues. As pointed out in [22] +(see also [4]), the so-called Dirichlet–Steklov eigenvalue problem + + + + + + + +−∆u = 0, +in Ω +u = 0, +on ∂Ω \ Γ +∂u +∂ν = σu, +on Γ +is equivalent to the eigenvalue problem of the Dirichlet-to-Neumann operator, which in fact admits +a sequence of positive eigenvalues +0 < σ1(Ω, Γ) ≤ σ2(Ω, Γ) ≤ . . . → +∞. +The lowest of them has the following variational characterization +(A.3) +σ1(Ω, Γ) = inf + + + + + + + +� +Ω +|∇u|2 dx +� +Γ +u2 dS +: u ∈ H1 +0,∂Ω\Γ(Ω) \ {0} + + + + + + + +. +By definition and (A.2), we have +1 +TΩ(Γ) = +� +Ω +|∇UΓ|2 dx +�� +Γ +UΓ dS +�2 ≥ +� +Ω +|∇UΓ|2 dx +Ld−1(Γ) +� +Γ +UΓ +2 dS +≥ σ1(Ω, Γ) +Ld−1(Γ) +where we applied Cauchy-Schwarz Inequality to gain the first inequality and (A.3) to reach the +second one. Summing up, we obtain +(A.4) +TΩ(Γ)σ1(Ω, Γ) ≤ Ld−1(Γ) +which can be considered a Dirichlet–Steklov version of the classical Polya inequality ([21]). Fi- +nally, Equation (A.1) has got relevant physical meanings. On one hand, it models the vertical +displacement of a membrane under an external pressure which is concentrated near the boundary +(see [5, Theorem 4.1] for the rigorous limit process): the considered membrane can move in the +vertical direction keeping a horizontal angle. On the other hand, solutions to (A.1) are stationary +solutions of the heat equation that models temperature in a homogeneous and isotropic heat con- +ductor. This is subjected to a constant heat flux through a small part of the boundary whereas +the temperature is kept constant in the remaining part. +Acknowledgments +R. Ognibene is partially supported by the project ERC VAREG - Variational approach to the +regularity of the free boundaries (grant agreement No. 853404) and by the INdAM-GNAMPA +2022 project Questioni di esistenza e unicit`a per problemi nonlocali con potenziali di tipo Hardy. +Part of this work was developed while R. Ognibene was in residence at Institut Mittag-Leffler +in Djursholm, Stockholm (Sweden) during the semester Geometric Aspects of Nonlinear Partial +Differential Equations in 2022, supported by the Swedish Research Council under grant no. 2016- +06596. +References +[1] Abatangelo, L., Felli, V., and L´ena, C. Eigenvalue variation under moving mixed Dirichlet-Neumann +boundary conditions and applications. ESAIM Control Optim. Calc. Var. 26 (2020), Paper No. 39, 47. +[2] Abatangelo, L., Felli, V., and Terracini, S. On the sharp effect of attaching a thin handle on the spectral +rate of convergence. J. Funct. Anal. 266, 6 (2014), 3632–3684. +[3] Abatangelo, L., L´ena, C., and Musolino, P. Ramification of multiple eigenvalues for the Dirichlet-Laplacian +in perforated domains. J. Funct. Anal. 283, 12 (2022), Paper No. 109718. +[4] Agranovich, M. S. On a mixed Poincar´e-Steklov type spectral problem in a Lipschitz domain. Russ. J. Math. +Phys. 13, 3 (2006), 239–244. + +22 +L. ABATANGELO AND R. OGNIBENE +[5] Arrieta, J. M., Jim´enez-Casas, A., and Rodr´ıguez-Bernal, A. Flux terms and Robin boundary conditions +as limit of reactions and potentials concentrating at the boundary. Rev. Mat. Iberoam. 24, 1 (2008), 183–211. +[6] Berger, M., Gauduchon, P., and Mazet, E. Le spectre d’une vari´et´e riemannienne. Lecture Notes in +Mathematics, Vol. 194. Springer-Verlag, Berlin-New York, 1971. +[7] Bers, L. Local behavior of solutions of general linear elliptic equations. Communications on Pure and Applied +Mathematics 8, 4 (nov 1955), 473–496. +[8] Brasco, L., Gonzalez, M., and Ispizua, M. A steklov version of the torsional rigidity, Preprint 2022, +https://arxiv.org/abs/2207.04816v1. +[9] Colin de Verdi`ere, Y. Sur la multiplicit´e de la premi`ere valeur propre non nulle du laplacien. Comment. +Math. Helv. 61, 2 (1986), 254–270. +[10] Collins, C. D., and Taylor, J. L. Eigenvalue convergence on perturbed Lipschitz domains for elliptic systems +with mixed general decompositions of the boundary. J. Differential Equations 265, 12 (2018), 6187–6209. +[11] Courtois, G. Spectrum of manifolds with holes. J. Funct. Anal. 134, 1 (1995), 194–221. +[12] Daners, D. Dirichlet problems on varying domains. J. Differential Equations 188, 2 (2003), 591–624. +[13] Felli, V., Ferrero, A., and Terracini, S. Asymptotic behavior of solutions to Schr¨odinger equations near +an isolated singularity of the electromagnetic potential. J. Eur. Math. Soc. (JEMS) 13, 1 (2011), 119–174. +[14] Felli, V., Noris, B., and Ognibene, R. Eigenvalues of the Laplacian with moving mixed boundary conditions: +the case of disappearing Dirichlet region. Calc. Var. Partial Differential Equations 60, 1 (2021), Paper No. 12, +33. +[15] Felli, V., Noris, B., and Ognibene, R. Eigenvalues of the Laplacian with moving mixed boundary conditions: +the case of disappearing Neumann region. J. Differential Equations 320 (2022), 247–315. +[16] Felli, V., and Ognibene, R. Sharp convergence rate of eigenvalues in a domain with a shrinking tube. J. +Differential Equations 269, 1 (2020), 713–763. +[17] Felli, V., and Terracini, S. Singularity of eigenfunctions at the junction of shrinking tubes, Part I. J. +Differential Equations 255, 4 (2013), 633–700. +[18] Gadyl’shin, R. Ramification of a multiple eigenvalue of the Dirichlet problem for the Laplacian under singular +perturbation of the boundary condition. Mathematical Notes 52, 4 (1992), 1020–1029. +[19] Gadyl’shin, R. R. The method of matching asymptotic expansions in a singularly perturbed boundary value +problem for the Laplace operator. Sovrem. Mat. Prilozh., 5, Asimptot. Metody Funkts. Anal. (2003), 3–32. +[20] Hardt, R., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., and Nadirashvili, N. Critical sets of +solutions to elliptic equations. J. Differential Geom. 51, 2 (1999), 359–373. +[21] P´olya, G., and Szeg¨o, G. Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, +No. 27. Princeton University Press, Princeton, N. J., 1951. +[22] Seo, D.-H. A shape optimization problem for the first mixed Steklov-Dirichlet eigenvalue. Ann. Global Anal. +Geom. 59, 3 (2021), 345–365. +[23] Taylor, J. L. Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains. J. Spectr. Theory +3, 3 (2013), 293–316. +Laura Abatangelo +Dipartimento di Matematica +Politecnico di Milano +Piazza Leonardo da Vinci 32, 20133 Milano, Italy +Email address: laura.abatangelo@polimi.it +Roberto Ognibene +Dipartimento di Matematica +Universit`a di Pisa +Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy +Email address: roberto.ognibene@dm.unipi.it + diff --git a/p9FKT4oBgHgl3EQfHS2t/content/tmp_files/load_file.txt b/p9FKT4oBgHgl3EQfHS2t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4f82789497b569620cf89f56a3c7e5a46248d89 --- /dev/null +++ b/p9FKT4oBgHgl3EQfHS2t/content/tmp_files/load_file.txt @@ -0,0 +1,925 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf,len=924 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='11729v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='AP] 27 Jan 2023 SHARP BEHAVIOR OF DIRICHLET–LAPLACIAN EIGENVALUES FOR A CLASS OF SINGULARLY PERTURBED PROBLEMS LAURA ABATANGELO AND ROBERTO OGNIBENE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of the perturbed eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We detect the proper quantity which sharply measures the perturbation’s magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It is a sort of torsional rigidity of the tube’s section relative to the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This allows us to sharply describe the asymptotic behavior of the perturbed spectrum, even when eigenvalues converge to a multiple one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The final asymptotics of eigenbranches depend on the local behavior near the junction of eigenfunctions chosen in a special way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The present techniques also apply when the perturbation of the Dirichlet eigenvalue problem consists in prescribing homogeneous Neumann boundary conditions on a small portion of the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Introduction and main results The object of the present paper is the eigenvalue variation of Dirichlet eigenvalues for a class of singularly perturbed domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Our main attention is devoted to the sharp effect of attaching a thin tube to a fixed bounded domain when its cross-section shrinks to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This class of problems covers also the case of the widely studied dumbbell domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In addition, the presented proofs fit also the case of moving mixed Dirichlet–Neumann boundary conditions, as the Neumann part tends to disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In both cases, the starting unperturbed problem is the classical eigenvalue problem for the Laplacian with Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' More precisely, we fix d ≥ 2 and a bounded, open and connected set Ω ⊆ Rd and we consider the problem of finding λ ∈ R and a nonzero ϕ: Ω → R such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) �−∆ϕ = λϕ, in Ω, ϕ = 0, on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This is considered in a weak sense: ϕ ∈ H1 0(Ω) \\ {0} is such that � Ω ∇ϕ · ∇v dx = λ � Ω ϕv dx, for all v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If this happens, we say that λ is an eigenvalue and ϕ is one of the corresponding eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' By classical spectral theory it is known that this problem admits a diverging sequence of positive eigenvalues, which we hereafter denote 0 < λ1 < λ2 ≤ · · · ≤ λn ≤ · · · → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We also denote by {ϕn}n≥1 a corresponding sequence of eigenfunctions, assumed to be orthonormal in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We now introduce a singular perturbation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1), which basically consists in attaching a thin tube to the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us assume that 0 ∈ ∂Ω and ∂Ω is flat in a neighbourhood of the origin, namely (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2) B′ r0 := {x ∈ Br0 : xd = 0} ⊆ ∂Ω and Br+ 0 := Br0 ∩ {xd > 0} ⊆ Ω for some r0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 1 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE Given a relatively open set Σ ⊆ B′ r0 with Lipschitz boundary and ε ∈ (0, 1), we consider the thin tube of section εΣ and fixed height equal to 1 attached at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If we denote Tε := εΣ × (−1, 0], our perturbed domain will be (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3) Ωε := Ω ∪ εΣ ∪ Tε and Ω ∩ Tε = εΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We then consider the eigenvalue problem for the Dirichlet-Laplacian on the perturbed domain Ωε (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4) �−∆ϕ = λεϕ, in Ωε, ϕ = 0, on ∂Ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Again by classical spectral theory, for ε ∈ (0, 1) this problem admits a sequence of eigenvalues tending to +∞, which will be denoted as 0 < λε 1 < λε 2 ≤ · · · ≤ λε n ≤ · · · → +∞, whereas {ϕε n}n≥1 will denote a corresponding sequence eigenfunctions, assumed to be orthonormal in L2(Ωε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The main goal of the present paper is understanding the behavior of the perturbed eigenvalues λε n as the tube radius ε tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Literature on this problem is very rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We refer to [16, Introduction] for a presentation of the established results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Here we just mention that such a large interest for the problem is due to physical and engineering motivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Spectral behavior of the Laplacian on thin branching domains appears in the theory of quantum graphs, which models propagation of waves in quasi one-dimensional systems (quantum wires and waveguides, photonic crystals, blood vessels and so on), as well as in the theory of elasticity and multistructure problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' First of all, as a consequence of the convergence in the sense of Mosco of the sequence of domains {Ωε}ε to the limit domain Ω (see the discussion in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2), classical results (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' [12]) ensure stability of the spectrum, in the sense that for any N ∈ N \\ {0} λε N → λN as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The analysis of the present paper originates from the main result in [16], which in turn generalizes the papers [2] and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In [16] the authors study the sharp asymptotic behavior of Dirichlet eigenvalues in a domain perturbed as described above (with some additional geometric assumption on the section of the tube) using Almgren-type monotonicity formulas, Courant-Fischer min-max characterization for eigenvalues and blow-up analysis for scaled eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Specifically, they restrict to the case in which the perturbed eigenvalues are converging to a simple eigenvalue of the limit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Due to the local nature of the singular perturbation taken into account, the eigenfunctions’ local behavior at 0 ∈ ∂Ω (namely the point where the thin branch is attached) plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Such behavior can be described as follows (see [7] or [13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3]): if ϕN is an eigenfunction of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1), there exists k ∈ N \\ {0} such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5) ϕN(rx) rk → ψk(x), in C1,α(B+ 1 ) as r → 0, for some ψk ∈ Pk odd, where Pk odd denotes the space of harmonic homogeneous polynomials of degree k, odd with respect to the last variable xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We point out that the polynomials in the class Pk odd, restricted to the (d−1)- dimensional unit sphere, are spherical harmonics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' eigenfunctions of the spherical Laplacian) vanishing on {xd = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If we denote (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6) Π := Rd + ∪ Σ ∪ T, where T := Σ × (−∞, 0) and by D1,2(Π) the completion of C∞ c (Π) with respect to the L2 norm of the gradient, the main result [16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1] establishes that, if Σ is starshaped with respect to the origin and if λN is a simple eigenvalue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1), then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7) λǫ N = λN − Ck,Σ εd−2+2k + o(εd−2+2k), as ε → 0, 3 where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8) Ck,Σ = −2 inf u∈D1,2(Π) �1 2 � Π |∇u|2 dx − � Σ u∂ψk ∂xd dx′ � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us now briefly comment on this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7), one can see that the local nature of the perturbation mainly emerges in the exponent k of the radius of the tube’s section ε: in this sense, the vanishing order k of the unique limit eigenfunction (up to multiplicative constants) at the junction point determines the rate of convergence of the perturbed eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The second factor which influences the asymptotic expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7) is the positive constant Ck,Σ, whose variational characterization (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8) sheds some light on its nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As already noticed for the first time in [2], this coefficient has to do with the ability of a membrane to respond to a vertical force acting on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The main drawback of [16] is the geometric assumptions on the tube’s section Σ and the hypothesis of simplicity of the limit eigenvalue λN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In this paper, we mainly address these two open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Concerning nontrivial multiplicity of eigenvalues, less is available in literature as compared to the simple case (we refer to [16] for the state of the art in this last instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In this regard, we mention the work by Taylor [23] which provides an estimate for the eigenvalue variation in a similar context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The distance between the perturbed and the limit eigenvalue is estimated by Cεa, where the rate a is independent of any eigenvalue and the constant C depends only on the distance between the limit eigenvalue to the nearby ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The same author provides a similar result in collaboration with Collins (see [10]) for problems with mixed Dirichlet–Neumann boundary conditions if the tube is attached at a point where Dirichlet condition is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Apart from these, no sharper result on eigenbranches is available in literature, for which eigenbranches could be distinguished each other by their different asymptotic behavior as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Splitting of eigenbranches is proved in [19, Chapter 3] but only in dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' To the best of our knowledge, no other result is available on this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Nevertheless, we strongly believe that this is a relevant topic of investigation: multiple eigenvalues appear in domains with symmetries and this is often the case in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Before stating our main results, we give the fundamental definition of the paper together with some remarks on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For f ∈ C1 � B+ r0ε � we introduce the functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='9) JΩε εΣ,f(u) = Jε f(u) := 1 2 � Ωε |∇u|2 dx − � εΣ u ∂f ∂xd dx′, defined for u ∈ H1 0(Ωε)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For any f ∈ C1 � B+ r0ε � we call the thin f-torsional rigidity of εΣ relative to Ωε the following quantity TΩε(εΣ, f) : = −2 inf u∈H1 0 (Ωε) Jε f(u), = sup u∈H1 0 (Ωε) � 2 � εΣ u ∂f ∂xd dx′ − � Ωε |∇u|2 dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If ∂f ∂xd = 1 on εΣ, we denote TΩε(εΣ) := TΩε(εΣ, f) and we call it the thin torsional rigidity of εΣ relative to Ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If we consider εΣ as a variable and Ωε and f as parameters, TΩε(εΣ, f) is a set-function and will play a crucial role in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It will be the right quantity to face the perturbation theory of eigenvalues in this framework as soon as f is a suitable relative eigenfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Realizing this is one of the main novelty of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Broadly speaking, this notion of torsional rigidity of the perturbing set plays a similar role as that of the capacity of a set in [3] (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1 for a more detailed explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In Section 2 we report some basic results concerning this quantity, such as existence and uniqueness of a minimizer for Jε f, equivalent formulations, monotonicity properties 1we note that Jε f can be defined for less regular functions f: it may be sufficient that ∂f ∂xd ∈ � H1/2 00 �′ provided the integral � εΣ u ∂f ∂xd dx′ is meant as a duality product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 4 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE and asymptotic behavior as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Here we just mention that there exists a unique function U Ωε εΣ,f = U ε f ∈ H1 0(Ωε) (depending also on Σ) achieving TΩε(εΣ, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Moreover, it weakly satisfies \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 −∆U ε f = 0, in Ωε \\ εΣ U ε f = 0, on ∂Ωε, ∂U ε f|Ω ∂xd − ∂U ε f|Tε ∂xd = − ∂f ∂xd , on εΣ, in the sense that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='10) � Ωε ∇U ε f · ∇ϕ dx = � εΣ ϕ ∂f ∂xd dx′, for all ϕ ∈ H1 0(Ωε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In addition, one can observe that the map f �→ U ε f is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' When ∂f ∂xd = 1 on εΣ, we drop the index f and we denote by U ε the unique function in H1 0(Ωε) achieving TΩε(εΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We call U ε f and U ε the thin f-torsion function and thin torsion function of εΣ (relative to Ωε), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Coming back to our problem, let λN be an eigenvalue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) with multiplicity m ≥ 1 and let us denote by E(λN) ⊆ H1 0(Ω) the associated m-dimensional eigenspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As already mentioned in the introduction, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , m}, λε N+i−1 → λN, as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Therefore we have exactly m eigenvalue branches departing from the multiple limit eigenvalue λN and some of them may a priori coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We investigate the asymptotic behaviors of λε N+i−1 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In order to find good approximations for perturbed eigenvalues we use a slight modification of a lemma by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Courtois [11], itself based on the work by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Colin de Verdi´ere [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It establishes essentially the possibility to approximate small eigenvalues of a quadratic form with eigenvalues of the same form restricted to a finite dimensional subspace of the form domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' One pays an error which can be estimated in terms of spectral projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For completeness, we report its statement in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3 as it appears in [3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Its proof is given in [3, Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Good approximations for perturbed eigenvalues rely on good approximations for perturbed eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' These will be suitable modifications of the limit eigenfunctions corresponding to the limit eigenvalue λN = · · · = λN+m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Best approximations will be Πεϕ = Φε ϕ := ϕ + U ε ϕ, with ϕ ∈ E(λN), where U ε ϕ is the thin ϕ-torsion function, with ϕ an eigenfunction relative to λN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='10), the function Φε ϕ ∈ H1 0(Ωε) weakly solves �−∆Φε ϕ = λNϕ, in Ωε, Φε ϕ = 0, on ∂Ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This means that Φε ϕ is in the domain of the perturbed operator, but −∆Φε ϕ acts as −∆ϕ in the sense of distributions on H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For further discussion see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If we apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3 in a suitable way (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1 for the details) we obtain our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We state it in the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , m}, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='11) λε N+i−1 = λN − µε i + o(χ2 ε) as ε → 0, where χ2 ε := sup{TΩε(εΣ, u) : u ∈ E(λN) and ∥u∥L2(Ω) = 1} and {µε i}m i=1 are the eigenvalues (taken in non-increasing order) of the quadratic form rε, defined for u, v ∈ E(λN) as rε(u, v) := � Ωε ∇U ε u · ∇U ε v dx + λN � Ωε U ε u U ε v dx, 5 where U ε u (U ε v) is the thin u-torsion function of εΣ (the thin v-torsion function of εΣ, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In fact, one can obviously even consider m = 1, for this is allowed in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In this case, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2 readily implies the main result in [16] (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8)), if supplied with the blow-up analysis for TΩε(εΣ, f) given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Although Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2 provides a good approximation for perturbed eigenvalues when the limit one is simple, it is not exhaustive for multiple ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For instance, for some i it may be µε i = o(χ2 ε): in this case (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='11) reduces to λε N+i−1 − λN = o(χ2 ε), providing no sharp asymptotic behavior but just an estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Nevertheless, we expect that eigenbranches’ asymptotic rates will depend on local behaviors at 0 of properly chosen limit eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In order to improve Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2 in this way, we introduce the following proposition, which is originally given in [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='10] in a slight different context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It sheds light on the proper choice of the limit eigenbasis: the limit eigenspace can be uniquely split in several subspaces where eigenfunctions share the same vanishing order at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It is called the order decomposition of E(λN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ([3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='10]) There exists an integer p ≥ 1, a decomposition of E(λN) into a sum of orthogonal subspaces E(λN) = E1 ⊕ · · · ⊕ Ep and an associated finite increasing sequence of integers 0 < k1 < · · · < kp such that, for all 1 ≤ j ≤ p, a function ϕ ∈ Ej \\ {0} has vanishing order kj at 0, that is ϕ(rx) rkj → ψkj(x) in C1,α(B+ 1 ), as ε → 0, for some harmonic polynomial ψkj, homogeneous of degree kj and odd with respect to xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In addition, such a decomposition is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Secondly, we will need a blow-up analysis for torsion functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us describe in a few words our procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' By the change of variables x �→ εx, we zoom in closely to the origin: the perturbed domain Ωε is transformed into 1 εΩε = 1 εΩ ∪ Σ ∪ 1 εTε, which is intuitively “converging” to Π (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6) as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Given the order decomposition as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4 and fixed j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , p}, we then consider the map Bj : Ej → Pkj odd, where Bjϕ is the harmonic homogeneous polynomial of degree kj which describes the local be- havior of ϕ ∈ Ej near the origin, namely ϕ(rx) rkj → (Bjϕ) (x) in C1,α(B+ 1 ), as r → 0 and Pkj odd denotes the space of harmonic polynomials homogeneous of degree kj odd with respect to the xd variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We observe that for ϕ ∈ Ej a simple change of variables gives TΩε(εΣ, ϕ) = −2 �1 2 � Ωε ��∇U ε ϕ ��2 dx − � εΣ U ε ϕ ∂ϕ ∂xd dx′ � = −2εd−2+2kj � 1 2 � 1 ε Ωε |∇ ˆU ε ϕ|2 dx − � Σ ˆU ε ϕ ∂ ˆϕε ∂xd dx′ � , where ˆU ε ϕ(x) := U ε ϕ(εx) εkj and ˆϕε(x) := ϕ(εx) εkj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE In view of this, it is reasonable to investigate the behavior of ˆU ε ϕ as ε → 0 and to expect that ε−d+2−2kjTΩε(εΣ, ϕ) admits a nontrivial, finite, limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This motivates the following quantity: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='12) TΠ(Σ, Ψ) := −2 inf �1 2 � Π |∇u|2 dx − � Σ u ∂Ψ ∂xd dx′ : u ∈ D1,2(Π) � , which we define for Ψ ∈ C1(B+ r0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If ∂Ψ ∂xd = 1 on Σ, we denote TΠ(Σ) := TΠ(Σ, Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Therefore, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8 will establish ε−d+2−kjTΩε(εΣ, ϕ) → TΠ(Σ, Bjϕ) as ε → 0, for all ϕ ∈ Ej, for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Hereafter we denote by U Π Σ,ϕ the function achieving TΠ(Σ, Bjϕ), for ϕ ∈ Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Taking into account Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4 and this blow-up analysis, we are able to prove the following result about the sharp asymptotic behavior of perturbed eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Before stating it, we need the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For 1 ≤ j ≤ p, we let Ej as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4 and we write mj := dim (Ej), so that m = m1 + · · · + mj + · · · + mp, whereas we denote by µj,1 ≥ · · · ≥ µj,ℓ ≥ · · · ≥ µj,mj > 0 Tj(u, v) = � Π ∇Uu · ∇Uv dx defined for u, v ∈ Ej ⊆ E(λN), where Uu = U Π Σ,u and Uv = U Π Σ,v achieve TΠ(Σ, Bju) and TΠ(Σ, Bjv), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We are now ready to state the main result of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=', m}, there holds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='13) λε N+i−1 = λN − µj,ℓ εd−2+2kj + o(εd−2+2kj), as ε → 0, where (j, ℓ) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 (1, i) if 1 ≤ i ≤ m1 (2, i − m1) if m1 + 1 ≤ i ≤ m1 + m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (p, i − (m − mp)) if m − mp + 1 ≤ i ≤ m This result concludes our analysis on this class of problems: it provides sharp asymptotics for any perturbed eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us now analyze more in depth the particular case when p = 1 and k1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This is relevant since the gradient of limit eigenfunctions vanish at most on a subset of ∂Ω ∩ {xd = 0} which has zero (d − 1)-dimensional measure (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Thus, the limit eigenfunctions vanish with order 1 for Ld−1-almost every point in ∂Ω ∩ {xd = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Broadly speaking, vanishing order 1 occurs generically with respect to the points in ∂Ω ∩ {xd = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We also observe that if d = 2 p = 1 can only uccur if m = 1 (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If u ∈ E(λN), we have that B1u = ∂u ∂xd (0) xd, that is u(rx) r → ∂u ∂xd (0) xd, in C1,α(B+ 1 ), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Hence, by linearity there holds U Π Σ,u = ∂u ∂xd (0) U Π Σ,xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 2We point out that a little abuse of notation has been made, since this minimization is made within D1,2(Π), which is strictly larger than H1 0(Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 7 Therefore, we can write down the quadratic form T1(u, v) = ∂u ∂xd (0) ∂v ∂xd (0) � Π ��∇U Π Σ,xd ��2 dx = ∂u ∂xd (0) ∂v ∂xd (0) TΠ(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In particular, it is possible to choose an eigenbasis {ϕN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , ϕN+m−1} for E(λN) which diagonal- izes T1, in such a way that µ1,i = �∂ϕN+i−1 ∂xd (0) �2 TΠ(Σ), for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , m, thus leading to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us assume Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4 holds with p = 1 and k1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then there exists a basis {ϕN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , ϕN+m−1} of E(λN), orthonormal in L2(Ω) and such that, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=', m}, there holds λε N+i−1 = λN − �∂ϕN+i−1 ∂xd (0) �2 TΠ(Σ) εd + o(εd), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We observe that this result recovers, in the case m = 1, what the authors obtained in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Finally, we would like to emphasize another particular instance, which concerns the behavior of the first eigenvalue, being one of the most widely studied set functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In this case, it is known that the limit eigenvalue is simple (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' m = 1) and that the corresponding eigenfunctions have nonzero gradient on any regular boundary point, being them positive in Ω (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' p = 1 and k1 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Hence, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let ϕ1 ∈ H1 0(Ω) be a normalized eigenfunction corresponding to λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then there holds λε 1 = λ1 − �∂ϕ1 ∂xd (0) �2 TΠ(Σ) εd + o(εd), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It is worth noticing that, in these two last results, the coefficients of the first term in the asymptotic expansion of the eigenvalue variation split as a product of two factors: one of them only depending on the behavior of the limit eigenfunctions at the origin and the other one only depending on the geometry of the set Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Section 2 is devoted to present basic properties of the thin f- torsional rigidity and contains preliminary result in view of the main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Section 3 contains the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2, whereas Section 4 contains the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Mixed Dirichlet–Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The preceeding arguments apply also when we perturb problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) by prescribing that eigenfunctions satisfy homogeneous Neumann boundary conditions on εΣ, in place of attaching a thin tube with section εΣ to the fixed domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The problem has been already studied in dimension 2 in [18], achieving a full asymptotic expansion of perturbed eigenvalues (see also [1] for related results) and in any dimension by [15] but only for simple eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This last paper is our starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us consider the weak form of the eigenvalue problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='14) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 −∆ξ = ˜λǫξ, in Ω ξ = 0, on ∂Ω \\ εΣ ∂ξ ∂ν = 0 on εΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' To this aim, we set the functional framework as it appears in [15], by introducing the space H1 0,∂Ω\\εΣ(Ω), defined as the closure in H1(Ω) of C∞ c (Ω∪εΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We say that ˜λε ∈ R is an eigenvalue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='14) if there exists ξε ∈ H1 0,∂Ω\\εΣ(Ω) \\ {0} (named eigenfunction) such that � Ω ∇ξε · ∇v dx = ˜λε � Ω ξεv dx for all v ∈ H1 0,∂Ω\\εΣ(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 8 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE For any ε ∈ (0, 1) there exists a non-decreasing sequence of positive eigenvalues 0 < ˜λε 1 < ˜λε 2 ≤ · · · ≤ ˜λε n ≤ · · · → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' When ε → 0 the Neumann region disappears and the Dirichlet region covers the entire boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' One then expects the eigenelements of the mixed Dirichlet-Neumann problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='14) to converge to the ones of the limit Dirichlet problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This problem revealed to be more involved than its counterpart, in which Neumann boundary condition are prescribed on a large part of ∂Ω and Dirichlet boundary conditions on a vanishing portion of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' A capacitary approach (such as the one developed in [14] for the case of disappearing Dirichlet region) turns out to be particularly effective when functions are required to vanish in “small” sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' this is basically related to the known fact that Sobolev spaces are not affected if their functions are prescribed to vanish on zero capacity sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' So, the case introduced in this subsection falls outside a capacitary context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Furthermore, to the best of our knowledge, in literature there is no analogue to the capacity, which can play a similar role in the converse case (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Because of this, in [15] the authors undertake the problem through another approach, based on Almgren-type monotonicity formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In this work they prove the convergence of the perturbed spectrum, as ε → 0, to the spectrum of the Dirichlet-Laplaciann (see [15, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3]), that is (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='15) ˜λǫ n → λn as ε → 0 for any n ∈ N \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Moreover, in case of simple limit eigenvalues, they provide an explicit asymptotic expansion of the perturbed eigenvalues, which is sharp only when the set Σ fulfills certain geometric assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In particular, they proved that, if Σ is strictly starshaped with respect to the origin, if λN is simple and if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5) holds, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='16) ˜λε N = λN − ˜Ck,Σ εd−2+2k + o(εd−2+2k), as ε → 0, where ˜Ck,Σ := −2 inf u∈D1,2(Rd +∪Σ) � 1 2 � Rd + |∇u|2 dx + � Σ u∂ψk ∂ν dx′ � and D1,2(Rd + ∪ Σ) denotes the completion of C∞ c (Rd + ∪ Σ) with respect to the L2 norm of the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We also observe that here ν = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , 0, −1) and ∂ψk ∂ν = − ∂ψk ∂xd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' With the very same method used to prove the results in the previous subsection, we are able to remove both the simplicity assumption on the limit eigenvalue λN and the geometric assumption on Σ and to prove a sharp asymptotic expansion for ˜λε N in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In the same spirit of the previous case, here we are able to detect the proper quantity which measures the magnitude of the perturbation and, consequently, the stability of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It is still a notion of torsional rigidity and plays the role of perfect counterpart of the capacity, in the present framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us first introduce the functional ˜JΩ εΣ,f(u) = ˜Jε f(u) := 1 2 � Ω |∇u|2 dx + � εΣ u ∂f ∂ν dx′, defined for u ∈ H1 0,∂Ω\\εΣ(Ω), where f ∈ C1(B+ r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In the same lines as in the previous subsection, we introduce the notion of relative torsional rigidity of a set which is suitable for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For any f ∈ C1(B+ r0) we call the boundary f-torsional rigidity of εΣ relative to Ω the following quantity TΩ(εΣ, f) : = −2 inf u∈H1 0,∂Ω\\εΣ(Ω) ˜Jε f(u) = −2 inf u∈H1 0,∂Ω\\εΣ(Ω) �1 2 � Ω |∇u|2 dx + � εΣ u ∂f ∂ν dx′ � If ∂f ∂ν = −1 on εΣ, we denote TΩ(εΣ) := TΩ(εΣ, f) and we call it the boundary torsional rigidity of εΣ relative to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 9 We point out that Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1 and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8 are completely matching each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As in the previous case, by standard minimization methods there exists ˜U ε f = ˜U Ω εΣ,f ∈ H1 0,∂Ω\\εΣ(Ω)\\{0} achieving TΩ(εΣ, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We call ˜U ε f the boundary f-torsion function of εΣ, relative to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In particular, ˜U ε f satisfies \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 −∆ ˜U ε f = 0, in Ω, ˜U ε f = 0, on ∂Ω \\ εΣ, ∂ ˜U ε f ∂ν = − ∂f ∂ν , on εΣ in a weak sense, that is ˜U ε f ∈ H1 0,∂Ω\\εΣ(Ω) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='17) � Ω ∇ ˜U ε f · ∇ϕ dx = − � εΣ ϕ∂f ∂ν dx′ for all ϕ ∈ H1 0,∂Ω\\εΣ(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let λN be an eigenvalue to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='14) with multiplicity m and E(λN) be the associated eigenspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Our first result on this problem is the analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , m}, ˜λε N+i−1 = λN − ˜µε i + o(˜χ2 ε) as ε → 0, where ˜χ2 ε := sup{TΩ(εΣ, u) : u ∈ E(λN) and ∥u∥L2(Ω) = 1} according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8 and {˜µε i}m i=1 are the eigenvalues (taken in non-increasing order) of the quadratic form ˜rε, defined for u, v ∈ E(λN) as ˜rε(u, v) := � Ω ∇ ˜U ε u · ∇ ˜U ε v + λN � Ω ˜U ε u ˜U ε v, where ˜U ε u ( ˜U ε v) is the boundary u-torsion function of εΣ (the boundary v-torsion function of εΣ, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Given the order decomposition of E(λN), as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4, and taking ϕ ∈ Ej (for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , p}), it is possible to perform a blow-up analysis for a suitable rescaling of the ϕ-boundary torsion function ˜U ε ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Namely, following the steps as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2 (outlined in the previous section), one can prove that ε−d+2−2kj ˜U ε ϕ(εx) as well as ε−d+2−2kkTΩ(εΣ, ϕ) admit nontrivial, finite limits as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It is then natural to introduce the following quantity, defined for Ψ ∈ C1(B+ 1 ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='18) TRd +(Σ, Ψ) := −2 � 1 2 � Rd + |∇u|2 dx + � Σ u∂Ψ ∂ν dx′ : u ∈ D1,2(Rd + ∪ Σ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If ∂Ψ ∂ν = −1 on Σ we denote TRd +(Σ) := TRd +(Σ, Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Hence, we are able to prove the following (which is the analogous of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , p} and let ϕ ∈ Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then TΩ(εΣ, ϕ) = εd−2+2kTRd +(Σ, Bjϕ) + o(εd−2+2kj), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In addition ε−d+2−kj ˜U ε ϕ(εx) → ˜U Rd + Σ,ϕ, in D1,2(Rd + ∪ Σ), as ε → 0, where ˜U Rd + Σ,ϕ ∈ D1,2(Rd + ∪ Σ) denotes the function achieving TRd +(Σ, Bjϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Of course, even in this case one can consider m = 1, for this is allowed in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In this way, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='9 provides immediately the main result as stated in [15] if supplied with Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 10 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE Taking into account the order decomposition Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4 for E(λN), and the blow-up anal- ysis for scaled torsion functions, we are able to improve the result of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Before stating the main theorem, we need the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For 1 ≤ j ≤ p, we let Ej as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4 and we recall that mj = dim (Ej) so that m = m1 + · · · + mj + · · · + mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Moreover, we denote by ˜µj,1 ≥ · · · ≥ ˜µj,ℓ ≥ · · · ≥ ˜µℓ,mj > 0 the eigenvalues of the quadratic form ˜Tj(u, v) = � Rd + ∇ ˜Uu · ∇ ˜Uv dx defined for u, v ∈ Ej ⊆ E(λN), where ˜Uu = ˜U Rd + Σ,u and ˜Uv = ˜U Rd + Σ,v achieve, respectively, TRd +(Σ, Bju) and TRd +(Σ, Bjv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We are now ready to state the main result of this section, which is the analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , m}, there holds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='19) ˜λε N+i−1 = λN − ˜µj,ℓ εd−2+2kj + o(εd−2+2kj), as ε → 0, where (j, ℓ) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 (1, i) if 1 ≤ i ≤ m1 (2, i − m1) if m1 + 1 ≤ i ≤ m1 + m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (p, i − (m − mp)) if m − mp + 1 ≤ i ≤ m For simplicity of exposition, we do not present here the proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='9 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='12, since they follow step by step the proofs of the case of domains with handles attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It will be sufficient to set all the arguments in the appropriate functional setting, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Also for this kind of perturbation, we think it is interesting to understand what happens in some particular cases, similarly to what we described for the attachment of a thin tube in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' More precisely, reasoning in a completely analogous way, if p = 1 and k1 = 1 one can see that it is possible to find an eigenbasis {ϕN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , ϕN+m−1} ⊆ E(λN), orthonormal in L2(Ω), in such a way that ˜µ1,i = �∂ϕN+i−1 ∂ν (0) �2 TRd +(Σ) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Before stating the result, we would like to observe that, in view of the characterization of the half-laplacian (−∆Rd−1) 1 2 on Rd−1 = ∂Rd + as a Dirichlet-to-Neumann map on Rd +, on can easily see that the quantity TRd +(Σ) coincides with 1 2-fractional torsional rigidity of Σ in Rd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Namely TRd +(Σ) = T 1 2 Rd−1(Σ), where T 1 2 Rd−1(Σ) := −2 inf �1 2 ∥u∥2 D 1 2 ,2(Rd−1) − � Σ u: u ∈ D 1 2 ,2 0 (Σ) � and D 1 2 ,2 0 (Σ) denotes the completion of C∞ c (Σ) with respect to the norm ∥u∥D 1 2 ,2(Rd−1) := � 1 (2π) d−1 2 � Rd−1 |ζ| |ˆu(ζ)|2 dζ � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Here by ˆu we denote the (normalized) Fourier transform of u in Rd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We thus have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us assume Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4 holds with p = 1 and k1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then there exists a basis {ϕN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , ϕN+m−1} of E(λN), orthonormal in L2(Ω) and such that, for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=', m}, there holds ˜λε N+i−1 = λN − �∂ϕN+i−1 ∂ν (0) �2 T 1 2 Rd−1(Σ) εd + o(εd), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 11 We can also investigate, as a remarkable instance, the perturbation of the first eigenvalue and obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let ϕ1 ∈ H1 0(Ω) be a normalized eigenfunction corresponding to λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then there holds λε 1 = λ1 − �∂ϕ1 ∂xd (0) �2 T 1 2 Rd−1(Σ) εd + o(εd), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Facts about TΩε(εΣ, f) In this section we collect some basic facts regarding the notion of thin f-torsional rigidity of εΣ introduced before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Basics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Firstly, we briefly mention the variational framework for TΩε(εΣ, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As already mentioned, by standard minimization methods, it can be proved that, for any f ∈ C1(B+ r0ε), there exists a unique U Ωε εΣ,f = U ε f ∈ H1 0(Ωε) \\ {0} such that Jf ε (U ε f ) = inf u∈H1 0 (Ωε) Jf ε (u), where Jf ε is as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In particular, U ε f satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) 0 = dJf ε (U ε f )[ϕ] = � Ωε ∇U ε f · ∇ϕ dx − � εΣ ϕ ∂f ∂xd dx′ for all ϕ ∈ H1 0(Ωε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Letting ϕ = U ε f in the previous equation we get � Ωε ��∇U ε f ��2 dx = � εΣ U ε f ∂f ∂xd dx′, hence obtaining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2) TΩε(εΣ, f) = � Ωε ��∇U ε f ��2 dx = � εΣ U ε f ∂f ∂xd dx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The first property deals with equivalent definitions for the thin f-torsional rigidity of εΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) is equivalent to the following (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3) TΩε(εΣ, f) = sup u∈H1 0 (Ωε)\\{0} �� εΣ u ∂f ∂xd dx′ �2 � Ωε |∇u|2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' By definition, TΩε(εΣ, f) = sup u∈H1 0 (Ωε)\\{0} sup t>0 � 2t � εΣ u ∂f ∂xd dx′ − t2 � Ωε |∇u|2 dx � and the inner supremum is actually attained at t = � εΣ u ∂f ∂xd dx′ � Ωε |∇u|2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Substituting this value into the previous equality leads to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We find useful to note that if ϕ ∈ E(λN) and uε is a perturbed eigenfunction then by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4) we have � εΣ ∂ϕ ∂xd uε = (λN − λε) � Ω ϕuε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE From the latter equality and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1 it follows the meaningful estimate λεTΩε(εΣ, ϕ) ≥ � (λN − λε) � Ω ϕuε �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' On the other hand, if we denote the bounded linear F : H1 0(Ωε) → H−1(Ωε) u �→ − � εΣ ∂ϕ ∂xd u, then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3) � TΩε(εΣ, ϕ) = ∥F∥∗, as the thin ϕ- torsion function U ε ϕ is the least energy element in F−1(∥F∥∗) ⊆ H1 0(Ωε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The next properties deal with its behavior as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If ε1 > ε2 then for any f ∈ C1(B+ r0ε1) we have TΩε1 (ε1Σ, f) ≥ TΩε2 (ε2Σ, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The statement is obvious thanks to the inclusion H1 0(Ωε2) ⊆ H1 0(Ωε1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For any f ∈ E(λN) we have that TΩε(εΣ, f) → 0 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Taking into account (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3), by Cauchy-Schwarz Inequality, the trace embedding H1(B+ r0) ֒→ L2(εΣ) and regularity of eigenfunctions we have that TΩ(εΣ, f) ≤ sup u∈H1 0 (Ωε)\\{0} � εΣ u2 dS � εΣ � ∂f ∂xd �2 dx′ � Ω |∇u|2 dx = � εΣ � ∂f ∂xd �2 dx′ sup u∈H1 0 (Ωε)\\{0} � εΣ u2 dx′ � Ω |∇u|2 dx (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4) ≤ Cd,Ω,r0 ���� ∂f ∂xd ���� 2 H1(B+ r0) sup u∈H1 0 (Ωε)\\{0} � εΣ u2 dx′ � Ωε |∇u|2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5) By scaling we have sup u∈H1 0 (Ωε)\\{0} � εΣ u2 dx′ � Ωε |∇u|2 dx ≤ sup u∈H1 0 (Ωε)\\{0} � εΣ u2 dx′ � B+ ε ∪Tε |∇u|2 dx ≤ sup u∈H1 0,∂(B+ 1 ∪T1)\\S+ 1 (B+ 1 ) ε � Σ u2 dx′ � B+ 1 ∪T1 |∇u|2 dx = CΣ ε (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6) where CΣ = sup u∈H1 0,∂(B+ 1 ∪T1)\\S+ 1 (B+ 1 ) � Σ u2 dx′ � B+ 1 ∪T1 |∇u|2 dx > 0 and, for any compact set K ⊆ B+ 1 , the space H1 0,K(B+ 1 ) is defined as the closure of C∞ c (B+ 1 \\ K) with respect to the H1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Actually, for regular K there holds H1 0,K(B+ 1 ) = {u ∈ H1(B+ 1 ): u = 0 on K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Invoking (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6) we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Blow-up analysis for the thin f-torsion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As already mentioned in the intro- duction, spectral stability for this problem is ensured by the results in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It is a consequence of the uniform convergence of the resolvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Nevertheless, in order to perform a blow-up analysis as ε → 0 for the thin torsion function, we need a fundamental notion of convergence of sets (or functional spaces): it is the so-called convergence in the sense of Mosco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In our setting of scaling handles it is established in [12, Section 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We report here the definition for future reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let Hε, H0 and H be Hilbert spaces such that Hε, H0 ⊆ H for all ε ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We say that Hε converges to H0 in the sense of Mosco in H if the following hold: (M1) if vε ∈ Hε for all ε ∈ (0, 1) and vε ⇀ v weakly in H, as ε → 0, then v ∈ H0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (M2) for any v ∈ H0 there exists a sequence {vε}ε∈(0,1) such that vε ∈ Hε for all ε ∈ (0, 1) and vε → v strongly in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We start this last subsection giving an important lemma for the forthcoming analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let f ∈ E(λN) and let U ε f ∈ H1 0(Ωε) be the thin f-torsion function of εΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then � Ωε |U ε f|2 dx = o(TΩε(εΣ, f)), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us assume by contradiction that there exists a sequence εn → 0 and a constant C > 0 such that � Ωε |U εn f |2 dx ≥ 1 C TΩε(Σεn, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We set Wn := U εn f ∥U εn f ∥L2(Ωε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We have ∥Wn∥L2(Ωε) = 1 and recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2) ∥∇Wn∥2 L2(Ωε) = 1 ∥U εn f ∥2 L2(Ωε) TΩε(Σεn, f) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' By the weak compactness of the unit ball of H1 0(Ωε0), the compactness of the inclusion H1 0(Ωε0) ⊂ L2(Ωε0) and thanks to the convergence of the perturbed domains in sense of Mosco (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5), there exists an increasing sequence of integers (nk)k≥1 and a function W ∈ H1 0(Ω) such that (Wnk)k≥1 converges to W when k goes to +∞, weakly in H1 0(Ωε0) and strongly in L2(Ωε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We have that at the same time ∥W∥L2(Ω) = 1 and � Ω ∇W · ∇ϕ = 0 for any ϕ ∈ H1 0(Ω), therefore W is identically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We have reached a contradiction and proved the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ We now turn to the very aim of the subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In order to give the blow-up result on the thin f-torsion function we start with an estimate on its energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let f ∈ E(λN) be such that f(εx) εk → ψk(x) in C1,α(B+ 1 ) as ε → 0, for some integer k ≥ 1 and some harmonic polynomial ψk, homogeneous of degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then TΩε(εΣ, f) = O(εd+2k−2), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We start from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Moreover, by assumption there holds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7) � εΣ � ∂f ∂xd �2 dx′ = εd+2k−3 � Σ � ∂ ∂xd �f(εx′) εk ��2 dx′ = O(εd+2k−3), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The conclusion follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ 14 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let U ε f ∈ H1 0(Ωε) be the thin f-torsion function of εΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Under the same assump- tions as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7 there holds ˆUε(x) := U ε f(εx) εk → U Π Σ,ψk(x) in D1,2(Π) as ε → 0, where U Π Σ,ψk achieves TΠ(Σ, ψk) as defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Moreover, TΩε(εΣ, f) = εd−2+2kTΠ(Σ, ψk) + o(εd−2+2k), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='7 we deduce that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8) � 1 ε Ω∪T |∇ ˆUε|2 dx ≤ C, for some C > 0 independent from ε, thus implying that { ˆUε}ε is bounded in D1,2(Π), if ˆUε is meant to be trivially extended in Π \\ ( 1 εΩ ∪ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then there exist a subsequence (still denoted by { ˆUε}ε) and a function W ∈ D1,2(Π) such that ˆUε ⇀ W in D1,2(Π), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='9) ˆUε → W in L2(Σ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='10) as ε → 0 by compactness of trace embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Now, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5) and the equation satisfied by ˆUε, one can easily derive the equation satisfied by W, which is � Π ∇W · ∇ϕ dx = � Σ ϕ∂ψk ∂xd for all ϕ ∈ D1,2(Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' By the uniqueness of the minimizer of TΠ(Σ, ψk) (see also [16, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2] and [17, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4]) we have that W = U Π Σ,ψk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Finally ε−d−2k+2TΩε(εΣ, f) = � 1 ε Ω∪T |∇ ˆUε|2 dx = � Σ ˆUε ∂ ∂xd �f(εx) εk � dx′ → � Σ U Π Σ,ψk ∂ψk ∂xd dx′ = � Π |∇U Π Σ,ψk|2 dx = TΠ(Σ, ψk) as ε → 0, and the proof is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Perturbation of eigenvalues Our subsequent analysis is close to [11, Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2] and [3, Section 3], except that we replace the standard capacity or the u-capacity with the thin f-torsional rigidity defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We introduce the quantity χε: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) χ2 ε := sup{TΩε(εΣ, u) : u ∈ E(λN) and ∥u∥L2(Ω) = 1} Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' There holds χε → 0, as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us pick u ∈ E(λN) such that ∥u∥L2(Ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If {uN+i−1}m i=1 ⊆ E(λN) denotes a L2(Ω)-orthonormal basis, we write u = �m i=1 ciuN+i−1, with �m i=1 c2 i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then by linearity, 15 Cauchy-Schwarz inequality, the trivial inequality (�m i=1 ai)2 ≤ m �m i=1 a2 i and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2) TΩε(εΣ, u) = ������ � 1≤i,j≤m cicj � Ωε ∇U ε uN+i−1 · ∇U ε uN+j−1 dx ������ ≤ � 1≤i,j≤m |ci||cj| �� Ωε |∇U ε uN+i−1|2 dx � 1 2 �� Ω |∇U ε uN+j−1|2 dx � 1 2 = � m � i=1 |ci| �� Ωε |∇U ε uN+i−1|2 dx � 1 2 �2 ≤ m � max 1≤i≤m � Ωε |∇U ε uN+i−1|2 dx � m � i=1 c2 i = m max 1≤i≤m TΩε(εΣ, uN+i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4 TΩε(εΣ, uN+i−1) → 0 for all 1 ≤ i ≤ m: the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ For ε > 0, we denote by Πε the linear mapping Πε : E(λN) → H1 0(Ωε) u �→ u + U ε u, where E(λN) and H1 0(Ωε) are considered to be endowed, respectively, with the L2(Ω) and L2(Ωε) norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If Mε := ∥I − Πε∥L(E(λN ),H1 0 (Ωε)), there holds Mε = o(χε), as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let v ∈ E(λN) such that ∥v∥L2(Ω) = 1 and let us write v = �m i=1 ciuN+i−1, for some {ci}m i=1 such that �m i=1 c2 i = 1, being {uN+i−1}m i=1 a basis of E(λN) orthonormal in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' By definition, we have (Πε − I)v = U ε v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Hence, by linearity and Cauchy-Schwarz inequality we find that ∥(Πε − I)v∥L(E(λN ),L2(Ωε)) = ∥U ε v∥L2(Ωε) ≤ m � i=1 |ci|∥U ε uN+i−1∥L2(Ωε) ≤ � m � i=1 c2 i � 1 2 � m � i=1 ∥U ε uN+i−1∥2 L2(Ωε) � 1 2 = � m � i=1 TΩε(εΣ, uN+i−1) ∥U ε uN+i−1∥2 L2(Ωε) TΩε(εΣ, uN+i−1) � 1 2 ≤ √m χε max 1≤i≤m ∥U ε N+i−1∥L2(Ωε) TΩε(εΣ, uN+i−1)1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6, the last term is o(χε), as ε → 0, and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ We observe that, in particular, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2 that Mε < 1, meaning that Πε is injective, for ε small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We will always assume this to be the case in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Application of the abstract lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We here recall the abstract result needed in order to find good approximation of perturbed eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3 ([3], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let (H, ∥ · ∥) be a Hilbert space and q be a quadratic form, semi-bounded from below (not necessarily positive), with domain D dense in H and with discrete spectrum {νi}i≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let {gi}i≥1 be an orthonormal basis of eigenvectors of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let N and m be positive integers, F an m-dimensional subspace of D and {ξF i }m i=1 the eigenvalues of the restriction of q to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Assume that there exist positive constants γ and δ such that (H1) 0 < δ < γ/ √ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (H2) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , m}, |νN+i−1| ≤ γ, νN+m ≥ γ and, if N ≥ 2, νN−1 ≤ −γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 16 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE (H3) |q(ϕ, g)| ≤ δ ∥ϕ∥ ∥g∥ for all g ∈ D and ϕ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then we have (i) ��νN+i−1 − ξF i �� ≤ 4 γ δ2 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (ii) ∥ΠN − I∥L(F,H) ≤ √ 2δ/γ, where ΠN is the projection onto the subspace of D spanned by {gN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , gN+m−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We are going to apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3 in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Here we follow the outline of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For ε > 0 small enough, we introduce the following set of definitions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5): Hε := L2(Ωε) and ∥·∥ := ∥·∥L2(Ωε) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2) Dε := H1 0(Ωε);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3) qε(u) := � Ωε |∇u|2 dx − λN � Ωε u2 dx, for all u ∈ Dε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4) Fε := Πε(E(λN)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5) By construction, the eigenvalues of qε are {λε i − λN}i≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We use the notation νε i := λε i − λN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' If ε is small enough, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2 implies that Πε is injective, so that Πε is bijective from E(λN) onto Fε and Fε is proved to be m-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Since λε i → λi for all i ∈ N \\ {0} and since λN is of multiplicity m, the assumption (H2) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3 is fulfilled for ε > 0 small enough if we take, for instance, γ := 1 2 min{λN − λN−1, λN+m − λN+m−1} when N ≥ 2 and, when N = 1 (in which case m = 1), γ := 1 2 (λ2 − λ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It remains to check whether condition (H3) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us choose u ∈ Fε and w ∈ Dε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Since Πε is injective, as a consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2, there exists a unique v ∈ E(λN) such that u = Πεv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Hence, we have qε(u, w) = � Ωε ∇(v + U ε v) · ∇w dx − λN � Ωε (v + U ε v)w dx = � Ω ∇v · ∇w dx + � Ωε ∇U ε v · ∇w dx − λN � Ω vw dx − λN � Ωε U ε vw dx = � εΣ ∂v ∂νΩ w dx′ + � Ωε ∇U ε v · ∇w dx − λN � Ωε U ε vw dx = − � εΣ ∂v ∂xd w dx′ + � εΣ ∂v ∂xd w dx′ − λN � Ωε U ε vw dx = −λN � Ωε U ε vw dx where we have used the facts that v is an eigenfunction relative to λN, the exterior normal vector to Ω on εΣ is νΩ = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , 0, −1) and U ε v satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We then obtain |qε(u, w)| = |qε(Πεv, w)| ≤ λN∥U ε v∥∥w∥ ≤ λNMε∥v∥∥w∥ ≤ λN Mε 1 − Mε ∥u∥∥w∥ because ∥U ε v∥L2(Ωε) = ∥v + U ε v − v∥L2(Ωε) = ∥(Πε − I)v∥L2(Ωε) and ∥v∥L2(Ωε) = ∥v − Πεv + Πεv∥L2(Ωε) ≤ Mε∥v∥L2(Ωε) + ∥u∥L2(Ωε), so that ∥v∥L2(Ωε) ≤ ∥u∥ 1 − Mε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2 then implies |qε(u, w)| ≤ δε∥u∥∥w∥, 17 for some δε > 0 such that δε = o(χε) as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We can now apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3 with δ = δε, which implies that for 1 ≤ i ≤ m (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6) λε N+i−1 = λN + ξε i + o(χ2 ε), where {ξε i }m i=1 are the eigenvalues of the restriction of qε to Fε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Analysis of the restricted quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We now need to study {ξε i }m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' To do so, we are now going to investigate how the quadratic form qε acts when restricted to the m-dimensional subspace Fε = Πε(E(λN)), still endowed with the L2(Ωε)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us introduce the following bilinear form rε(u, v) := � Ωε ∇U ε u · ∇U ε v dx + λN � Ωε U ε u U ε v dx, defined for u, v ∈ E(λN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For all ϕi, ϕj ∈ E(λN), qε (Πεϕi, Πεϕj) = −rε(ϕi, ϕj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For simplicity, in the sequel we write U ε i in place of U ε ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We have qε (Πεϕi, Πεϕj) = � Ωε ∇(ϕi + U ε i ) · ∇(ϕj + U ε j ) dx − λN � Ωε (ϕi + U ε i ) (ϕj + U ε j ) dx = � Ωε ∇ϕi · ∇U ε j dx + � Ωε ∇ϕj · ∇U ε i dx + � Ωε ∇U ε i · ∇U ε j dx − λN � Ωε ϕi U ε j dx − λN � Ωε ϕj U ε i dx − λN � Ωε U ε i U ε j dx, where we have used the fact that ϕi, ϕj are both eigenfunctions relative to λN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Note that the integral involving ϕi or ϕj taken over Ωε are the same if taken over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Integrating by parts we obtain � Ω ∇ϕi · ∇U ε j dx = λN � Ωε ϕi U ε j dx + � εΣ ∂ϕi ∂νΩ U ε j dx′, where νΩ = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , 0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We can go ahead obtaining qε (Πεϕi, Πεϕj) = − � εΣ ∂ϕi ∂xd U ε j dx′ − � εΣ ∂ϕj ∂xd U ε i dx′ + � Ωε ∇U ε i · ∇U ε j dx′ − λN � Ωε U ε i U ε j dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Taking into account (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) with U ε j and U ε i as test functions we obtain � Ωε ∇U ε i · ∇U ε j dx = � εΣ ∂ϕi ∂xd U ε j dx′ = � εΣ ∂ϕj ∂xd U ε i dx′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In this way, qε (Πεϕi, Πεϕj) = − � Ωε ∇U ε i · ∇U ε j dx − λN � Ωε U ε i U ε j dx′, and the proof is concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As detailed in [3, Appendix C], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4 implies that ξε j = µε j + o(χε2) as ε → 0, where µε j denote the eigenvalues of the form rε(·, ·) defined on E(λN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Therefore, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6) and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 18 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Ramification of eigenvalues The aim of the present section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In order to investigate the occurrence of ramification of multiple eigenvalues, we need to study µε j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' the eigenvalues of the form rε(·, ·) defined on E(λN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As already mentioned in the introduction, we expect that as ε > 0 multiple eigenvalues split according to the order of vanishing of suitably chosen limit eigenfunctions at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Hence, we first introduce the aforementioned order decomposition of E(λN), which drives us towards the choice of the proper limit eigenbasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Secondly, we iteratively apply the abstract result Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3, by choosing smaller and smaller approximating spaces F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Order decomposition of the eigenspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For clarity of exposition, we report here the statement of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4, which consists of a slight variation of [3, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' There exists a decomposition of E(λN) into a sum of orthogonal subspaces E(λN) = E1 ⊕ · · · ⊕ Ep, for some integer p ≥ 1, and an associated finite increasing sequence of integers 0 < k1 < · · · < kp such that, for all 1 ≤ j ≤ p, a function in Ej \\ {0} has the order of vanishing kj at 0, that is ϕ(rx) rkj → ψkj(x) in C1,α(B+ 1 ), as ε → 0, for some harmonic polynomial ψkj, homogeneous of degree kj and odd with respect to xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In addition, such a decomposition is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Given any u ∈ E(λN), let us consider its restriction to B+ R = {x ∈ BR : xd > 0}, for R < r0 sufficiently small so that B+ R ⊆ Ω (with r0 as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2)), let us extend it to BR oddly with respect to xd, and let us call it ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then −∆˜u = λN ˜u, in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Hence, in view of classical regularity results, ˜u is analytic at 0 and any truncation of its Taylor expansion at 0 is odd with respect to xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' For any k ∈ N, let us define the mapping Πk : E(λN) → Rodd k [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , Xd] that associates to a function its (upper) Taylor expansion at 0, truncated to order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Here Rodd k [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , Xd] is the the space of polynomials odd with respect to xd of degree at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The proof can then proceed as in [3, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let E(λN) = E1 ⊕ · · · ⊕ Ep, be the order decomposition of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Then the dimension of Ej is at most the dimension of the space of spherical harmonics in d variables of degree kj (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=', [6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 159–165]) vanishing on {xd = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Explicitly, dim(Ej) ≤ �kj + d − 2 kj � + �kj + d − 3 kj − 1 � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As a consequence, in the case d = 2 dim(Ej) ≤ 1 for all 1 ≤ j ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Eigenvalues µε j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us denote by E(λN) = E1 ⊕ · · · ⊕ Ep the order decomposition of the eigenspace E(λN) (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1), with 0 < k1 < · · · < kp the associated finite sequence of vanishing orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Up to a change of basis, we can therefore assume, in the course of the proof, that the orthonormal basis {uN+i−1}m i=1 has a form which is convenient for our computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The final result will not depend on this choice of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' More precisely, we introduce the bilinear forms Tℓ(u, v) = � Π ∇U Π Σ,u · ∇U Π Σ,v for u, v ∈ Eℓ ⊆ E(λN) 19 where U Π Σ,u and U Π Σ,v achieve TΠ(Σ, Bℓu) = −2 inf w∈D1,2(Π) �1 2 � Π |∇w|2 dx − � Σ w∂Bℓu ∂xd dx′ � and TΠ(Σ, Bℓv) = −2 inf w∈D1,2(Π) �1 2 � Π |∇w|2 dx − � Σ w∂Bℓv ∂xd dx′ � respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We can assume that the orthonormal basis {uN+i−1}m i=1 agrees with the order de- composition and diagonalizes each of the quadratic forms Tℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Explicitly, this means that, for all ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , p}, Eℓ = span{uN+m1+···+mℓ−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' , uN+m1+···+mℓ−1+mℓ−1} and, for all 1 ≤ s < t ≤ mℓ, Tℓ(uN+m1+···+mℓ−1+s−1, uN+m1+···+mℓ−1+t−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It follows that, for all 1 ≤ s ≤ mℓ, Tℓ(uN+m1+···+mℓ−1+s−1, uN+m1+···+mℓ−1+s−1) = µℓ,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' According to Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5, we start from the lowest rate of convergence k1 and we look for the m1 largest eigenvalues (as ε → 0) of the matrix of the quadratic form rε in the basis {uN+i−1}m i=1, namely Aε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='6 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='8 that Aε = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 µ1,1 εd−2+2k1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 0 µ1,m1 εd−2+2k1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 + o � εd−2+2k1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Using the min-max characterization of eigenvalues, we conclude that, for 1 ≤ i ≤ m1, µε i = µ1,i εd−2+2k1 + o � εd−2+2k1� and, for m1 + 1 ≤ i ≤ m, µε i = o � εd−2+2k1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='5 and the fact that χ2 ε and εd−2+2k1 are of the same order, tell us that the same estimates hold for the difference λε N−1+i − λN: for 1 ≤ i ≤ m1 νε N−1+i = −µ1,iεd−2+2k1 + o � εd−2+2k1� and, for m1 + 1 ≤ i ≤ m, νε N−1+i = o � εd−2+2k1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The rest of the proof consists of a step-by-step procedure, in which we rescale the quadratic form qε and apply the same arguments in order to identify successive groups of eigenvalues converging to λN with the same rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let us sketch the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We set, for u, v ∈ Dε, qε 2(u, v) ≡ 1 εd−2+2k1 qε(u, v), and we define the subspace F ε 2 = Πε(E2 ⊕ · · · ⊕ Ep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The eigenvalues of qε 2 are � 1 εd−2+2k1 νε i � i≥1 = 1 εd−2+2k1 {λε i − λN}i≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We know from the first step that, for 1 ≤ i ≤ m1, lim ε→0 1 εd−2+2k1 νε N−1+i = −µ1,i < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' It follows immediately that there exists γ > 0 such that, for ε > 0 small enough, ���� 1 εd−2+2k1 νε N−1+i ���� ≤ γ for m1 + 1 ≤ i ≤ m and 1 εd−2+2k1 νε N+m ≥ 2γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 20 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ABATANGELO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' OGNIBENE whereas in case N ≥ 2 even 1 εd−2+2k1 νε N−1 ≤ −2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Repeating the arguments of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1, we can show that for all v ∈ F ε 2 and w ∈ Dε, |qε 2(v, w)| ≤ o ��εd−2+2k2 εd−2+2k1 �1/2� ∥v∥∥w∥ = o � εk2−k1� ∥v∥∥w∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Using the arguments in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2 and in the first step, we conclude that, for 1 + m1 ≤ i ≤ m1 + m2, 1 εd−2+2k1 νε N−1+i = −µ2,i−m1 ε2k2−2k1 + o � ε2k2−2k1� and, for m1 + m2 + 1 ≤ i ≤ m, 1 εd−2+2k1 νε N−1+i = o �εd−2+2k2 εd−2+2k1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This gives us finally, for m1 + 1 ≤ i ≤ m1 + m2, νε N−1+i = −µ2,i−m1 εd−2+2k2 + o � εd−2+2k2� and, for m1 + m2 + 1 ≤ i ≤ m, νε N−1+i = o � εd−2+2k2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Carrying on the procedure for ℓ from 3 to m, we reach the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' By 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2, in dimension d = 2 the eigenfunctions associated to a multiple eigenvalue have necessarily different vanishing order at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We then recover the result proved in [19, Section 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Remarks on mixed Dirichlet–Neumann boundary conditions The case of mixed Dirichlet–Neumann boundary conditions possesses additional features worthy to come to light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Firstly, let us recall the following definition, already introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Let Γ ⊆ ∂Ω be a relatively open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We call boundary torsional rigidity of Γ relative to Ω the quantity TΩ(Γ) := −2 inf �1 2 � Ω |∇u|2 dx − � Γ u dS : u ∈ H1 0,∂Ω\\Γ(Ω) � which coincides to the energy of the unique weak solution of the problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 −∆UΓ = 0, in Ω ∂UΓ ∂ν = 1, on Γ UΓ = 0, on ∂Ω \\ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We also recall that H1 0,∂Ω\\Γ(Ω) denotes the closure of C∞ c (Ω ∪ Γ) in H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' We would like to stress that Problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) has to do with the so-called boundary torsional rigidity of Ω as it is introduced in [8], there denotated by T (Ω, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As explained in [8, Section 2], T (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' δ) is modeled on the trace Sobolev embedding W 1,2(Ω) ֒→ L1(∂Ω) and it is closely related to the Steklov eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' In this case it is worthwhile to mention that, equivalently (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2) TΩ(Γ) = sup ϕ∈H1 0,∂Ω\\Γ(Ω)\\{0} �� Γ ϕ dS �2 � Ω |∇ϕ|2 dx , 21 for it is related to the best constant for the Sobolev embedding H1 0,∂Ω\\Γ(Ω) ֒→ L1(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Moreover, TΩ(Γ) is related to the lowest of the so-called Dirichlet–Steklov eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' As pointed out in [22] (see also [4]), the so-called Dirichlet–Steklov eigenvalue problem \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 −∆u = 0, in Ω u = 0, on ∂Ω \\ Γ ∂u ∂ν = σu, on Γ is equivalent to the eigenvalue problem of the Dirichlet-to-Neumann operator, which in fact admits a sequence of positive eigenvalues 0 < σ1(Ω, Γ) ≤ σ2(Ω, Γ) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' The lowest of them has the following variational characterization (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3) σ1(Ω, Γ) = inf \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 � Ω |∇u|2 dx � Γ u2 dS : u ∈ H1 0,∂Ω\\Γ(Ω) \\ {0} \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' By definition and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='2), we have 1 TΩ(Γ) = � Ω |∇UΓ|2 dx �� Γ UΓ dS �2 ≥ � Ω |∇UΓ|2 dx Ld−1(Γ) � Γ UΓ 2 dS ≥ σ1(Ω, Γ) Ld−1(Γ) where we applied Cauchy-Schwarz Inequality to gain the first inequality and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='3) to reach the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Summing up, we obtain (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='4) TΩ(Γ)σ1(Ω, Γ) ≤ Ld−1(Γ) which can be considered a Dirichlet–Steklov version of the classical Polya inequality ([21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Fi- nally, Equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) has got relevant physical meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' On one hand, it models the vertical displacement of a membrane under an external pressure which is concentrated near the boundary (see [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1] for the rigorous limit process): the considered membrane can move in the vertical direction keeping a horizontal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' On the other hand, solutions to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='1) are stationary solutions of the heat equation that models temperature in a homogeneous and isotropic heat con- ductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' This is subjected to a constant heat flux through a small part of the boundary whereas the temperature is kept constant in the remaining part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Acknowledgments R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Ognibene is partially supported by the project ERC VAREG - Variational approach to the regularity of the free boundaries (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 853404) and by the INdAM-GNAMPA 2022 project Questioni di esistenza e unicit`a per problemi nonlocali con potenziali di tipo Hardy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Part of this work was developed while R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Ognibene was in residence at Institut Mittag-Leffler in Djursholm, Stockholm (Sweden) during the semester Geometric Aspects of Nonlinear Partial Differential Equations in 2022, supported by the Swedish Research Council under grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' 2016- 06596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' References [1] Abatangelo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=', Felli, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=', and L´ena, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' Eigenvalue variation under moving mixed Dirichlet-Neumann boundary conditions and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content=' ESAIM Control Optim.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='ognibene@dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} +page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FKT4oBgHgl3EQfHS2t/content/2301.11729v1.pdf'} diff --git a/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf b/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7f726d1155f3997db6927273db0678f7a2dfaaed --- /dev/null +++ b/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6dc3594042fce4b6bae97942fc9a268e2fa8dab913465002a62c218b77a30ae8 +size 193191 diff --git 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anisotropic bulk dispersion and spin-polarized topological +surface states in CoTe2 +Atasi Chakraborty,1, ∗ Jun Fujii,2, ∗ Chia-Nung Kuo,3, 4 Chin Shan +Lue,3, 4 Antonio Politano,5, † Ivana Vobornik,2, ‡ and Amit Agarwal1, § +1Department of Physics, Indian Institute of Technology - Kanpur, Kanpur 208016, India +2Istituto Officina dei Materiali (IOM)-CNR, Laboratorio TASC, +in Area Science Park, S.S.14, Km 163.5, I-34149 Trieste, Italy. +3Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan +4Taiwan Consortium of Emergent Crystalline Materials, +Ministry of Science and Technology, Taipei 10601, Taiwan +5Dipartimento di Scienze Fisiche e Chimiche (DSFC), +Universit`a dell’Aquila, Via Vetoio 10, I-67100 L’Aquila, Italy +We present CoTe2 as a new type-II Dirac semimetal supporting Lorentz symmetry violating Dirac +fermions in the vicinity of the Fermi energy. By combining first principle ab-initio calculations with +experimental angle-resolved photo-emission spectroscopy results, we show the CoTe2 hosts a pair +of type-II Dirac fermions around 90 meV above the Fermi energy. In addition to the bulk Dirac +fermions, we find several topological band inversions in bulk CoTe2, which gives rise to a ladder +of spin-polarized surface states over a wide range of energies. +In contrast to the surface states +which typically display Rashba type in-plane spin splitting, we find that CoTe2 hosts novel out- +of-plane spin polarization as well. +Our work establishes CoTe2 as a potential candidate for the +exploration of Dirac fermiology and applications in spintronic devices, infrared plasmonics, and +ultrafast optoelectronics. +I. +INTRODUCTION +The broad class of layered transition metal dichalco- +genides (TMDs) has attracted significant attention in +the last decades due to their novel electronic, optical, +and topological properties, combined with their poten- +tial for various applications1–7. Owing to the weak inter- +layer van der Waals interaction, TMDs offer easy exfo- +liation of isolated monolayers which host different phys- +ical properties from their bulk counterpart. Interesting +examples of this include quantum spin Hall effect, super- +conductivity, charge density wave, and various topologi- +cal phases8–21. The physical and chemical properties of +TMDs can be tuned by the selection of the constituents, +the crystal structures, and the layer thicknesses22–28. +Specifically, among the TMX2 family of TMDs, PdTe2, +PtTe2, PtSe2, and NiTe2 have attracted notable inter- +est due to observation of Lorentz- symmetry violating, +type-II Dirac fermions associated with a tilted Dirac cone +near the Fermi energy18,28–39. +The Lorentz-symmetry +breaking type-II Dirac fermions have electronic, optical, +and other physical properties which are different from +those found in other topological semi-metals. The elec- +tronic band-structure and spin-polarize topological sur- +face states in these materials have been thoroughly inves- +tigated by combining realistic ab-initio calculations with +spin-resolved and conventional angle-resolved photoemis- +sion spectroscopy (ARPES) experiments. +However, the electronic properties of another prospec- +tive candidate material in the series, CoTe2 are yet to +be explored40. +CoTe2 can crystallize in both trigonal +(P¯3m1) and orthorhombic (Pnn2 and Pnnm) forms. +Among these, the centrosymmetric trigonal 1T-CoTe2 +has recently been shown to be a highly efficient electro- +catalyst for water splitting41,42. In this paper, for the +first time, we present a detailed investigation of the +electronic structure of 1T-CoTe2 by combining first- +principles calculations with spin-polarized ARPES ex- +periments. +We find that similar to other TMX2 com- +pounds, CoTe2 is also a topological semimetal supporting +a type-II Dirac crossing in the vicinity of the Fermi en- +ergy. +In addition to the bulk electronic structure, we +demonstrate that CoTe2 hosts a ladder of topological +surface states arising from several topological band in- +versions in the bulk electronic structure. These give rise +to spin-polarized Dirac surface states, with a large spec- +tral weight. +We probe this via spin-ARPES measure- +ments and the measured spin-polarized states are con- +sistent with our spin-dependent spectral function calcu- +lations. Interestingly, we find that some of the surface +states, away from the ¯Γ point, have an out-of-plane spin +polarization. +The rest of the paper is organized as follows. We de- +scribe the crystal structure and computational details in +Sec. II, followed by the details of the spin-ARPES mea- +surements in Sec. III. In Sec. IV, we explore the band +structure and geometry of the Fermi surface (FS) in +CoTe2. We study the origin of the Dirac states, multiple +band inversions, and their origin in CoTe2 employing the +ARPES measurement combined with ab initio electronic +structure calculations in Sec. V. In Sec. VI, we discuss the +spin-polarized surface states and the existence of unique +out-of-plane spin-polarized states in CoTe2 calculations. +We summarize our findings in Sec. VII. +arXiv:2301.11550v1 [cond-mat.mtrl-sci] 27 Jan 2023 + +2 +FIG. 1. +The side (a) and top (b) view of the CoTe2 crystal. In presence of SOC, the band dispersion of the experimental +(orange) and relaxed (green) structures are plotted along the high symmetry paths, marked in the Brillouin zone shown in (d). +The type II Dirac crossings near the Fermi energy of the experimental (Dexp) and relaxed (Drel) structure are marked with +red and black arrows. (e) The x-ray diffraction peak structure for CoTe2. The inset shows Laue pattern of the (0001)-oriented +CoTe2 single crystals, clearly indicating its purity and the threefold symmetry along the (001) direction. +TABLE I. Comparison of the experimental and theoretically +relaxed lattice parameters. +Experimental +Ref a +Relaxed +a/b (˚A) +3.791(9) +3.804 +3.778 +c (˚A) +5.417(0) +5.405 +5.618 +a Topological Quantum Chemistry Database +II. +CRYSTAL STRUCTURE AND +THEORETICAL METHODS +Bulk CoTe2 crystallizes in CdI2-type trigonal struc- +ture that belongs to the space group P¯3m1 (164). Each +unit cell of CoTe2 has one Co atom and two Te atoms. +To obtain the minimum-energy structure for CoTe2, we +performed the symmetry-protected cell volume and ion +relaxation using the conjugate-gradient algorithm until +the Hellman-Feynman forces on each atom were less than +the tolerance value of 10−4 eV/˚A . The cell volume of the +experimental structure increased by 2.5% as a result of +the relaxation. The comparison of lattice parameters be- +tween experimental and theoretically relaxed structures +is presented in Table I. +The trigonally distorted CoTe6 octahedra accommo- +dating the nearest neighbor Co-Te bonds (∼ 2.55˚A ) +form an edge shared geometrical network on the crys- +tallographic a−b plane [see Fig. 1 (a) and (b)]. Adjacent +mono-layers, stacked along the c axis, interact via weak +Van-der Waals interaction. +Fig. 1 (d) shows the cor- +responding bulk and (001) surface Brillouin zones (BZs) +along with the high-symmetry points. The CoTe2 crystal +structure possesses threefold rotational symmetry around +the z-axis (C3), inversion symmetry I, and the three mir- +ror symmetries M100, M010, and M110. Fig. 1 (e) shows +the experimental X-ray diffraction pattern for CoTe2. +The observation of sharp white spots in the Laue diffrac- +tion pattern in the inset of Fig. 1 (e) confirms the high +quality of the CoTe2 crystals cleaved along the (0001) di- +rection. The presence of the threefold rotation symmetry +is also evident. +To perform the ab-initio calculations, we used the den- +sity functional theory (DFT) in the plane wave basis +set. We used the Perdew-Burke-Ernzerhof (PBE)43 im- +plementation of the generalized gradient approximation +(GGA) for the exchange-correlation. This was combined +with the projector augmented wave potentials44,45 as im- +plemented in the Vienna ab initio simulation package +(VASP)46,47. GGA calculations are carried out with and +without Coulomb correlation (Hubbard U) and spin-orbit +coupling (SOC). The SOC is included in the calculations +as a second variational form to the original Hamilto- +nian. +The kinetic energy cutoff of the plane wave ba- +sis for the DFT calculations was chosen to be 450 eV. +A Γ-centered 12 × 12 × 8 Monkhorst-Pack48 k-point grid +was used to perform the momentum-space calculations +for the Brillouin zone (BZ) integration of bulk. To cal- +culate the surface spectral function for finite geometry +slabs of CoTe2, we construct the tight-binding model +Hamiltonian by deploying atom-centered Wannier func- +tions within the VASP2WANNIER9049 codes. Utilizing +the obtained tight-binding model, we calculate the sur- +face spectral function using the iterative Green’s function +method, as implemented in the WannierTools package50. + +(a) +(C) +2 +OTe +(eV) +e +P-3m1 +E +Intensity (arb. units) +0 +(d) +K +2 +< +K +M +A +H +L +A +10 +20 +30 +40 +50 +60 +70 +20(degree)3 +FIG. 2. +Side (a) and top (b) view of the 3D FS. The projected FS at E = EF on the (c) Kx − Kz plane along Γ − K, and +(d) the Ky − Kz plane along Γ − M directions. The experimentally measured 2D energy contours over Kx − Ky plane at fixed +values (e) Kz =0.03c∗, (f) 0.16 c∗, (g) 0.29c∗, and (h) 0.42 c∗, where c∗ = 2π/c. The theoretical FS cuts for specific Kz planes +are plotted on top of the corresponding experimental results. +III. +ARPES AND SPIN-ARPES +MEASUREMENTS +ARPES and Spin-ARPES measurements were per- +formed at low energy (LE) branch of the APE-NFFA +beamline51 at the Elettra synchrotron facility (Trieste, +Italy), which is equipped with VESPA 52 as an electron +spin polarization detector. +The details of the experi- +mental geometry, like the electron analyzer slit opening +and incoming photon direction with respect to the ana- +lyzer lens axis, can be found in Ref. [52]. To determine +the inner potential (V0) of CoTe2 (0001) experimentally, +angle-resolved valence band spectra and FS maps were +measured for the photon energy range between 13 eV +and 85 eV with 2 linear polarizations (s- and p- polariza- +tion). Spin-ARPES maps were acquired for two-photon +energies (hν= 19 eV and 75 eV). The energy and angu- +lar resolutions for the Spin-ARPES measurements were +set to 100 meV and 1.5◦, respectively. The clean (0001) +surface of CoTe2 was obtained by the cleavage of the sin- +gle crystal in situ in an ultra-high vacuum. The sample +temperature during the ARPES and Spin-ARPES mea- +surements was kept at 78 K. +IV. +ELECTRONIC BAND-STRUCTURE AND +THE FS GEOMETRY +The ionic balance of the chemical formula of CoTe2, +suggests that the Co and Te atoms are in 3d34s0 and +5s25p6 configurations, respectively. +As a consequence, +we expect the Co-d and Te-p orbitals to have a major +contribution at the Fermi energy (EF ). We present the +bulk band-dispersion in presence of SOC, for the exper- +imental structure, and also for the relaxed structure in +Fig. 1(c). The experimental electronic band dispersion +in Fig. 1(c), clearly shows the existence of a couple of +tilted Dirac-like crossings just above EF , along the Γ-A +high symmetry direction. We find that the position of +the Dirac point (DP) is sensitive to small variations of +the structural parameters. It shifts from ∼ 0.68 eV to +∼0.92 eV above EF due to the small change in the struc- +tural parameters on relaxation. Since the Γ-A path is +an invariant subspace of the three-fold rotational crystal +symmetry (C3), the Dirac cone is protected by the ro- +tational symmetry. This is similar to the Dirac crossing +in NiTe2 and other related materials in the same space +group18,35,53. Two accidental linear band crossings, one +exactly at EF on the high symmetry A point and another +one at ∼2.2 eV above EF along the Γ − A path become +gapped due to relaxation, as highlighted in Fig. 1(c). +The geometry of the FS and its evolution with change + +(a) +(b) +(c) 0.5 +(d) +0.5 +Kz (2n/c) +(2/c) +0 +Kz +-0.5 +0.5 +0.5 +0.5 +0 +Kx +0.5 +0 +0.5 +[FK) Kx (A-1) +[FM Ky (A-1) +1.0 +e +1.0 +1.0 +(6) +1.0 +(h) +0.5 +0.5 +0.5 +0.5 +(A-1) +0 +0 +0- +0 +-0.5 +-0.5 +-0.5 +-0.5 +-1.0 +0.5 +-1.0- +-1.0 +-0.5 +-0.5 +1.0 +1.0 +-1.0 +0 +1.0 +-1.0 +0 +0.5 +-1.0 +-0.5 +0.5 +1.0 +-1.0-0.5 +Kx (A-1) +Kx (A-1) +Kx (A-1) +Kx (A-1)4 +in the Fermi energy is shown in Fig. 2 for the relaxed +structure. The 3D FS for E = EF is shown in Fig. 2(a). +The projection of the FS on a plane perpendicular to the +Kz axis in Fig. 2(b) clearly shows three distinct types of +band contributions at the FS, each having two-fold de- +generacy. Figures 2(c) and (d) capture the projection of +the FS on the Kx − Kz plane along the Γ − K line, and +the Ky − Kz plane along the Γ − M direction. The FS +along the Γ − M path is highly anisotropic as seen in +Fig. 2(d). Clearly, bulk CoTe2 has a strong momentum- +dependent anisotropic FS (see Appendix B for details), +which is also expected from the presence of type-II Dirac +fermions in the system. To investigate the FS modula- +tions along the Kz direction, we have shown the energy +contours at different Kz values in panels (e)-(h) of Fig. 2. +The theoretically calculated (solid line) and the experi- +mentally measured (mud color scale) 2D energy contours +within the Wigner Seitz cell are shown in Fig. 2(e)-(h) +over the Kx − Ky plane for different Kz values. Differ- +ent Kz values are probed in the ARPES experiment by +changing the energy of the incident photon beam. Using +the free electron final state model54, we have +k⊥ = 1 +ℏ +� +2m (V0 + Ekin cos2 θ). +(1) +Here, V0 is the inner potential, Ekin is the kinetic en- +ergy of a photoelectron and θ denotes the emission an- +gled from the sample surface normal. For the different +panels of Fig. 2(e)-(h), we have hν (corresponding Kz) += 75 eV (0.03 c∗), 70 eV (0.16 c∗), 65 eV (0.29 c∗), and +60 eV (0.42 c∗), respectively where c∗ = 2π/c. We have +applied V0 = 11 eV, to calculate the Kz values. +The experimental FS demonstrates the transforma- +tion of its symmetry from sixfold at Kz = 0 to three- +fold for Kz > 0, which is consistent with the theoret- +ical calculations. +For Kz = 0 [Fig . 2(e)], all three +(the hexapetalus flower-shaped, hexagonal, and circular) +states are observed and well matched to the calculated +FS. The hexapetalus flower-shaped states in Fig. 2(e) is +transformed into the trefoil in Fig. 2(f). Due to the exper- +imental geometry and the corresponding matrix-element +effect, the measured FS shows an anisotropic distribu- +tion in the photoemission intensity. The photoemission +intensity is higher along one of the three ¯ +M-¯Γ- ¯ +M direc- +tions and lower along the two other ¯ +M-¯Γ- ¯ +M directions. +This effect reduces the clarity of the three-fold symme- +try in the FS, measured for Fig. 2(g) and (h). However, +the strong modulation of the FS on changing Kz is clear, +and it is broadly consistent with the 3D FS distribution +of Fig. 2(a). +We now focus on the FS, in the vicinity of the DP. The +side and top view of the 3D FS distribution within the +Wigner-Seitz cell at E = EDP for the relaxed structure +is presented in Fig. 3(a) and (b), respectively. The pres- +ence of three contributing bands, each having twofold +Kramer’s degeneracy, can be clearly seen. +The outer- +most part of the FS arises from the electron pocket of +the first unoccupied band of CoTe2, as seen in Fig. 1(c). +The type-II nature of the DSM phase is also confirmed +by the fact that the Dirac point appears at the four- +fold degenerate touching point of the other electron and +hole pockets in the middle, as marked by the red arrow +in the FS in Fig. 3(a). +The energy contours over the +Kx −Kz and the Ky −Kz planes, for E = EDP is shown +in Fig. 3(c) and (d), respectively. Our calculations reveal +a prominent Dirac crossing located at Kz ∼ ±0.25 c∗. +The anisotropic nature of the FS along the Ky direction +persists even at the DP. The in-plane projection of the +energy contours at the DP is presented in Fig. 3(e)-(i), +for five different out-of-plane distances (or Kz values). At +Kz = 0 c∗, we observe a hexapetalus flower shape along +with a small circle at its center. The electron pockets at +Kz = 0 transform into an isolated bean-shaped pattern +with increasing Kz magnitude as seen in Fig. 3(g)-(i). At +Kz approaching the vicinity of bulk DP, the central con- +tour converges to a tiny circle while the hexagonal outer +contour acquires an almost triangular shape [see Fig. 3(g) +and (h)]. Finally, at Kz = 0.3c∗, the FS cut appears as +two contours centered around the origin, which exhibit +a circular and triangular shape for the inner and outer +contours, respectively. In addition, there are small pock- +ets along three of the six A−H lines [see Fig. 3(h)]. The +energy contour at negative Kz values with the same mag- +nitude shows the rest of the three small pockets along the +other A − H lines. In Fig. 3(i) the inner contours van- +ish and we only see three small pockets along to A − H +direction. +V. +ORIGIN OF DIRAC STATES, BAND +INVERSION, AND THE SURFACE STATES +The presence of type-II Dirac fermions in the bulk dis- +persion of CoTe2 suggests the strong possibility of finding +topologically protected surface states near the Fermi en- +ergy. Additionally, the bulk bands of CoTe2 also support +several other topological band inversions in its bulk. In +Fig. 4(a), the orbital-resolved band structure along the +Γ-A path shows that the linear crossings near EF are +mainly composed of the Te-5p orbitals. The Dirac band +crossing near 0.92 eV above Fermi energy arises from the +interplay of the Te px + py and the Te pz orbitals. Ad- +ditionally, we find that these orbitals also contribute to +multiple band inversion gaps along different high sym- +metry paths including Γ − A [see Fig. 4(a)]. To under- +stand the origin of the Dirac band crossing, we show the +systematic evolution of the energy levels of the Te-5p or- +bital manifold in Fig. 4(e). The six degenerate p orbital +splits into lower (upper) lying three-fold bonding (anti- +bonding) orbitals due to inter-site hybridization. +The +presence of local trigonal distortion of the Co-Te octa- +hedra further lifts the degeneracy of the bonding/anti- +bonding p orbitals breaking it into singly degenerate a1g +(pz) and doubly degenerate eπ +g (px, py) orbitals. Includ- +ing the SOC splits the p orbitals into fourfold Jeff = 3/2 +and two-fold Jeff = 1/2 pseudo spin basis as shown in the + +5 +FIG. 3. +Side (a) and top (b) view of the 3D FS distribution. +The planar projection of the constant energy surface at +E − EF = 0.94 eV, on the (c) Kx − Kz, and the (d) Ky − Kz surface. The 2D energy contours within the Wigner Seitz cell +(marked by dotted line) in Kx−Ky plane at fixed Kz values of (e) Kz = 0, (f) Kz = 0.15 c∗, (g) Kz = 0.25 c∗, (h) Kz = 0.30 c∗, +and (i) Kz = 0.42 c∗, where c∗ = 2π/c. The 2D plane of (g), which hosts the Dirac point, is marked in Fig. 1 (d). +fourth column of Fig. 4(e). The last column of Fig. 4(e) +highlights the effect of the dispersion along the Γ − A +direction. The bulk type-II Dirac point arises from the +crossing of the bonding and anti-bonding states of the +Jeff = 3/2 orbitals. +The ladder of multiple band inversions and the Dirac +point in the bulk band structure points to the existence +of topologically protected surface states in CoTe2. This +is confirmed by our experiments and theoretical calcula- +tions. The measured ARPES results and the correspond- +ing theoretical spectral function of the relaxed structure +are shown along the high symmetry ¯K-¯Γ- ¯K and ¯ +M-¯Γ- ¯ +M +directions in Fig. 4(c), (d) and Fig. 4(g), (h), respectively. +The pattern of the spectral function and position of the +surface Dirac cone matches well between the theoretical +calculations and experimental plots. However, the other +sharp spectral functions [purple, and yellow arrows in +Fig. 4(c), (d) and Fig. 4(g)] arising from the bulk and sur- +face states are slightly off in energy (see Appendix A for +detailed discussions). This can be due to several reasons +including i) small variations in the structural parameters, +ii) some ambiguity in the pseudopotential for capturing +core states, iii) some impurities or stacking faults in the +crystal which are not included in theoretical calculations, +amongst others. We also note that as the Bulk Dirac cone +is significantly above the Fermi energy, it cannot be di- +rectly observed or mapped via our occupied state ARPES +data. +The ARPES measurements were done with hν = 75 +eV, which corresponds to Kz ∼ 0c∗. +Therefore these +spectra capture the bulk bands along with the surface +states. The prominent features corresponding to the sur- +face states, in the measured ARPES spectrum and the +calculated spectral function are marked by thick arrows. +Despite some discrepancies in the binding energy of a few +states, the experimental and the theoretical results show +good qualitative agreement. The small energy difference +in the location of the surface states possibly arises due +to structural effects or from the surface potential which +is not included in our theoretical calculations. +The Dirac cone in the surface states is located at the +¯Γ point at an energy 0.49 eV below the Fermi energy. +The presence of a topological band inversion near EF , +as marked by an arrow in Fig. 4 (a) gives rise to this +surface Dirac crossing observed in ARPES. A similar +surface Dirac cone, which has relatively broad features +in ARPES experiments compared to theoretical calcula- +tions, has also been observed in other isostructural com- +pounds such as NiTe2 and in PtTe2. +Other than the +Dirac cone at the ¯Γ point, several sharp non-trivial sur- +face states appear near the high symmetry ¯ +M point and +along the ¯Γ− ¯K path. These arise from the multiple band +inversions throughout the BZ. We find the surface states +to be symmetric along both the ¯K-¯Γ- ¯K and the ¯ +M-¯Γ- ¯ +M +directions. +The Fermi arc states at constant energy are plotted in +Fig. 4 (b) at E−EF = 0 eV and in Fig. 4 (f) at E−EF = +−0.49 eV. At the Fermi energy, circular arcs of the sharp + +(a) +(b) +(c) 0.5 +(d)0.5 +(2/c) +[2/c) +Kz +0 +1 +0.5 +0.5 +-0.5 +0.5 +0 +0.5 +n +0.5 +[FK] Kx (A-1) +[FM) Ky (A-1) +0.5 +0.5 +-1.0 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +0.5 +0 +0.5 +0.5 +0.5 +Kx (A-1, +Kx (A-1) +Kx (A-1) +Kx (A-16 +FIG. 4. +(a) Multiple band inversion arising from the Te-p orbital manifold along the Γ − A direction. The band inversion +gap near the Fermi energy (IBG) is marked with an arrow. The Fermi arc states at constant energy (b) E − EF = 0, and (f) +at E − EF = −0.49 eV. The theoretically calculated momentum resolved spectral density plot along the (c) ¯Γ- ¯K and the (g) +¯Γ- ¯ +M paths. The experimentally measured ARPES plots along the (d) ¯Γ- ¯K and (h) ¯Γ- ¯ +M paths. (e) To highlight the origin of +the bulk type-II Dirac fermions, we show the schematic of the level diagram of the Te-5p orbitals in presence of a local crystal +field and SOC. +surface states appear around the ¯Γ point. In contrast, a +prominent peak is observed exactly at the ¯Γ point Fig. 4 +(f) which captures the dominant surface Dirac, crossing +along with a few less intense circular arcs along the ¯Γ- ¯K +paths. There is another set of high-intensity surface arc +states around −1.4 eV below EF [see Fig. 4 (c) and (g)], +which disperse symmetrically around the ¯Γ point. +VI. +SPIN POLARIZED SURFACE STATES +The demonstration of topological surface states in +CoTe2 inspires the exploration of their spin-polarization. +To study the spin polarization of the surface states and +the surface Dirac cone, we experimentally measured the +spin-polarized ARPES spectrum of CoTe2, as shown in +Fig. 5(b), (d) and (f). The component of probed spin +components is chosen to be perpendicular to the direction +of the dispersion. The corresponding theoretically calcu- +lated spin resolved spectral function is shown in Fig. 5(a), +(c), and (e). +Figures 5(b) and (d) display the mea- +sured spin-resolved band structures superimposed onto +the measured spin-integrated band structures shown in +Figs. 4(d) and (h), along ¯K-¯Γ- ¯K and +¯ +M-¯Γ- ¯ +M, respec- +tively. As was seen in the experimental plots of Fig. 4(d) +and (h), the surface Dirac cone spectral intensity is rel- +atively low compared to the observed bulk states for the +photon energy hν = 75 eV. Accordingly, its contribution +to the measured spin-ARPES spectra is also small. To +improve the resolution of the spin information of the sur- +face Dirac cone, the Spin-ARPES spectra in Fig. 5(d) +were measured with hν = 19 eV. The crossing of the +up-spin (red) and the down-spin (blue) bands are well +observed around the energy of the surface Dirac point, +matching well with the calculated spin texture shown in +Fig. 5(c). This confirms the helical nature of the spin- +momentum locking around the surface Dirac point and +its topological origin. The signs of measured and the cal- +culated spin polarization is reversed for Kx/y → −Kx/y, +in all panels. This implies that the spin polarization is +not due to the breaking of time-reversal symmetry. +Interestingly, we also observe a significant contribution +of the out-of-plane component in our spin ARPES experi- +ments and calculations for the ¯K-¯Γ- ¯K direction, as shown +in Fig. 5(e) and (f). The measurement is done with inci- +dent photons with energy hν = 75 eV. The corresponding +out-of-plane spin component for the ¯ +M-¯Γ- ¯ +M direction is +negligibly small. The scale of the in-plane and the out- +of-plane spin polarization in all the panels is identical. +Note that due to the presence of time reversal and in- +version symmetry in CoTe2, the spin polarization of the +bulk states is forbidden. Even an isolated monolayer of +CoTe2 preserves the inversion and the time-reversal sym- +metries. Thus, an isolated monolayer of CoTe2 will also + +(a) 3 +(b) +-2 +0 +2 +E-Eε=0 eV +(d) +K +(c) +0 +(eV) +2 +1 +Te (px+py) +-0.5 +(eV) +Te (pz) +E +(A-1) +0 +i +-1.0 +K +出 +4 +-1.5 +IBG +0 +0 +2.0 +-1 +0 +1.0-0.50.00.51.0 +-1.0-0.50.00.5 +51.0 +A +Kx (A-1) +Kx (A-1) +Kx (A-1) +0 +(e) +A +(f) +2 +p2 +E-Ee=-0.49 eV +(g) +M +0 +[3/2,1/2) +1 +(eV) +pxy +: +(A-1) +-0.5 +.3/2,.3/2) +pxy +E +-1.0 +Te1-p +.13/2.3/2) +[3/2,3/2) +-1.5 +[3/2,1/2 +j3/2,3/2 +-1 +pxyz +EF +-2.0 +[1/2,1/2) +-1 +0 +1 +zd +-0.50.0 +0.5 +-0.5 +0.0 +0.5 +Kx (A-1) +Ky (A-1) +S7 +Ky (A-1)7 +FIG. 5. Spin polarization of the surface bands along the high symmetry (a) ¯K-¯Γ- ¯K and (c) +¯ +M-¯Γ- ¯ +M directions. The spin- +ARPES measurements for (b) ¯K-¯Γ- ¯K and the (d) ¯ +M-¯Γ- ¯ +M directions. The spin components are orthogonal to the corresponding +momentum directions. (e) The theoretical and (f) experimentally measured out-of-plane spin polarization along the ¯K-¯Γ- ¯K +direction. CoTe2 supports spin-polarized surface states over a wide range of energies in the entire BZ. +not support spin-polarized states. However, in a system +of finite size, the inversion symmetry is broken for the +atomic layers near the surface even for bulk centrosym- +metric systems. This is what allows for spin polarization +of the surface states (both in-plane and out-of-plane) in +a finite slab of CoTe2, and other Dirac semimetals. An- +other interesting point is that the surface states near the +¯Γ point primarily arise from the topological bulk band +inversions, and these lead to Dirac surface states which +have an in-plane Rashba-like spin momentum locking. +This can be clearly seen in Fig. 5(e), where the out-of- +plane spin states are completely absent near the ¯Γ point. +VII. +CONCLUSIONS +In summary, based on the ARPES experiments com- +bined with detailed first principle calculations, we show +that CoTe2 hosts a pair of type-II Dirac nodes. +The +Dirac node is located along the Γ − A axis around 0.92 +eV above the Fermi energy, and they support Lorentz +symmetry violating Dirac fermions. We find that in addi- +tion to the Dirac fermions, bulk CoTe2 also hosts several +topological band inversions which give rise to a ladder of +spin-polarized surface states over a wide range of ener- +gies. The surface states corresponding to the bulk band +inversions form a surface Dirac cone at the ¯Γ point, which +has Rashba-type in-plane spin splitting. Interestingly, we +find that some surface states in CoTe2 also support an +out-of-plane spin polarization. Our study highlights that +CoTe2 supports interesting bulk and surface Dirac fermi- +ology, which should be explored further in transport, op- +tical, plasmonic, and optoelectronic experiments. +VIII. +ACKNOWLEDGEMENT +A.C. acknowledges the Indian Institute of Technol- +ogy, Kanpur, and the Science and Engineering Re- +search Board (SERB) National Postdoctoral Fellowship +(PDF/2021/000346), India for financial support. +We +thank Debasis Dutta and Barun Ghosh for the useful +discussions. We acknowledge the Science and Engineer- +ing Research Board (SERB) and the Department of Sci- +ence and Technology (DST) of the Government of In- +dia for financial support. We thank CC-IITK for pro- +viding the High-Performance Computing facility. +This +work has been partly performed in the framework of +the nanoscience foundry and fine analysis (NFFA-MIUR +Italy, Progetti Internazionali) facility. +Appendix A: Scaled ARPES +The prominent bulk and surface states (except the +Dirac crossing) of theoretically calculated spectral func- +tion and experimentally measured ARPES plots in +Fig. 4(c), (d) and in Fig. 4(g)] have an energy differ- +ence of ∼500 meV. This can arise from various factors +as discussed in section V. For example, a similar discrep- +ancy of energy is reported for a related compound PtSe2 +in Ref.55. An energy scale factor of 1.05 and an energy +offset of -0.1 eV is necessary for the PtSe2 compound to +correctly match the energy between theoretical and ex- +perimental ARPES results. Similarly in our calculation, +an energy scaling of 0.7 can be used to best fit the ex- +perimental plot (see Fig. 6). + +(a) +(b) +(c) +(d) +(e) +(f) +→K +K←--T- +-→K +M←I +→M +M+--T--→M +K+--T---→K +K+--T--→K +0.0 +-0.5 +(eV) +-1.0- +0.25 +0.25 +40.25 +0.25 +0.25 +in-plane +0.25 +in-plane +out-of-plane +-1.5. +-1.0-0.5 +0.0 +0.5 +1.0 -1.0 -0.5 0.00.51.0 +-0.5 +0.0 +0.5 +-1.0-0.50.0.0.5 +1.0 -1.0 -0.5 0.0,0.51.0 +Kx (1/A) +Kx (1/A) +Ky (1/A) +Ky (1/A) +Kx (1/A) +Kx (1/A)8 +FIG. 6. +Theoretically calculated (a) and experimental (b) +momentum resolved spectral function plot ¯ +M-¯Γ- ¯ +M directions. +The theoretical spectral function incorporates an energy scal- +ing factor of 0.7 to best match the experimental data. +Appendix B: Fermi surface anisotropy +In this section, we have compared the theoretically cal- +culated and experimentally observed Fermi surface maps +on Kx-Kz and Ky-Kz planes. 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King, Nature Materials +17, 21 (2018). + diff --git a/uNFJT4oBgHgl3EQfdyz5/content/tmp_files/load_file.txt b/uNFJT4oBgHgl3EQfdyz5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d686cd27fd2b391bfc3d85b2579f4a866cce9a4b --- /dev/null +++ b/uNFJT4oBgHgl3EQfdyz5/content/tmp_files/load_file.txt @@ -0,0 +1,1265 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf,len=1264 +page_content='Observation of highly anisotropic bulk dispersion and spin-polarized topological surface states in CoTe2 Atasi Chakraborty,1, ∗ Jun Fujii,2, ∗ Chia-Nung Kuo,3, 4 Chin Shan Lue,3, 4 Antonio Politano,5, † Ivana Vobornik,2, ‡ and Amit Agarwal1, § 1Department of Physics, Indian Institute of Technology - Kanpur, Kanpur 208016, India 2Istituto Officina dei Materiali (IOM)-CNR, Laboratorio TASC, in Area Science Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='14, Km 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5, I-34149 Trieste, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan 4Taiwan Consortium of Emergent Crystalline Materials, Ministry of Science and Technology, Taipei 10601, Taiwan 5Dipartimento di Scienze Fisiche e Chimiche (DSFC), Universit`a dell’Aquila, Via Vetoio 10, I-67100 L’Aquila, Italy We present CoTe2 as a new type-II Dirac semimetal supporting Lorentz symmetry violating Dirac fermions in the vicinity of the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' By combining first principle ab-initio calculations with experimental angle-resolved photo-emission spectroscopy results, we show the CoTe2 hosts a pair of type-II Dirac fermions around 90 meV above the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In addition to the bulk Dirac fermions, we find several topological band inversions in bulk CoTe2, which gives rise to a ladder of spin-polarized surface states over a wide range of energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In contrast to the surface states which typically display Rashba type in-plane spin splitting, we find that CoTe2 hosts novel out- of-plane spin polarization as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Our work establishes CoTe2 as a potential candidate for the exploration of Dirac fermiology and applications in spintronic devices, infrared plasmonics, and ultrafast optoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' INTRODUCTION The broad class of layered transition metal dichalco- genides (TMDs) has attracted significant attention in the last decades due to their novel electronic, optical, and topological properties, combined with their poten- tial for various applications1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Owing to the weak inter- layer van der Waals interaction, TMDs offer easy exfo- liation of isolated monolayers which host different phys- ical properties from their bulk counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Interesting examples of this include quantum spin Hall effect, super- conductivity, charge density wave, and various topologi- cal phases8–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The physical and chemical properties of TMDs can be tuned by the selection of the constituents, the crystal structures, and the layer thicknesses22–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Specifically, among the TMX2 family of TMDs, PdTe2, PtTe2, PtSe2, and NiTe2 have attracted notable inter- est due to observation of Lorentz- symmetry violating, type-II Dirac fermions associated with a tilted Dirac cone near the Fermi energy18,28–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The Lorentz-symmetry breaking type-II Dirac fermions have electronic, optical, and other physical properties which are different from those found in other topological semi-metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The elec- tronic band-structure and spin-polarize topological sur- face states in these materials have been thoroughly inves- tigated by combining realistic ab-initio calculations with spin-resolved and conventional angle-resolved photoemis- sion spectroscopy (ARPES) experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' However, the electronic properties of another prospec- tive candidate material in the series, CoTe2 are yet to be explored40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' CoTe2 can crystallize in both trigonal (P¯3m1) and orthorhombic (Pnn2 and Pnnm) forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Among these, the centrosymmetric trigonal 1T-CoTe2 has recently been shown to be a highly efficient electro- catalyst for water splitting41,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In this paper, for the first time, we present a detailed investigation of the electronic structure of 1T-CoTe2 by combining first- principles calculations with spin-polarized ARPES ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We find that similar to other TMX2 com- pounds, CoTe2 is also a topological semimetal supporting a type-II Dirac crossing in the vicinity of the Fermi en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In addition to the bulk electronic structure, we demonstrate that CoTe2 hosts a ladder of topological surface states arising from several topological band in- versions in the bulk electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' These give rise to spin-polarized Dirac surface states, with a large spec- tral weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We probe this via spin-ARPES measure- ments and the measured spin-polarized states are con- sistent with our spin-dependent spectral function calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Interestingly, we find that some of the surface states, away from the ¯Γ point, have an out-of-plane spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We de- scribe the crystal structure and computational details in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' II, followed by the details of the spin-ARPES mea- surements in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' IV, we explore the band structure and geometry of the Fermi surface (FS) in CoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We study the origin of the Dirac states, multiple band inversions, and their origin in CoTe2 employing the ARPES measurement combined with ab initio electronic structure calculations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' VI, we discuss the spin-polarized surface states and the existence of unique out-of-plane spin-polarized states in CoTe2 calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We summarize our findings in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='11550v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='mtrl-sci] 27 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The side (a) and top (b) view of the CoTe2 crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In presence of SOC, the band dispersion of the experimental (orange) and relaxed (green) structures are plotted along the high symmetry paths, marked in the Brillouin zone shown in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The type II Dirac crossings near the Fermi energy of the experimental (Dexp) and relaxed (Drel) structure are marked with red and black arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' (e) The x-ray diffraction peak structure for CoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The inset shows Laue pattern of the (0001)-oriented CoTe2 single crystals, clearly indicating its purity and the threefold symmetry along the (001) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Comparison of the experimental and theoretically relaxed lattice parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Experimental Ref a Relaxed a/b (˚A) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='791(9) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='804 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='778 c (˚A) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='417(0) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='405 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='618 a Topological Quantum Chemistry Database II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' CRYSTAL STRUCTURE AND THEORETICAL METHODS Bulk CoTe2 crystallizes in CdI2-type trigonal struc- ture that belongs to the space group P¯3m1 (164).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Each unit cell of CoTe2 has one Co atom and two Te atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' To obtain the minimum-energy structure for CoTe2, we performed the symmetry-protected cell volume and ion relaxation using the conjugate-gradient algorithm until the Hellman-Feynman forces on each atom were less than the tolerance value of 10−4 eV/˚A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The cell volume of the experimental structure increased by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5% as a result of the relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The comparison of lattice parameters be- tween experimental and theoretically relaxed structures is presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The trigonally distorted CoTe6 octahedra accommo- dating the nearest neighbor Co-Te bonds (∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='55˚A ) form an edge shared geometrical network on the crys- tallographic a−b plane [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1 (a) and (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Adjacent mono-layers, stacked along the c axis, interact via weak Van-der Waals interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1 (d) shows the cor- responding bulk and (001) surface Brillouin zones (BZs) along with the high-symmetry points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The CoTe2 crystal structure possesses threefold rotational symmetry around the z-axis (C3), inversion symmetry I, and the three mir- ror symmetries M100, M010, and M110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1 (e) shows the experimental X-ray diffraction pattern for CoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The observation of sharp white spots in the Laue diffrac- tion pattern in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1 (e) confirms the high quality of the CoTe2 crystals cleaved along the (0001) di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The presence of the threefold rotation symmetry is also evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' To perform the ab-initio calculations, we used the den- sity functional theory (DFT) in the plane wave basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We used the Perdew-Burke-Ernzerhof (PBE)43 im- plementation of the generalized gradient approximation (GGA) for the exchange-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This was combined with the projector augmented wave potentials44,45 as im- plemented in the Vienna ab initio simulation package (VASP)46,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' GGA calculations are carried out with and without Coulomb correlation (Hubbard U) and spin-orbit coupling (SOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The SOC is included in the calculations as a second variational form to the original Hamilto- nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The kinetic energy cutoff of the plane wave ba- sis for the DFT calculations was chosen to be 450 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' A Γ-centered 12 × 12 × 8 Monkhorst-Pack48 k-point grid was used to perform the momentum-space calculations for the Brillouin zone (BZ) integration of bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' To cal- culate the surface spectral function for finite geometry slabs of CoTe2, we construct the tight-binding model Hamiltonian by deploying atom-centered Wannier func- tions within the VASP2WANNIER9049 codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Utilizing the obtained tight-binding model, we calculate the sur- face spectral function using the iterative Green’s function method, as implemented in the WannierTools package50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' (a) (C) 2 OTe (eV) e P-3m1 E Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' units) 0 (d) K 2 < K M A H L A 10 20 30 40 50 60 70 20(degree)3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Side (a) and top (b) view of the 3D FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The projected FS at E = EF on the (c) Kx − Kz plane along Γ − K, and (d) the Ky − Kz plane along Γ − M directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The experimentally measured 2D energy contours over Kx − Ky plane at fixed values (e) Kz =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='03c∗, (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='16 c∗, (g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='29c∗, and (h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='42 c∗, where c∗ = 2π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The theoretical FS cuts for specific Kz planes are plotted on top of the corresponding experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' ARPES AND SPIN-ARPES MEASUREMENTS ARPES and Spin-ARPES measurements were per- formed at low energy (LE) branch of the APE-NFFA beamline51 at the Elettra synchrotron facility (Trieste, Italy), which is equipped with VESPA 52 as an electron spin polarization detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The details of the experi- mental geometry, like the electron analyzer slit opening and incoming photon direction with respect to the ana- lyzer lens axis, can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' To determine the inner potential (V0) of CoTe2 (0001) experimentally, angle-resolved valence band spectra and FS maps were measured for the photon energy range between 13 eV and 85 eV with 2 linear polarizations (s- and p- polariza- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Spin-ARPES maps were acquired for two-photon energies (hν= 19 eV and 75 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The energy and angu- lar resolutions for the Spin-ARPES measurements were set to 100 meV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The clean (0001) surface of CoTe2 was obtained by the cleavage of the sin- gle crystal in situ in an ultra-high vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The sample temperature during the ARPES and Spin-ARPES mea- surements was kept at 78 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' ELECTRONIC BAND-STRUCTURE AND THE FS GEOMETRY The ionic balance of the chemical formula of CoTe2, suggests that the Co and Te atoms are in 3d34s0 and 5s25p6 configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' As a consequence, we expect the Co-d and Te-p orbitals to have a major contribution at the Fermi energy (EF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We present the bulk band-dispersion in presence of SOC, for the exper- imental structure, and also for the relaxed structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The experimental electronic band dispersion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1(c), clearly shows the existence of a couple of tilted Dirac-like crossings just above EF , along the Γ-A high symmetry direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We find that the position of the Dirac point (DP) is sensitive to small variations of the structural parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' It shifts from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='68 eV to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='92 eV above EF due to the small change in the struc- tural parameters on relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Since the Γ-A path is an invariant subspace of the three-fold rotational crystal symmetry (C3), the Dirac cone is protected by the ro- tational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This is similar to the Dirac crossing in NiTe2 and other related materials in the same space group18,35,53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Two accidental linear band crossings, one exactly at EF on the high symmetry A point and another one at ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='2 eV above EF along the Γ − A path become gapped due to relaxation, as highlighted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The geometry of the FS and its evolution with change (a) (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 Kz (2n/c) (2/c) 0 Kz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0 Kx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 [FK) Kx (A-1) [FM Ky (A-1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 (6) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 (h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 (A-1) 0 0 0- 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 Kx (A-1) Kx (A-1) Kx (A-1) Kx (A-1)4 in the Fermi energy is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2 for the relaxed structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The 3D FS for E = EF is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The projection of the FS on a plane perpendicular to the Kz axis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(b) clearly shows three distinct types of band contributions at the FS, each having two-fold de- generacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Figures 2(c) and (d) capture the projection of the FS on the Kx − Kz plane along the Γ − K line, and the Ky − Kz plane along the Γ − M direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The FS along the Γ − M path is highly anisotropic as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Clearly, bulk CoTe2 has a strong momentum- dependent anisotropic FS (see Appendix B for details), which is also expected from the presence of type-II Dirac fermions in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' To investigate the FS modula- tions along the Kz direction, we have shown the energy contours at different Kz values in panels (e)-(h) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The theoretically calculated (solid line) and the experi- mentally measured (mud color scale) 2D energy contours within the Wigner Seitz cell are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(e)-(h) over the Kx − Ky plane for different Kz values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Differ- ent Kz values are probed in the ARPES experiment by changing the energy of the incident photon beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Using the free electron final state model54, we have k⊥ = 1 ℏ � 2m (V0 + Ekin cos2 θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' (1) Here, V0 is the inner potential, Ekin is the kinetic en- ergy of a photoelectron and θ denotes the emission an- gled from the sample surface normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' For the different panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(e)-(h), we have hν (corresponding Kz) = 75 eV (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='03 c∗), 70 eV (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='16 c∗), 65 eV (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='29 c∗), and 60 eV (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='42 c∗), respectively where c∗ = 2π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We have applied V0 = 11 eV, to calculate the Kz values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The experimental FS demonstrates the transforma- tion of its symmetry from sixfold at Kz = 0 to three- fold for Kz > 0, which is consistent with the theoret- ical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' For Kz = 0 [Fig .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(e)], all three (the hexapetalus flower-shaped, hexagonal, and circular) states are observed and well matched to the calculated FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The hexapetalus flower-shaped states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(e) is transformed into the trefoil in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Due to the exper- imental geometry and the corresponding matrix-element effect, the measured FS shows an anisotropic distribu- tion in the photoemission intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The photoemission intensity is higher along one of the three ¯ M-¯Γ- ¯ M direc- tions and lower along the two other ¯ M-¯Γ- ¯ M directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This effect reduces the clarity of the three-fold symme- try in the FS, measured for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(g) and (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' However, the strong modulation of the FS on changing Kz is clear, and it is broadly consistent with the 3D FS distribution of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We now focus on the FS, in the vicinity of the DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The side and top view of the 3D FS distribution within the Wigner-Seitz cell at E = EDP for the relaxed structure is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3(a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The pres- ence of three contributing bands, each having twofold Kramer’s degeneracy, can be clearly seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The outer- most part of the FS arises from the electron pocket of the first unoccupied band of CoTe2, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The type-II nature of the DSM phase is also confirmed by the fact that the Dirac point appears at the four- fold degenerate touching point of the other electron and hole pockets in the middle, as marked by the red arrow in the FS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The energy contours over the Kx −Kz and the Ky −Kz planes, for E = EDP is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3(c) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Our calculations reveal a prominent Dirac crossing located at Kz ∼ ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='25 c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The anisotropic nature of the FS along the Ky direction persists even at the DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The in-plane projection of the energy contours at the DP is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3(e)-(i), for five different out-of-plane distances (or Kz values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' At Kz = 0 c∗, we observe a hexapetalus flower shape along with a small circle at its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The electron pockets at Kz = 0 transform into an isolated bean-shaped pattern with increasing Kz magnitude as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3(g)-(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' At Kz approaching the vicinity of bulk DP, the central con- tour converges to a tiny circle while the hexagonal outer contour acquires an almost triangular shape [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3(g) and (h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Finally, at Kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='3c∗, the FS cut appears as two contours centered around the origin, which exhibit a circular and triangular shape for the inner and outer contours, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In addition, there are small pock- ets along three of the six A−H lines [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3(h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The energy contour at negative Kz values with the same mag- nitude shows the rest of the three small pockets along the other A − H lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3(i) the inner contours van- ish and we only see three small pockets along to A − H direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' ORIGIN OF DIRAC STATES, BAND INVERSION, AND THE SURFACE STATES The presence of type-II Dirac fermions in the bulk dis- persion of CoTe2 suggests the strong possibility of finding topologically protected surface states near the Fermi en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Additionally, the bulk bands of CoTe2 also support several other topological band inversions in its bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(a), the orbital-resolved band structure along the Γ-A path shows that the linear crossings near EF are mainly composed of the Te-5p orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The Dirac band crossing near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='92 eV above Fermi energy arises from the interplay of the Te px + py and the Te pz orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Ad- ditionally, we find that these orbitals also contribute to multiple band inversion gaps along different high sym- metry paths including Γ − A [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' To under- stand the origin of the Dirac band crossing, we show the systematic evolution of the energy levels of the Te-5p or- bital manifold in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The six degenerate p orbital splits into lower (upper) lying three-fold bonding (anti- bonding) orbitals due to inter-site hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The presence of local trigonal distortion of the Co-Te octa- hedra further lifts the degeneracy of the bonding/anti- bonding p orbitals breaking it into singly degenerate a1g (pz) and doubly degenerate eπ g (px, py) orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Includ- ing the SOC splits the p orbitals into fourfold Jeff = 3/2 and two-fold Jeff = 1/2 pseudo spin basis as shown in the 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Side (a) and top (b) view of the 3D FS distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The planar projection of the constant energy surface at E − EF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='94 eV, on the (c) Kx − Kz, and the (d) Ky − Kz surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The 2D energy contours within the Wigner Seitz cell (marked by dotted line) in Kx−Ky plane at fixed Kz values of (e) Kz = 0, (f) Kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='15 c∗, (g) Kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='25 c∗, (h) Kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='30 c∗, and (i) Kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='42 c∗, where c∗ = 2π/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The 2D plane of (g), which hosts the Dirac point, is marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' fourth column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The last column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(e) highlights the effect of the dispersion along the Γ − A direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The bulk type-II Dirac point arises from the crossing of the bonding and anti-bonding states of the Jeff = 3/2 orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The ladder of multiple band inversions and the Dirac point in the bulk band structure points to the existence of topologically protected surface states in CoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This is confirmed by our experiments and theoretical calcula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The measured ARPES results and the correspond- ing theoretical spectral function of the relaxed structure are shown along the high symmetry ¯K-¯Γ- ¯K and ¯ M-¯Γ- ¯ M directions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(c), (d) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(g), (h), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The pattern of the spectral function and position of the surface Dirac cone matches well between the theoretical calculations and experimental plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' However, the other sharp spectral functions [purple, and yellow arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(c), (d) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(g)] arising from the bulk and sur- face states are slightly off in energy (see Appendix A for detailed discussions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This can be due to several reasons including i) small variations in the structural parameters, ii) some ambiguity in the pseudopotential for capturing core states, iii) some impurities or stacking faults in the crystal which are not included in theoretical calculations, amongst others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We also note that as the Bulk Dirac cone is significantly above the Fermi energy, it cannot be di- rectly observed or mapped via our occupied state ARPES data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The ARPES measurements were done with hν = 75 eV, which corresponds to Kz ∼ 0c∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Therefore these spectra capture the bulk bands along with the surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The prominent features corresponding to the sur- face states, in the measured ARPES spectrum and the calculated spectral function are marked by thick arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Despite some discrepancies in the binding energy of a few states, the experimental and the theoretical results show good qualitative agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The small energy difference in the location of the surface states possibly arises due to structural effects or from the surface potential which is not included in our theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The Dirac cone in the surface states is located at the ¯Γ point at an energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='49 eV below the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The presence of a topological band inversion near EF , as marked by an arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4 (a) gives rise to this surface Dirac crossing observed in ARPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' A similar surface Dirac cone, which has relatively broad features in ARPES experiments compared to theoretical calcula- tions, has also been observed in other isostructural com- pounds such as NiTe2 and in PtTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Other than the Dirac cone at the ¯Γ point, several sharp non-trivial sur- face states appear near the high symmetry ¯ M point and along the ¯Γ− ¯K path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' These arise from the multiple band inversions throughout the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We find the surface states to be symmetric along both the ¯K-¯Γ- ¯K and the ¯ M-¯Γ- ¯ M directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The Fermi arc states at constant energy are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4 (b) at E−EF = 0 eV and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4 (f) at E−EF = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='49 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' At the Fermi energy, circular arcs of the sharp (a) (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 (d)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 (2/c) [2/c) Kz 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 [FK] Kx (A-1) [FM) Ky (A-1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 Kx (A-1, Kx (A-1) Kx (A-1) Kx (A-16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' (a) Multiple band inversion arising from the Te-p orbital manifold along the Γ − A direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The band inversion gap near the Fermi energy (IBG) is marked with an arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The Fermi arc states at constant energy (b) E − EF = 0, and (f) at E − EF = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='49 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The theoretically calculated momentum resolved spectral density plot along the (c) ¯Γ- ¯K and the (g) ¯Γ- ¯ M paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The experimentally measured ARPES plots along the (d) ¯Γ- ¯K and (h) ¯Γ- ¯ M paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' (e) To highlight the origin of the bulk type-II Dirac fermions, we show the schematic of the level diagram of the Te-5p orbitals in presence of a local crystal field and SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' surface states appear around the ¯Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In contrast, a prominent peak is observed exactly at the ¯Γ point Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4 (f) which captures the dominant surface Dirac, crossing along with a few less intense circular arcs along the ¯Γ- ¯K paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' There is another set of high-intensity surface arc states around −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='4 eV below EF [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4 (c) and (g)], which disperse symmetrically around the ¯Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' SPIN POLARIZED SURFACE STATES The demonstration of topological surface states in CoTe2 inspires the exploration of their spin-polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' To study the spin polarization of the surface states and the surface Dirac cone, we experimentally measured the spin-polarized ARPES spectrum of CoTe2, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 5(b), (d) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The component of probed spin components is chosen to be perpendicular to the direction of the dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The corresponding theoretically calcu- lated spin resolved spectral function is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 5(a), (c), and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Figures 5(b) and (d) display the mea- sured spin-resolved band structures superimposed onto the measured spin-integrated band structures shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(d) and (h), along ¯K-¯Γ- ¯K and ¯ M-¯Γ- ¯ M, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' As was seen in the experimental plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(d) and (h), the surface Dirac cone spectral intensity is rel- atively low compared to the observed bulk states for the photon energy hν = 75 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Accordingly, its contribution to the measured spin-ARPES spectra is also small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' To improve the resolution of the spin information of the sur- face Dirac cone, the Spin-ARPES spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 5(d) were measured with hν = 19 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The crossing of the up-spin (red) and the down-spin (blue) bands are well observed around the energy of the surface Dirac point, matching well with the calculated spin texture shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This confirms the helical nature of the spin- momentum locking around the surface Dirac point and its topological origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The signs of measured and the cal- culated spin polarization is reversed for Kx/y → −Kx/y, in all panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This implies that the spin polarization is not due to the breaking of time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Interestingly, we also observe a significant contribution of the out-of-plane component in our spin ARPES experi- ments and calculations for the ¯K-¯Γ- ¯K direction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 5(e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The measurement is done with inci- dent photons with energy hν = 75 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The corresponding out-of-plane spin component for the ¯ M-¯Γ- ¯ M direction is negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The scale of the in-plane and the out- of-plane spin polarization in all the panels is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Note that due to the presence of time reversal and in- version symmetry in CoTe2, the spin polarization of the bulk states is forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Even an isolated monolayer of CoTe2 preserves the inversion and the time-reversal sym- metries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Thus, an isolated monolayer of CoTe2 will also (a) 3 (b) 2 0 2 E-Eε=0 eV (d) K (c) 0 (eV) 2 1 Te (px+py) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 (eV) Te (pz) E (A-1) 0 i 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 K 出 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 IBG 0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 A Kx (A-1) Kx (A-1) Kx (A-1) 0 (e) A (f) 2 p2 E-Ee=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='49 eV (g) M 0 [3/2,1/2) 1 (eV) pxy : (A-1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='3/2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='3/2) pxy E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 Te1-p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='13/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='3/2) [3/2,3/2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 [3/2,1/2 j3/2,3/2 1 pxyz EF 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 [1/2,1/2) 1 0 1 zd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 Kx (A-1) Ky (A-1) S7 Ky (A-1)7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Spin polarization of the surface bands along the high symmetry (a) ¯K-¯Γ- ¯K and (c) ¯ M-¯Γ- ¯ M directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The spin- ARPES measurements for (b) ¯K-¯Γ- ¯K and the (d) ¯ M-¯Γ- ¯ M directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The spin components are orthogonal to the corresponding momentum directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' (e) The theoretical and (f) experimentally measured out-of-plane spin polarization along the ¯K-¯Γ- ¯K direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' CoTe2 supports spin-polarized surface states over a wide range of energies in the entire BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' not support spin-polarized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' However, in a system of finite size, the inversion symmetry is broken for the atomic layers near the surface even for bulk centrosym- metric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This is what allows for spin polarization of the surface states (both in-plane and out-of-plane) in a finite slab of CoTe2, and other Dirac semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' An- other interesting point is that the surface states near the ¯Γ point primarily arise from the topological bulk band inversions, and these lead to Dirac surface states which have an in-plane Rashba-like spin momentum locking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This can be clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 5(e), where the out-of- plane spin states are completely absent near the ¯Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' CONCLUSIONS In summary, based on the ARPES experiments com- bined with detailed first principle calculations, we show that CoTe2 hosts a pair of type-II Dirac nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The Dirac node is located along the Γ − A axis around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='92 eV above the Fermi energy, and they support Lorentz symmetry violating Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We find that in addi- tion to the Dirac fermions, bulk CoTe2 also hosts several topological band inversions which give rise to a ladder of spin-polarized surface states over a wide range of ener- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The surface states corresponding to the bulk band inversions form a surface Dirac cone at the ¯Γ point, which has Rashba-type in-plane spin splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Interestingly, we find that some surface states in CoTe2 also support an out-of-plane spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Our study highlights that CoTe2 supports interesting bulk and surface Dirac fermi- ology, which should be explored further in transport, op- tical, plasmonic, and optoelectronic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' ACKNOWLEDGEMENT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' acknowledges the Indian Institute of Technol- ogy, Kanpur, and the Science and Engineering Re- search Board (SERB) National Postdoctoral Fellowship (PDF/2021/000346), India for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We thank Debasis Dutta and Barun Ghosh for the useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We acknowledge the Science and Engineer- ing Research Board (SERB) and the Department of Sci- ence and Technology (DST) of the Government of In- dia for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' We thank CC-IITK for pro- viding the High-Performance Computing facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This work has been partly performed in the framework of the nanoscience foundry and fine analysis (NFFA-MIUR Italy, Progetti Internazionali) facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Appendix A: Scaled ARPES The prominent bulk and surface states (except the Dirac crossing) of theoretically calculated spectral func- tion and experimentally measured ARPES plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(c), (d) and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 4(g)] have an energy differ- ence of ∼500 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' This can arise from various factors as discussed in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' For example, a similar discrep- ancy of energy is reported for a related compound PtSe2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' An energy scale factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='05 and an energy offset of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='1 eV is necessary for the PtSe2 compound to correctly match the energy between theoretical and ex- perimental ARPES results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Similarly in our calculation, an energy scaling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='7 can be used to best fit the ex- perimental plot (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) →K K←--T- →K M←I →M M+--T--→M K+--T---→K K+--T--→K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='25 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='25 in-plane 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='25 in-plane out-of-plane 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='0 Kx (1/A) Kx (1/A) Ky (1/A) Ky (1/A) Kx (1/A) Kx (1/A)8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Theoretically calculated (a) and experimental (b) momentum resolved spectral function plot ¯ M-¯Γ- ¯ M directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The theoretical spectral function incorporates an energy scal- ing factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='7 to best match the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Appendix B: Fermi surface anisotropy In this section, we have compared the theoretically cal- culated and experimentally observed Fermi surface maps on Kx-Kz and Ky-Kz planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 7, we have plot- ted the theoretically calculated 2D energy contours at E = EF on top of the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Here the anisotropy, as discussed in section IV is evident from the differences between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 7 (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 7 (b) plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The experimental Kx −Kz and Ky −Kz maps are taken with the photon energy range between 55 eV and 85 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The projected experimental Fermi surface at E = EF on the (a) Kx − Kz plane along where Kx is along the Γ-K direction, and on the (b) Ky − Kz plane with Ky being along the Γ-M direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' The black lines are the theoretically calculated 2D energy contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' ∗ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' † antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='politano@univaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='it ‡ ivana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='vobornik@elettra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='trieste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='it § amitag@iitk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='in 1 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Yin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Tang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Chhowalla, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Chen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Wee, Chemical Society Reviews 50, 10087 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Zhang, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Markovi´c, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Mazzola, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Feng, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Sunko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Riley, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Meevasana, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Fujii, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Vobornik, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Hoesch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Sasagawa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} 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Alessandro, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Kuo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Lue, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Caputi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Bansil, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Galdi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Agarwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Politano, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Cupolillo, ACS Applied Nano Mate- rials (2022), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='1021/acsanm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='2c04340.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Lue, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Pellegrini, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Agarwal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 121, 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Sankar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Zhigadlo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Qi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Qian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Chou, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Xu, Chem- istry of Materials 30, 4823 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' 38 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} +page_content=' Xu, L.' metadata={'source': 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+page_content=' King, Nature Materials 17, 21 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNFJT4oBgHgl3EQfdyz5/content/2301.11550v1.pdf'} diff --git a/udE1T4oBgHgl3EQfQwM3/content/tmp_files/2301.03043v1.pdf.txt b/udE1T4oBgHgl3EQfQwM3/content/tmp_files/2301.03043v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1375760ab45b45bbf66db2e4f5c02b42b2174ace --- /dev/null +++ b/udE1T4oBgHgl3EQfQwM3/content/tmp_files/2301.03043v1.pdf.txt @@ -0,0 +1,1132 @@ +XDQN: Inherently Interpretable DQN through Mimicking +Andreas Kontogiannis +National Technical University of Athens +Greece +andr.kontog@gmail.com +George Vouros +University of Piraeus +Greece +georgev@unipi.gr +ABSTRACT +Although deep reinforcement learning (DRL) methods have been +successfully applied in challenging tasks, their application in real- +world operational settings is challenged by methods’ limited ability +to provide explanations. Among the paradigms for explainability in +DRL is the interpretable box design paradigm, where interpretable +models substitute inner constituent models of the DRL method, thus +making the DRL method “inherently" interpretable. In this paper +we explore this paradigm and we propose XDQN, an explainable +variation of DQN, which uses an interpretable policy model trained +through mimicking. XDQN is challenged in a complex, real-world +operational multi-agent problem, where agents are independent learn- +ers solving congestion problems. Specifically, XDQN is evaluated in +three MARL scenarios, pertaining to the demand-capacity balancing +problem of air traffic management. XDQN achieves performance +similar to that of DQN, while its abilities to provide global models’ +interpretations and interpretations of local decisions are demon- +strated. +KEYWORDS +Deep Reinforcement Learning, Mimic Learning, Explainability +ACM Reference Format: +Andreas Kontogiannis and George Vouros. 2023. XDQN: Inherently Inter- +pretable DQN through Mimicking. +1 +INTRODUCTION +Deep Reinforcement Learning (DRL) has mastered decision making +policies in various difficult control tasks [11] [18] [15], games [22] +[13] and other real-time applications [14] [37]. Despite the remark- +able performance of DRL models, the knowledge of mastering these +tasks remains implicit in deep neural networks. Thus, its application +in real-world operational settings is challenged by methods’ limited +ability to provide explanations at global (policy) and local (individ- +ual decisions) levels. This lack of interpretability makes it difficult +to trust DRL for solving safety-critical real-world tasks. However, +besides the inability of DRL models to provide interpretations on the +selection of actions in specific circumstances, they are also unable +to provide information about the evolution of models during the +training process. These challenges are naturally further extended to +multi-agent settings, in which different agents empowered by multi- +agent reinforcement learning (MARL) methods aim at learning a +joint optimal policy towards solving a target task. +To address some of the aforementioned challenges, one may +follow different paradigms for the provision of explanations: The +interpretable box design paradigm is one of them where interpretable +models substitute inner components of DRL [35]. Additionally, +Under submission. +mimic learning has been proposed, so as to infer interpretable mod- +els that mimic the behavior of well-trained deep neural networks +[2, 5]. In the DRL case, mimic learning aims to replace the closed- +box DRL controller with an interpretable one, able to mimic the +decisions made by the former [3, 19, 35]. A mimic learner tries to +optimize fidelity [35], which is determined by comparing the mimic +controller’s actions with the actions selected by the DRL model. To +extract knowledge from deep neural networks, recent work [3, 19] +has applied mimic learning with tree representations, using decision +trees: Criteria used for splitting tree nodes provide a tractable way +to explain the predictions made by the controller. +Typically, mimic learning approaches require already well-trained +complex policy networks (which we refer to as mature networks), +whose behavior are mimicking to support interpretability. In real- +world scenarios, this could be quite impractical, since the training +overhead required to train the mimic models can often be a very +time-consuming and costly process, especially for large state-action +spaces and for multi-agent settings. Another limitation of such ap- +proaches is that they solely aim at providing explainability on the +predictions of only the mature DRL model, ignoring completely the +training process of this model. In other words, in these approaches, +the mimic learner can only provide explanations about the policy of +the inferred DRL controller, but not about the patterns and behaviors +learned throughout the training process. +To deal with these challenges, in this paper we propose eXplainable +Deep Q-Network (XDQN), which is an explainable variation of the +well-known DQN [22] method. In XDQN, our goal is to provide +inherent explainability of DQN via mimic learning in an online man- +ner, by replacing the complex deep Q-network with an interpretable +mimic learner in testing. In so doing, XDQN does not require the +existence of a well-trained model to train an interpretable one. In +particular, we train a mimic learner in parallel with the deep neural +network (Q-network) of DQN in an online setting, where: at a train- +ing step the DRL model uses the mimic learner to compute the target +values of the Q-network needed for its training, while the mimic +learner learns to behave as the DRL model, but in an explainable +way. Since the mimic learner is trained and updated while the DQN +policy model is trained, we can keep multiple “snapshots” of the +model evolution through time, offering interpretability on these in- +termediate models, and insights about the patterns and behaviors +that DQN learns during training. +To evaluate our method’s utility in real-world operational settings, +XDQN is challenged in a complex, real-world multi-agent problem, +where agents solve airspace congestion problems. Agents in this +setting are trained via parameter sharing following the centralized +training, decentralized execution paradigm. We summarize the main +contributions of this paper below: +arXiv:2301.03043v1 [cs.LG] 8 Jan 2023 + +• To our knowledge, this work is the first that provides DQN +with inherent interpretability through mimic learning without +requiring the existence of a well-trained DRL model. +• We propose XDQN, an explainable variation of DQN, in +which an interpretable mimic learner is trained in parallel +with the Q-network of DQN and plays the role of the target +Q-network of DQN. +• Experimentally, we show that XDQN can perform similarly +to DQN, demonstrating good play performance and fidelity +to DQN decisions in complex, real-world operational multi- +agent problems. +• We demonstrate the ability of XDQN to provide global (pol- +icy) and local (in specific circumstances) explanations regard- +ing agents’ decisions, also while models are being trained. +2 +BACKGROUND +2.1 +Markov Decision Process +We consider a sequential decision making setup, in which an agent +interacts with an environment 𝐸 over discrete time steps. At a given +timestep, the agent perceives features regarding a state 𝑠𝑡 ∈ 𝑆, where +𝑆 is the state space. The agent then chooses an action 𝑎𝑡 from a +discrete set 𝐴 and observes a reward 𝑟𝑡 generated by the environment. +The agent’s behavior is determined by a policy 𝜋, which maps +states to a probability distribution over the actions, that is 𝜋 : 𝑆 → +𝑃(𝐴). Apart from an agent’s policy, the environment 𝐸 may also be +stochastic. We model it as a Markov Decision Process (MDP) with +a state space 𝑆, action space 𝐴, an initial state distribution 𝑝(𝑠1), +transition dynamics 𝑝(𝑠𝑡+1|𝑠𝑡) and a reward function 𝑟 (𝑠𝑡,𝑎𝑡,𝑠𝑡+1). +For brevity, we write 𝑟𝑡 = 𝑟 (𝑠𝑡,𝑎𝑡,𝑠𝑡+1). +The agent aims to maximize the expected discounted cumulative +reward, which is formulated as 𝐺𝑡 = �∞ +𝜏=𝑡 𝛾𝜏−𝑡𝑟𝜏. Here, 𝛾 ∈ (0, 1) +is a discount factor which trades-off the importance of immediate +and future rewards. Considering that an agent acts under a stochastic +policy 𝜋, the Q-function (state-action value) of a pair (𝑠,𝑎) is defined +as follows +𝑄𝜋 (𝑠,𝑎) = E [𝐺𝑡 | 𝑠𝑡 = 𝑠,𝑎𝑡 = 𝑎, 𝜋] +(1) +which can also be computed recursively with bootstrapping: +𝑄𝜋 (𝑠,𝑎) = E +� +𝑟𝑡 + 𝛾E𝑎∼𝜋 (𝑠𝑡+1) [𝑄𝜋 (𝑠𝑡+1,𝑎)] | 𝑠𝑡 = 𝑠,𝑎𝑡 = 𝑎, 𝜋 +� +(2) +The Q-function measures the value of choosing a particular action +when the agent is in this state. We define the optimal policy 𝜋∗ under +which the agent receives the optimal 𝑄∗(𝑠,𝑎) = 𝑚𝑎𝑥𝜋𝑄𝜋 (𝑠,𝑎). For +a given state 𝑠, under the optimal policy 𝜋∗, the agent selects action +𝑎 = 𝑎𝑟𝑔𝑚𝑎𝑥𝑎′∈𝐴𝑄∗(𝑠,𝑎′). Therefore, it follows that the optimal +Q-function satisfies the Bellman equation: +𝑄∗(𝑠,𝑎) = E +� +𝑟𝑡 + 𝛾𝑚𝑎𝑥𝑎𝑄∗(𝑠𝑡+1,𝑎) | 𝑠𝑡 = 𝑠,𝑎𝑡 = 𝑎, 𝜋 +� +. +(3) +2.2 +Deep Q-Networks +To deal with a high dimensional state space, the state-action value +function can be approximated by an online deep Q-network (DQN +[22]); i.e. a deep neural network 𝑄(𝑠,𝑎;𝜃) with weight parameters 𝜃. +To estimate the parameters 𝜃, at iteration 𝑖 the expected mean squared +loss between the estimated Q-value of a state-action pair and its +temporal difference target, produced by a fixed and separate target +Q-network 𝑄(𝑠,𝑎;𝜃−) with weight parameters 𝜃−, is minimized. +Formally: +𝐿𝑖 (𝜃𝑖) = E +� +𝑌 𝐷𝑄𝑁 +𝑖 +− 𝑄(𝑠,𝑎;𝜃) +� +, +(4) +with +𝑌 𝐷𝑄𝑁 +𝑖 += 𝑟𝑡 + 𝛾 max +𝑎∈𝐴 𝑄(𝑠𝑡+1,𝑎;𝜃−) +(5) +In order to train DQN and estimate 𝜃, we could use the standard +Q-learning update algorithm. Nevertheless, the Q-learning estimator +performs very poorly in practice. To stabilize the training procedure +of DQN, Mnih et. al [22] freezed the parameters, 𝜃−, of the target +Q-network for a fixed number of training iterations while updating +the online Q-network with gradient descent steps with respect to 𝜃. +In addition to the target network, during the learning process, +DQN uses an experience replay buffer [22], which is an accumulative +dataset, 𝐷𝑡, of state transitions - in the form of (𝑠, 𝑎, 𝑟, 𝑠′) - from +past episodes. In a training step, instead of only using the current +state transition, the Q-Network is trained by sampling mini-batches +of past transitions from 𝐷 uniformly, at random. Therefore, the loss +can be written as follows: +𝐿𝑖 (𝜃𝑖) = E(𝑠,𝑎,𝑟,𝑠′)∼U(𝐷) +� +(𝑌 𝐷𝑄𝑁 +𝑖 +− 𝑄(𝑠,𝑎;𝜃))2� +. +(6) +The main advantage of using an experience replay buffer is that +uniform sampling reduces the correlation among the experience +samples used for training the Q-network. The replay buffer also +improves data efficiency through reusing the experience samples in +multiple training steps. +Instead of sampling mini-batches of past transitions uniformly +from the experience replay buffer, a further improvement over DQN +results from using a prioritized experience replay buffer [30]. It aims +at increasing the probability of sampling those past transitions from +the experience replay that are expected to be more useful in terms of +absolute temporal difference error. +2.3 +Mimic Learning for Deep Reinforcement +Learning +Recent work on mimic learning [7, 19] has shown that rule-based +models, like decision trees, or shallow feed-forward neural networks +can mimic a not linear function inferred by a deep neural network +with millions of parameters. We present two known settings for +mimicking the Q-function of a DRL model. +2.3.1 +Experience Training. In the experience training setting +[7, 19], all the state-action pairs ⟨𝑠,𝑎⟩ of a DRL training process +are collected in a time horizon 𝑇. Then, to obtain the corresponding +Q-values, these pairs are provided as input into a DRL model. The +final set {(⟨𝑠1,𝑎1⟩,𝑄1), ...(⟨𝑠𝑇,𝑎𝑇 ⟩,𝑄𝑇 )} of tuples is used as the +experience training dataset. +2.3.2 +Active Play. The main problem with the experience training +is that suboptimal state-action pairs are collected through training, +making it more difficult for a learner to mimic the behavior of the +DRL model. To address this challenge, active play [19] uses a mature +DRL model to generate state-action pairs to construct the training +dataset of an active mimic learner. The training data is collected in + +an online manner through queries, in which the active learner selects +the actions, given the states, and the mature DRL model provides +the estimated Q-values. These Q-values are then used to update the +active learner’s parameters on minibatches of the collected dataset. +3 +EXPLAINABLE DEEP Q-NETWORK +(XDQN) +In this work, we are interested in providing interpretability in deep +Q-learning through mimicking the behavior of DQN. To this aim, +we propose eXplainable Deep Q-Network (XDQN) 1, which is an +explainable variation of DQN [22]. XDQN aims at inferring the +parameters of the online Q-network and the parameters of a mimic +learner concurrently, in an online manner, with the latter substituting +the target Q-network of DQN. +Formally, let 𝜃 be the parameters of the online Q-network and 𝜙𝑋 +be the parameters of the mimic learner. In XDQN, the mimic learner +is used to estimate the state-action value function and select the best +action for the next state in the XDQN target: +𝑌𝑋𝐷𝑄𝑁 +𝑖 += 𝑟𝑡 + 𝛾 max +𝑎∈𝐴 𝑄 (𝑠𝑡+1,𝑎;𝜙𝑋 ) +(7) +Similar to DQN, 𝜙𝑋 are updated every 𝑇𝑢 number of timesteps. The +full training procedure of XDQN is presented in Algorithm 1. +In contrast to DQN in which we simply copy the parameters +𝜃 of the online Q-network to update the parameters of the target +Q-network, here we perform mimic learning on 𝑄(𝑠,𝑎,𝜃) (steps +17-20). To update 𝜙𝑋 we train the mimic learner on minibatches +of the experience replay buffer 𝐵 by minimizing the Mean Squared +Error (MSE) loss function using 𝑄(𝑠,𝑎,𝜃) to estimate the soft labels +(Q-values) of the state-action pairs in the minibatches. Formally the +optimization problem for each update of 𝜙𝑋 can be written as: +min +𝜙𝑋 +E(𝑠,𝑎)∼𝐵 +� +(𝑄(𝑠,𝑎;𝜙𝑋 ) − 𝑄(𝑠,𝑎;𝜃))2� +(8) +In our experiments, we utilize a prioritized experience replay [30] +as the replay buffer 𝐵, as described in Section 2. Similarly to active +play, when updating 𝜙𝑋 , to ensure that the state-action pairs of the +minibatches provide up-to-date target values with respect to 𝜃, we +use records from the replay buffer that were stored during the 𝐾 +latest training steps. +It is worth noting that at each update of 𝜙𝑋 the hyperparameter +𝐾 for past transitions plays a similar role as the discounted factor 𝛾 +plays for future rewards, but from the mimic learner’s perspective. +Building upon the experience training and active play paradigms, +XDQN can leverage the benefits of both of them. In particular, +the hyperparameter 𝐾 manages the trade-off between experience +training and active play in XDQN. If 𝐾 is large, the mimic model +learns from state-action pairs that may have been collected through +more suboptimal instances of 𝜃; deploying however data-augmented +versions of Q-value. On the other hand, if 𝐾 is small, it learns from +the most recent instances of 𝜃; making use of up-to-date Q-values. +Nevertheless, opting for very small values of 𝐾 could lead to less +stable mimic training, due to the smaller number of minibatches that +can be produced for updating 𝜙𝑋 , while using large 𝐾 can result in a +very slow training process. +1The implementation code will be made available in the final version of the manuscript. +From all the above, we note that 𝜃 (Q-network) and 𝜙𝑋 (mimic +learner) are highly dependent. To update 𝜃, Q-network uses the +mimic learner model with 𝜙𝑋 to compute the target soft labels +(target Q-values), while to update 𝜙𝑋 the mimic learner uses the +original Q-network with parameters 𝜃 to compute the respective +target soft labels (online Q-values). Since XDQN produces different +instances of 𝜙𝑋 throughout training, it can eventually output multiple +interpretable mimic learner models (up to the number of 𝜙𝑋 updates), +with each one of them corresponding to a different training timestep. +Assuming that all these mimic learner instances are interpretable +models, XDQN can also provide explainability on how a DRL model +learns to solve the target task. +Finally, after Q-network (𝜃) and mimic learner (𝜙𝑋 ) have been +trained, without requiring to learn 𝜃 before 𝜙𝑋 , we can discard the +online Q-network and use the mimic learner model as the controller. +Therefore, in testing, given a state, the interpretable mimic learner +selects the action that profits the highest Q-value, being also able to +provide explainability. +4 +EXPERIMENTAL SETUP +In this section, we demonstrate the effectiveness of XDQN through +experiments on real-world data. In all experiments we utilize a Gra- +dient Boosting Regressor [36] as the mimic learner, so as to exploit +its boosting ability to learn effectively by exploiting instances gener- +ated by the deep Q-network. Although most decision tree algorithms, +being rule-based models, are naturally interpretable models [3, 19], +this is not the case for a Gradient Boosting Regressor, since the +boosting structure makes it very difficult to provide explainability. +However, following the work in [38], we are able to enrich the Gra- +dient Boosting Regressor mimic learner with the ability to provide +explainability as follows: Given a state-action pair as an input of the +mimic learner, we can measure the contribution of each state feature +to the predicted Q-value. Therefore, our mimic learner is expected +not only to mimic effectively the behavior of the DRL controller, but +also, to give local and global explanations on its decisions. +Overall, we are interested in comparing the performance of XDQN +with that of DQN in real-world environments where the latter has +been state-of-the-art, and also designing appropriate experimental +setups, aiming at studying XDQN interpretability. In so doing, we +evaluate XDQN on real-world operational multi-agent experimen- +tal scenarios, pertaining to the demand-capacity balancing (DCB) +problem of air traffic management (ATM), which we describe next. +4.1 +Real-world demand-capacity problem setting +The current ATM system is based on time-based operations resulting +in DCB [17] problems. To solve the DCB issues at the pre-tactical +stage of operations, the ATM system opts for methods that generate +delays and costs for the entire system. In ATM, the airspace consists +of a set of 3D sectors where each one these is characterized by a +specific capacity. This is the number of flights that cross the sector +during a specific period (e.g. of 20 minutes). The main challenge of +dealing with the DCB problem in ATM is to reduce the number of +cases where the demand of airspace use exceeds its capacity. These +cases are called hotspots. +Recent work has transformed the DCB challenge to a multi-agent +RL problem by formulating the setting as a multi-agent MDP [17]. + +Algorithm 1 eXplainable Deep Q-Network (XDQN) +1: Initialize replay buffer 𝐵 with capacity N +2: Initialize 𝜃 and 𝜙𝑋 +3: Initialize timestep count 𝑐 = 0 +4: for episode 1, M do +5: +Augment 𝑐 = 𝑐 + 1 +6: +Initialize state 𝑠1 +7: +With probability 𝜖 select a random action 𝑎𝑡, otherwise 𝑎𝑡 = 𝑎𝑟𝑔𝑚𝑎𝑥𝑎𝑄(𝑠𝑡,𝑎;𝜃) +8: +Execute action 𝑎𝑡 and observe next state 𝑠𝑡+1 and reward 𝑟𝑡 +9: +Store transition (𝑠𝑡,𝑎𝑡,𝑠𝑡+1,𝑟𝑡) in 𝐵 +10: +Sample a minibatch of transitions (𝑠𝑖,𝑎𝑖,𝑠𝑖+1,𝑟𝑖) from 𝐵 +11: +if 𝑠𝑖+1 not terminal then +12: +Set 𝑌𝑋𝐷𝑄𝑁 +𝑖 += 𝑟𝑖 + 𝛾 max𝑎∈𝐴 𝑄 (𝑠𝑖+1,𝑎;𝜙𝑋 ) +13: +else +14: +Set 𝑌𝑋𝐷𝑄𝑁 +𝑖 += 𝑟𝑖 +15: +end if +16: +Perform a gradient descent step on +� +𝑌𝑋𝐷𝑄𝑁 +𝑖 +− 𝑄(𝑠𝑖,𝑎𝑖;𝜃) +�2 +w.r.t. 𝜃 +17: +if 𝑐 mod 𝑇𝑢 = 0 then +18: +Initialize 𝜙𝑋 +19: +Sample a minibatch of transitions (𝑠𝑖,𝑎𝑖,𝑠𝑖+1,𝑟𝑖) from 𝐵 that were stored at most 𝑐 − 𝐾 timesteps before +20: +Perform mimic learning update on (𝑄(𝑠,𝑎;𝜙𝑋 ) − 𝑄(𝑠,𝑎;𝜃))2 w.r.t 𝜙𝑋 +21: +end if +22: end for +We follow the work and the experimental setup of [15–17, 31, 32] +and encourage the reader to see the problem formulation [17] in +details. In this setting, we consider a society of agents, where each +agent is a flight (related to a specific aircraft) that needs to coordi- +nate its decisions, so as to resolve hotspots that occur, jointly with +other society agents. Agents’ local states comprise 81 state variables +related to: (a) the delay (in the range of 0, ..., maxDelay) set by the +referring agent, (b) the number of hotspots in which the agent is +involved in, (c) the sectors that it crosses, (d) the minutes that the +agent is within each sector it crosses, (e) the periods in which the +agent joins in hotspots in sectors, and (f) the minute of the day that +the agent takes off. The tuple containing all agents’ local states is the +joint global state. Q-learning [33] agents has been shown to achieve +remarkable performance on this task [15]. In our experiments, all +agents share parameters and replay buffer and act independently. +A DCB scenario comprises multiple flights crossing various +airspace sectors in a time horizon of 24h. This time horizon is segre- +gated into simulation time steps. At each simulation time step (equal +to 10 minutes of real time), given only the local state, each agent se- +lects an action which is related to its preference to add ground delay +regulating its flight, in order to resolve hotspots in which it partici- +pates. The set of local actions for each agent contains |maxDelay+1| +actions, at each simulation time step. We use maxDelay = 10. The +joint (global) action is a tuple of local actions selected by the agents. +Similarly, we consider local rewards and joint (global) rewards. The +local reward is related to the cost per minute within a hotspot, the +total duration of the flight (agent) in hotspots as well as to the delay +that a flight has accumulated up to the simulation timestep [15]. +4.2 +Evaluation Metrics and Methods +For the evaluation of the proposed method, first, we make use of +two known evaluation metrics: (a) play performance [19] of the +online deep Q-network, and (b) fidelity [35] of the mimic learner. +Play performance measures how well the deep Q-network performs +with the mimic learner estimating its temporal difference targets, +while fidelity measures how well the mimic learner matches the +predictions of the online deep Q-network. +As far as play performance is concerned, we aim at minimizing +the number of hotspots, the average delay per flight and the number +of delayed flights. As for fidelity, we use two metric scores: (a) the +mean absolute error (MAE) and (b) the accuracy score. Given a +minibatch of states, we calculate the MAE of this minibatch for +any action as the mean absolute difference between the Q-values +estimated by the mimic learner and the Q-values estimated by the +deep Q-network for that action. More formally, for a minibatch of +states 𝐷𝑠, the MAE𝑖 of action 𝑎𝑖 is denoted as: +𝑀𝐴𝐸𝑖 = +1 +|𝐷𝑠 | +∑︁ +𝑠 ∈𝐷𝑠 +|𝑄(𝑠,𝑎𝑖;𝜙𝑋 ) − 𝑄(𝑠,𝑎𝑖;𝜃)| +(9) +It is worth noting that minimizing the MAE of the mimic learner is +very important for training XDQN. Since deep Q-network updates +its parameters 𝜃 by using the mimic model to provide the target +Q-values, large MAEs can lead deep Q-network to overestimate bad +states and understimate the good ones, and thus, find very diverging +policies that completely fail to solve the task. +To calculate the accuracy score, again given a minibatch of states, +for each state we compare the action selected by the mimic model +and the online Q-network. Accuracy measures the percentage of the + +Scenario +DQN +XDQN +Final Hotspots +Average Delay +Delayed Flights +Final Hotspots +Average Delay +Delayed Flights +20190705 +38.4 +13.04 +1556.5 +39.0 +13.19 +1618.54 +20190708 +4.6 +11.4 +1387.2 +6.0 +11.73 +1331.58 +20190714 +4.8 +10.72 +1645.2 +7.0 +13.46 +1849.49 +Table 1: Comparison of testing performance of DQN and XDQN on the three experimental ATM scenarios +predictions of the two estimators that agree with each other, consid- +ering that both models select the action with the highest estimated +Q-value. +Second, we design appropriate experiments and illustrate XDQN’s +local and global interpretability. We focus on providing aggregated +interpretations, focusing on the contribution of features to local deci- +sions and to the overall policy: This, as suggested by ATM operators, +is beneficial towards understanding decisions, helping them to in- +crease their confidence to the solutions proposed, and mastering +the inherent complexity in such a multi-agent setting, as solutions +may be due to complex phenomena that are hard to be traced [15]. +Specifically, in this work, local explainability measures state fea- +tures’ importance on a specific instance (i.e. a single state-action +pair), demonstrating which features contribute to the selection of +a particular action over the other available ones. Global explain- +ability aggregates feature importance on particular action selections +over many different instances and aims to explain the overall policy +of mimic learner. Third, we demonstrate global explainability of +the DRL model through the whole training process, addressing the +question of how a DRL model learns to solve the target task. +4.3 +Experimental Scenarios and Settings +Experiments were conducted on three in total scenarios. Each of +these scenarios corresponds to a date in 2019 with heavy traffic in the +Spanish airspace. In particular, the date scenarios, on which we as- +sess our models, are 20190705, 20190708 and 20190714. However, +to bootstrap the training process we utilize a deep Q-network pre- +trained in various scenarios, also including 20190705 and 20190708. +In the training process, the deep Q-network is further trained ac- +cording to the method we propose. The experimental scenarios were +selected based on the number of hotspots and the average delay +generated in the ATM system within the duration of the day, which +shows the difficulty of the scenario. We note that for each scenario +we ran five separate experiments and average results. +Table 2 presents information on the three experimental scenarios. +In particular, the flights column indicates the total number of flights +(represented by agents) during the specific day. The initial hotspots +column indicates the number of hotspots appearing in the initial +state of the scenario. The flights in hotspots column indicates the +number of flights in at least one of the initial hotspots. Note that +all three scenarios display populations of agents of similar size, +with 20190708 having the smaller population and the least initial +hotspots. +Scenario +Flights +Initial Hotspots +Flights in Hotspots +20190705 +6676 +100 +2074 +20190708 +6581 +79 +1567 +20190714 +6773 +92 +2004 +Table 2: The three experimental Air Traffic Management +(ATM) scenarios +Figure 1: Episodic reward in the three evaluated ATM scenarios +4.4 +Implementation Details +In our implementation setting we utilize a deep multilayer perceptron +as the Q-network. In particular, we use an 𝜖-greedy policy, which at +the start of exploration has 𝜖 equal to 0.9 decaying by 0.01 every 15 +episodes until reaching the minimum of 0.04. The total number of +episodes are set to 1600 and the update target frequency is set to 9 +episodes. In the exploitation mode, we set 𝜖 equal to 0.04. We set +the maximum depth of the Gradient Boosting Regressor equal to +45 and the number of minimum samples for a split equal to 20. We +also use the mean squared error as the splitting criterion. To train a +single decision tree for all different actions, we create a non binary +splitting rule of the root based on the action size of the task, so that +the state-action pairs sharing the same action match the same subtree +of the splitting root. Empirically, we set the memory capacity of +the experience replay for the mimic learner, i.e. the hyperparameter +𝐾, equal to the 1/20 of the product of three other hyperparameters, +namely the total number of timesteps per episode (set to 1440), the +update target frequency (set to 9) and the number of agents (set to +7000). Thus, 𝐾 is set to 4536000 steps. +4.5 +Evaluation of play performance +Table 1 demonstrates the performance of DQN and XDQN on the +three experimental scenarios. The final hotspots column indicates +the number of unresolved hotspots in the final state: It must be noted + +20190708 +20190714 +0.8 +20190705 +0.7 +Reward +0.6 +0.5 +0.4 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +EpisodesAction (Delay Option) +XDQN mimic models +X0705 +X0708 +X0714 +0 +0.279 +0.237 +0.291 +1 +1.766 +1.971 +1.942 +2 +0.910 +0.928 +1.002 +3 +0.575 +0.661 +0.640 +4 +0.639 +0.748 +0.725 +5 +1.893 +2.096 +2.121 +6 +1.590 +1.766 +1.715 +7 +1.610 +1.816 +1.733 +8 +0.449 +0.514 +0.497 +9 +0.740 +0.849 +0.823 +10 +1.292 +1.525 +1.461 +Table 3: Evaluation of the average Mean Absolute Errors (MAE) of the trained mimic models over all mimic updates +that these hotspots may have emerged due to delays assigned to +flights and may be different than the hotspots at the beginning of +each scenario. The average delay per flight column shows the total +minutes of delay imposed, divided by the number of flights in the +specific scenario. The delayed flights column indicates the number +of flights affected by more than four minutes of delay, as it is done +by operators. +We observe that XDQN performs similar to DQN in all three +evaluated metric scores. In particular, DQN slightly outperforms +XDQN in terms of the final hotspots and average delay in all three +scenarios. Nonetheless, XDQN achieves to decrease the number of +the delayed flights in one scenario, while it demonstrates competitive +performance on the others. Figure 1 shows the episodic reward of +XDQN over time: XDQN manages to reach convergent behavior in +all scenarios by retaining high episodic rewards. +4.6 +Evaluation of fidelity +As discussed in Subsection 4.2, for the fidelity evaluation, we mea- +sure the mean absolute error (MAE) and the accuracy score. Given +the DCB experimental scenarios, we train three different mimic mod- +els; namely X0705, X0708 and X0714. Table 3 reports the average +MAE for each decided action over all mimic learning updates. We +observe that all errors range in relatively small quantities, given +that in testing, the absolute Q-values hovered around 200. As we +highlighted above, this is very important for stabilizing the training +process of XDQN, since we need very accurate mimic Q-value pre- +dictions, ideally equal to the ones generated by the deep Q-network. +To further assess the fidelity of XDQN mimic learner, Table 4 il- +lustrates the average accuracy scores over all mimic learning updates. +Since a Gradient Boosting Regressor mimic learner is a boosting +algorithm, it produces sequential decision trees that can successfully +seperate the state space and approximate well the predictions of the +deep Q-network function. We observe that the mimic learner and +the deep Q-network agree with each other to a very good extent; +namely from approximately 81% to 91%. Therefore, we expect the +mimic learner to be able to accumulate the knowledge from the deep +Q-network with high fidelity. +Scenario +Accuracy (%) +20190705 +88.45 +20190708 +81.89 +20190714 +90.88 +Table 4: The accuracy scores of mimic models +4.7 +Local and Global Explainability +In the DCB setting, it is important for the operator to understand how +the system reaches decisions on regulations (i.e. assignment of delays +to flights): This, as already pointed out, should be done at a level of +abstraction that would allow them to increase their confidence to the +solutions proposed, mastering the inherent complexity of the setting. +Therefore, we are mainly interested in receiving explanations about +which state features contribute to the selection of delay actions over +the no-delay action (i.e. action equal to 0). +First, we demonstrate the ability of the mimic learner to provide +local explainability. As already said, local explainability involves +showing which state features contribute to the selection of a par- +ticular action over the other available ones in a specific state. To +this aim, we work on pairs of actions - let 𝑎1 and 𝑎2 - and calculate +the differences of feature contributions in selecting 𝑎1 and 𝑎2 in a +single state. To highlight only the most significant differences, we +focus only on those features whose differences are above a threshold. +Empirically, we set this threshold equal to 0.5. Figure 2 illustrates +local explainability on a given state in which action "2" was selected. +Figure 2 provides the differences of feature contributions to the +estimation of Q-values when selecting action "0" against selecting +action "2" (denoted by “0-2"). We observe that the features that +contributed more to the selection of the delay action "2" were those +with index 32 (i.e. The sector in which the last hotspot occurs), 2 (i.e. +the sector in which the first hotpot occurs) and 62 (i.e. the minutes +that the flight spends crossing the last sector). +Finally, we demonstrate XDQN global explainability by aggre- +gating the importance of features on particular action selections over +many different state-action instances. In particular, we are interested +in measuring the state feature contributions to the selection of delay +actions (i.e. actions in the range [1, 10]) over the no-delay action (i.e. + +Feature Index +Feature Meaning +ACD +0 +Delay the corresponding flight has accumulated up to this point +Positive +1 +Total number of hotspots the corresponding flight participates in +Positive +3 +The sector in which the second hotspot the corresponding flight participates occurs +Positive +63 +The minute of day the flight takes off given the delay (CTOT) +Negative +64 +The minutes the flight remains in the first sector it crosses +Negative +68 +The minutes the flight remains in the fifth sector it crosses +Negative +Table 5: Demonstration of the most significant state features in terms of average contribution difference (ACD) in selecting the +no-delay action versus a delay action. A positive ACD means that the corresponding state feature on average contributes more to +the selection of the no-delay action “0". On the contrary, a negative ACD means that the corresponding state feature on average +contributes to the selection of a delay action “1 - 10". +Figure 2: Illustration of significant differences of feature con- +tributions to Q-value in selecting action "0" and action "2" in +a single state, in which action "2" was selected. Positive differ- +ences mean that the respective state features have a greater con- +tribution to Q-value when action "0" is selected, rather than +when action "2" is selected. Negative differences have the oppo- +site meaning. +action "0") in the overall policy. To this aim, we work on all possible +pairs of actions, with one action always being the no-delay action +and the other one being a delay action, and average the differences +of feature contributions to estimating the Q-value in selecting those +actions over many different state-action instances with the same se- +lected delay action. Table 5 shows the most significant state features +in terms of average contribution difference (ACD) in selecting the +no-delay action versus a delay action. To select those features, we +initially filter the most significant ones, namely the features whose +absolute ACD is greater than a threshold, for each action in the +range [1, 10]) over the no-delay action (i.e. action "0"), and present +the three most common features with positive and negative ACD. +We observe that features with index 0, 1 and 3 contribute more to +the selection of the no-delay action. On the contrary, features with +indexes 64, 63 and 68 contribute more to the selection of a no-delay +action. +Last but not least, we demonstrate how global explainability +evolves through the training process, addressing the question of +how a DRL model learns to solve the target task. To this aim, we +measure the absolute average feature contribution (AAFC) to Q- +value at different training episodes for the selection of each action. +Figure 3 illustrates the evolution of global explainability for selecting +Figure 3: Illustration of the evolution of features’ contribu- +tions for selecting the no-delay action ("0") and a delay one +("2") through 5 representative training episodes (360th, 720th, +1100th, 1400th and 1600th) in terms of absolute average feature +contribution (AAFC) to Q-value for the eight features with high- +est AAFC values in the final model (episode 1600) in the selec- +tion of the aforementioned actions. +the no-delay action and a delay action through 5 representative +training episodes (360th, 720th, 1100th, 1400th and 1600th), in +terms of AAFC to Q-value for the eight features with highest AAFC +values in the final model (episode 1600) in the selection of the + +15 +10 +5 +0 +-5 +-10 +32 +2 +62 +63 +64 +67 +70 +66 +69 +68 +1 +33 +0 +State Feature30 +Episode 360 +Average Contribution to Q-value for Action 0 +25 +Episode 720 +Episode 1100 +20 +Episode 1400 +Episode 1600 +15 +10 +5 +0 +-10 +-15 +0 +1 +2 +62 +3 +32 +64 +33 +State Feature30 +Episode 360 +Average Contribution to Q-value for Action 2 +2 +25 +Episode 720 +Episode 1100 +20 +Episode 1400 +Episode 1600 +15 +10 +5 +0 +-5 +-10 +-15 +0 +32 +1 +33 +62 +2 +63 +66 +State Featureaforementioned actions. We observe that for both evaluated actions +most of the features show an increasing/decreasing trend in their +average contribution to Q-value over time, such as those with indices +0, 1 and 63. It is worth noting that although the features with indexes +0 and 1 have been highlighted as the most significant for the selection +of the no-delay action, they have also significant but less contribution +to a delay action as well. +5 +RELATED WORK +Explainability in Deep Reinforcement Learning (DRL) is an emer- +gent area whose necessity is related to the fact that DRL agents solve +sequential tasks, acting in the real-world, in operational settings +where safety, criticality of decisions and the necessity for trans- +parency (i.e. explainability with respect to real-world pragmatic con- +straints [35]) is the norm. However, DRL methods use closed-boxes +whose functionality is intertwined and are not interpretable: This +may hinder DRL methods explainability. In this paper we address +this problem by proposing an interpretable DQN method comprising +two models which are trained jointly: An interpretable mimicking +model and a deep policy model. The later offers training samples to +the mimicking one and the former interpretable model offers target +action values for the other to improve its predictions. At the end of +the training process, the mimicking model has the capacity to pro- +vide high-fidelity interpretations to the decisions of the deep policy +model. This is a specific example for interpreting DRL methods, +according to the interpretable box design paradigm: This paradigm +follows the conjecture (stated for instance in [28]) that there is high +probability that the accuracy of closed boxes can be approximated +by well designed interpretable models. In this work, following this +paradigm, we train an interpretable model via mimicking, in par- +allel to the online Q network. Distillation could be another option +[29], but in this work we explore mimicking as a process to train +inherently interpretable models, such as decision trees. +There are many proposals for interpreting deep NNs models, +through distillation and mimicking approaches. These approaches +differ in several dimensions: (a) the targeted representation (e.g., +decision trees in DecText [6], logistic model trees (LMTs) in ref- +erence [9], or Gradient Boosting Trees in reference [7]), (b) to the +different splitting rules used towards learning a comprehensive repre- +sentation, (c) to the actual method used for building the interpretable +model (e.g., [9] uses the LogiBoost method, reference [6] proposes +the DecText method, while the approach proposed in reference [7] +proposes a pipeline with an external classifier, (d) on the way of +generating samples to expand the training dataset. These methods +can be used towards interpreting constituent individual DRL models +employing (deep) NNs. The interested reader is encouraged to read +a thorough review on these methods provided in [4, 12, 24, 28]. +For DRL, authors in [19] introduce Linear Model U-trees (LMUTs) +to approximate predictions for DRL agents. An LMUT is learned by +an on-line algorithm that is well-suited for an active play setting. The +use of LMUTs is compared against using CART, M5 with regression +tree, Fast Incremental Model Tree (FIMT) and with Adaptive Filters +(FIMT-AF). The use of decision trees as interpretable policy models +trained through mimicking has been also investigated in [21], in con- +junction to using a causal model representing agent’s objectives and +opportunity chains. However, the decision tree in this work is used +to infer the effects of actions approximating the causal model of the +environment. Similarly to what we do here, the decision tree policy +model is trained concurrently with the RL policy model, assuming a +model-free RL algorithm and exploiting state-action samples using +an experience replay buffer. In [8] authors illustrate how Soft Deci- +sion Trees (SDT) [10] can be used in spatial settings as interpretable +policy models. SDT are hybrid classification models of binary trees +of predetermined depth, and neural networks. However their inherent +interpretability is questioned given their structure. Other approaches +train interpretable models other than trees, such as the Abstracted +Policy Graphs (APGs) proposed in [34], assuming a well-trained +policy model. APGs can offer interpretable representations of poli- +cies, concisely summarizing them, so that individual decisions can +be explained in the context of expected future transitions. +Approaches following the interpretable box design paradigm also +use use attention models for visual agents [1, 23], and interpretable +policy models in a rather direct way [25, 27]. +In contrast to the above mentioned approaches, XDQN can be +applied to any setting with arbitrary state features, where the in- +terpretable model formed using Gradient Boosting Regressors is +trained jointly to a deep one through mimicking in an active play +setting, following the DQN algorithm. It is worth noting that experi- +mentally, instead of Gradient Boosting Regressors, we also tested +naturally interpretable Linear Trees (such as LMUTs [19]); i.e. de- +cision trees with linear models in their leaves). However, such ap- +proaches completely failed to solve the task, demonstrating quite +low play performance with very large mean absolute errors. +As far as explanations are concerned, we opted for features’ con- +tributions to the Q-values, in a rather aggregated way, using the +residue of each Gradient Boosting Regressor node, as done in [38]. +This approach, as shown in [38], reports advantages over using well +known feature importance calculation methods, avoiding linearity as- +sumptions made by LIME [26] and bias in areas where features have +high variance, and also avoiding taking all tree paths into account in +case of outliers, as done by SHAP [20]. +6 +CONCLUSION AND FUTURE WORK +In this work, we address the challenging issue of training inter- +pretable policy models for solving real-world problems, such as the +multi-agent demand-capacity balancing problem pertaining to air +traffic management. To this aim, we have trained interpretable deep +Q-learning models through mimic learning without requiring the ex- +istence of already well-trained deep Q-networks. Experimentally, we +have shown that the proposed interpretable XDQN method, utilizing +a Gradient Boosting Regressor as the mimic learner, performs on a +par with DQN in terms of play performance whereas demonstrating +high fidelity. +Further work on XDQN is to design, evaluate and compare var- +ious explainable mimic models that can effectively substitute the +target Q-Network. Moreover, the proposed mimicking paradigm is +generic, and can be naturally extended to many well-known DRL +algorithms. Thus, future steps should also aim at benchmarking our +methodology utilizing state-of-the-art DRL in various experimental +settings. + +ACKNOWLEDGMENTS +Acknowledgements will appear in the final version of this manu- +script. +REFERENCES +[1] Raghuram Mandyam Annasamy and Katia Sycara. 2019. 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Information Sciences 589 (2022), +199–212. https://doi.org/10.1016/j.ins.2021.12.111 + diff --git a/udE1T4oBgHgl3EQfQwM3/content/tmp_files/load_file.txt b/udE1T4oBgHgl3EQfQwM3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe46a6334899fafca4a5178256eb1d327f3b046e --- /dev/null +++ b/udE1T4oBgHgl3EQfQwM3/content/tmp_files/load_file.txt @@ -0,0 +1,742 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf,len=741 +page_content='XDQN: Inherently Interpretable DQN through Mimicking Andreas Kontogiannis National Technical University of Athens Greece andr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='kontog@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='com George Vouros University of Piraeus Greece georgev@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='gr ABSTRACT Although deep reinforcement learning (DRL) methods have been successfully applied in challenging tasks, their application in real- world operational settings is challenged by methods’ limited ability to provide explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Among the paradigms for explainability in DRL is the interpretable box design paradigm, where interpretable models substitute inner constituent models of the DRL method, thus making the DRL method “inherently" interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In this paper we explore this paradigm and we propose XDQN, an explainable variation of DQN, which uses an interpretable policy model trained through mimicking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' XDQN is challenged in a complex, real-world operational multi-agent problem, where agents are independent learn- ers solving congestion problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Specifically, XDQN is evaluated in three MARL scenarios, pertaining to the demand-capacity balancing problem of air traffic management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' XDQN achieves performance similar to that of DQN, while its abilities to provide global models’ interpretations and interpretations of local decisions are demon- strated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' KEYWORDS Deep Reinforcement Learning, Mimic Learning, Explainability ACM Reference Format: Andreas Kontogiannis and George Vouros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' XDQN: Inherently Inter- pretable DQN through Mimicking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 1 INTRODUCTION Deep Reinforcement Learning (DRL) has mastered decision making policies in various difficult control tasks [11] [18] [15], games [22] [13] and other real-time applications [14] [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Despite the remark- able performance of DRL models, the knowledge of mastering these tasks remains implicit in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Thus, its application in real-world operational settings is challenged by methods’ limited ability to provide explanations at global (policy) and local (individ- ual decisions) levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' This lack of interpretability makes it difficult to trust DRL for solving safety-critical real-world tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' However, besides the inability of DRL models to provide interpretations on the selection of actions in specific circumstances, they are also unable to provide information about the evolution of models during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' These challenges are naturally further extended to multi-agent settings, in which different agents empowered by multi- agent reinforcement learning (MARL) methods aim at learning a joint optimal policy towards solving a target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To address some of the aforementioned challenges, one may follow different paradigms for the provision of explanations: The interpretable box design paradigm is one of them where interpretable models substitute inner components of DRL [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Additionally, Under submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' mimic learning has been proposed, so as to infer interpretable mod- els that mimic the behavior of well-trained deep neural networks [2, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In the DRL case, mimic learning aims to replace the closed- box DRL controller with an interpretable one, able to mimic the decisions made by the former [3, 19, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' A mimic learner tries to optimize fidelity [35], which is determined by comparing the mimic controller’s actions with the actions selected by the DRL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To extract knowledge from deep neural networks, recent work [3, 19] has applied mimic learning with tree representations, using decision trees: Criteria used for splitting tree nodes provide a tractable way to explain the predictions made by the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Typically, mimic learning approaches require already well-trained complex policy networks (which we refer to as mature networks), whose behavior are mimicking to support interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In real- world scenarios, this could be quite impractical, since the training overhead required to train the mimic models can often be a very time-consuming and costly process, especially for large state-action spaces and for multi-agent settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Another limitation of such ap- proaches is that they solely aim at providing explainability on the predictions of only the mature DRL model, ignoring completely the training process of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In other words, in these approaches, the mimic learner can only provide explanations about the policy of the inferred DRL controller, but not about the patterns and behaviors learned throughout the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To deal with these challenges, in this paper we propose eXplainable Deep Q-Network (XDQN), which is an explainable variation of the well-known DQN [22] method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In XDQN, our goal is to provide inherent explainability of DQN via mimic learning in an online man- ner, by replacing the complex deep Q-network with an interpretable mimic learner in testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In so doing, XDQN does not require the existence of a well-trained model to train an interpretable one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In particular, we train a mimic learner in parallel with the deep neural network (Q-network) of DQN in an online setting, where: at a train- ing step the DRL model uses the mimic learner to compute the target values of the Q-network needed for its training, while the mimic learner learns to behave as the DRL model, but in an explainable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Since the mimic learner is trained and updated while the DQN policy model is trained, we can keep multiple “snapshots” of the model evolution through time, offering interpretability on these in- termediate models, and insights about the patterns and behaviors that DQN learns during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To evaluate our method’s utility in real-world operational settings, XDQN is challenged in a complex, real-world multi-agent problem, where agents solve airspace congestion problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Agents in this setting are trained via parameter sharing following the centralized training, decentralized execution paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We summarize the main contributions of this paper below: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='03043v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='LG] 8 Jan 2023 To our knowledge, this work is the first that provides DQN with inherent interpretability through mimic learning without requiring the existence of a well-trained DRL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We propose XDQN, an explainable variation of DQN, in which an interpretable mimic learner is trained in parallel with the Q-network of DQN and plays the role of the target Q-network of DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Experimentally, we show that XDQN can perform similarly to DQN, demonstrating good play performance and fidelity to DQN decisions in complex, real-world operational multi- agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We demonstrate the ability of XDQN to provide global (pol- icy) and local (in specific circumstances) explanations regard- ing agents’ decisions, also while models are being trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 2 BACKGROUND 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='1 Markov Decision Process We consider a sequential decision making setup, in which an agent interacts with an environment 𝐸 over discrete time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' At a given timestep, the agent perceives features regarding a state 𝑠𝑡 ∈ 𝑆, where 𝑆 is the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The agent then chooses an action 𝑎𝑡 from a discrete set 𝐴 and observes a reward 𝑟𝑡 generated by the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The agent’s behavior is determined by a policy 𝜋, which maps states to a probability distribution over the actions, that is 𝜋 : 𝑆 → 𝑃(𝐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Apart from an agent’s policy, the environment 𝐸 may also be stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We model it as a Markov Decision Process (MDP) with a state space 𝑆, action space 𝐴, an initial state distribution 𝑝(𝑠1), transition dynamics 𝑝(𝑠𝑡+1|𝑠𝑡) and a reward function 𝑟 (𝑠𝑡,𝑎𝑡,𝑠𝑡+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' For brevity, we write 𝑟𝑡 = 𝑟 (𝑠𝑡,𝑎𝑡,𝑠𝑡+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The agent aims to maximize the expected discounted cumulative reward, which is formulated as 𝐺𝑡 = �∞ 𝜏=𝑡 𝛾𝜏−𝑡𝑟𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Here, 𝛾 ∈ (0, 1) is a discount factor which trades-off the importance of immediate and future rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Considering that an agent acts under a stochastic policy 𝜋, the Q-function (state-action value) of a pair (𝑠,𝑎) is defined as follows 𝑄𝜋 (𝑠,𝑎) = E [𝐺𝑡 | 𝑠𝑡 = 𝑠,𝑎𝑡 = 𝑎, 𝜋] (1) which can also be computed recursively with bootstrapping: 𝑄𝜋 (𝑠,𝑎) = E � 𝑟𝑡 + 𝛾E𝑎∼𝜋 (𝑠𝑡+1) [𝑄𝜋 (𝑠𝑡+1,𝑎)] | 𝑠𝑡 = 𝑠,𝑎𝑡 = 𝑎, 𝜋 � (2) The Q-function measures the value of choosing a particular action when the agent is in this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We define the optimal policy 𝜋∗ under which the agent receives the optimal 𝑄∗(𝑠,𝑎) = 𝑚𝑎𝑥𝜋𝑄𝜋 (𝑠,𝑎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' For a given state 𝑠, under the optimal policy 𝜋∗, the agent selects action 𝑎 = 𝑎𝑟𝑔𝑚𝑎𝑥𝑎′∈𝐴𝑄∗(𝑠,𝑎′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Therefore, it follows that the optimal Q-function satisfies the Bellman equation: 𝑄∗(𝑠,𝑎) = E � 𝑟𝑡 + 𝛾𝑚𝑎𝑥𝑎𝑄∗(𝑠𝑡+1,𝑎) | 𝑠𝑡 = 𝑠,𝑎𝑡 = 𝑎, 𝜋 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' (3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2 Deep Q-Networks To deal with a high dimensional state space, the state-action value function can be approximated by an online deep Q-network (DQN [22]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' a deep neural network 𝑄(𝑠,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃) with weight parameters 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To estimate the parameters 𝜃, at iteration 𝑖 the expected mean squared loss between the estimated Q-value of a state-action pair and its temporal difference target, produced by a fixed and separate target Q-network 𝑄(𝑠,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃−) with weight parameters 𝜃−, is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Formally: 𝐿𝑖 (𝜃𝑖) = E � 𝑌 𝐷𝑄𝑁 𝑖 − 𝑄(𝑠,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃) � , (4) with 𝑌 𝐷𝑄𝑁 𝑖 = 𝑟𝑡 + 𝛾 max 𝑎∈𝐴 𝑄(𝑠𝑡+1,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃−) (5) In order to train DQN and estimate 𝜃, we could use the standard Q-learning update algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Nevertheless, the Q-learning estimator performs very poorly in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To stabilize the training procedure of DQN, Mnih et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' al [22] freezed the parameters, 𝜃−, of the target Q-network for a fixed number of training iterations while updating the online Q-network with gradient descent steps with respect to 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In addition to the target network, during the learning process, DQN uses an experience replay buffer [22], which is an accumulative dataset, 𝐷𝑡, of state transitions - in the form of (𝑠, 𝑎, 𝑟, 𝑠′) - from past episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In a training step, instead of only using the current state transition, the Q-Network is trained by sampling mini-batches of past transitions from 𝐷 uniformly, at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Therefore, the loss can be written as follows: 𝐿𝑖 (𝜃𝑖) = E(𝑠,𝑎,𝑟,𝑠′)∼U(𝐷) � (𝑌 𝐷𝑄𝑁 𝑖 − 𝑄(𝑠,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' (6) The main advantage of using an experience replay buffer is that uniform sampling reduces the correlation among the experience samples used for training the Q-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The replay buffer also improves data efficiency through reusing the experience samples in multiple training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Instead of sampling mini-batches of past transitions uniformly from the experience replay buffer, a further improvement over DQN results from using a prioritized experience replay buffer [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' It aims at increasing the probability of sampling those past transitions from the experience replay that are expected to be more useful in terms of absolute temporal difference error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='3 Mimic Learning for Deep Reinforcement Learning Recent work on mimic learning [7, 19] has shown that rule-based models, like decision trees, or shallow feed-forward neural networks can mimic a not linear function inferred by a deep neural network with millions of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We present two known settings for mimicking the Q-function of a DRL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='1 Experience Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In the experience training setting [7, 19], all the state-action pairs ⟨𝑠,𝑎⟩ of a DRL training process are collected in a time horizon 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Then, to obtain the corresponding Q-values, these pairs are provided as input into a DRL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The final set {(⟨𝑠1,𝑎1⟩,𝑄1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='(⟨𝑠𝑇,𝑎𝑇 ⟩,𝑄𝑇 )} of tuples is used as the experience training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2 Active Play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The main problem with the experience training is that suboptimal state-action pairs are collected through training, making it more difficult for a learner to mimic the behavior of the DRL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To address this challenge, active play [19] uses a mature DRL model to generate state-action pairs to construct the training dataset of an active mimic learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The training data is collected in an online manner through queries, in which the active learner selects the actions, given the states, and the mature DRL model provides the estimated Q-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' These Q-values are then used to update the active learner’s parameters on minibatches of the collected dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 3 EXPLAINABLE DEEP Q-NETWORK (XDQN) In this work, we are interested in providing interpretability in deep Q-learning through mimicking the behavior of DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To this aim, we propose eXplainable Deep Q-Network (XDQN) 1, which is an explainable variation of DQN [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' XDQN aims at inferring the parameters of the online Q-network and the parameters of a mimic learner concurrently, in an online manner, with the latter substituting the target Q-network of DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Formally, let 𝜃 be the parameters of the online Q-network and 𝜙𝑋 be the parameters of the mimic learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In XDQN, the mimic learner is used to estimate the state-action value function and select the best action for the next state in the XDQN target: 𝑌𝑋𝐷𝑄𝑁 𝑖 = 𝑟𝑡 + 𝛾 max 𝑎∈𝐴 𝑄 (𝑠𝑡+1,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜙𝑋 ) (7) Similar to DQN, 𝜙𝑋 are updated every 𝑇𝑢 number of timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The full training procedure of XDQN is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In contrast to DQN in which we simply copy the parameters 𝜃 of the online Q-network to update the parameters of the target Q-network, here we perform mimic learning on 𝑄(𝑠,𝑎,𝜃) (steps 17-20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To update 𝜙𝑋 we train the mimic learner on minibatches of the experience replay buffer 𝐵 by minimizing the Mean Squared Error (MSE) loss function using 𝑄(𝑠,𝑎,𝜃) to estimate the soft labels (Q-values) of the state-action pairs in the minibatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Formally the optimization problem for each update of 𝜙𝑋 can be written as: min 𝜙𝑋 E(𝑠,𝑎)∼𝐵 � (𝑄(𝑠,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜙𝑋 ) − 𝑄(𝑠,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃))2� (8) In our experiments, we utilize a prioritized experience replay [30] as the replay buffer 𝐵, as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Similarly to active play, when updating 𝜙𝑋 , to ensure that the state-action pairs of the minibatches provide up-to-date target values with respect to 𝜃, we use records from the replay buffer that were stored during the 𝐾 latest training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' It is worth noting that at each update of 𝜙𝑋 the hyperparameter 𝐾 for past transitions plays a similar role as the discounted factor 𝛾 plays for future rewards, but from the mimic learner’s perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Building upon the experience training and active play paradigms, XDQN can leverage the benefits of both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In particular, the hyperparameter 𝐾 manages the trade-off between experience training and active play in XDQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' If 𝐾 is large, the mimic model learns from state-action pairs that may have been collected through more suboptimal instances of 𝜃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' deploying however data-augmented versions of Q-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' On the other hand, if 𝐾 is small, it learns from the most recent instances of 𝜃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' making use of up-to-date Q-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Nevertheless, opting for very small values of 𝐾 could lead to less stable mimic training, due to the smaller number of minibatches that can be produced for updating 𝜙𝑋 , while using large 𝐾 can result in a very slow training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 1The implementation code will be made available in the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' From all the above, we note that 𝜃 (Q-network) and 𝜙𝑋 (mimic learner) are highly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To update 𝜃, Q-network uses the mimic learner model with 𝜙𝑋 to compute the target soft labels (target Q-values), while to update 𝜙𝑋 the mimic learner uses the original Q-network with parameters 𝜃 to compute the respective target soft labels (online Q-values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Since XDQN produces different instances of 𝜙𝑋 throughout training, it can eventually output multiple interpretable mimic learner models (up to the number of 𝜙𝑋 updates), with each one of them corresponding to a different training timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Assuming that all these mimic learner instances are interpretable models, XDQN can also provide explainability on how a DRL model learns to solve the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Finally, after Q-network (𝜃) and mimic learner (𝜙𝑋 ) have been trained, without requiring to learn 𝜃 before 𝜙𝑋 , we can discard the online Q-network and use the mimic learner model as the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Therefore, in testing, given a state, the interpretable mimic learner selects the action that profits the highest Q-value, being also able to provide explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 4 EXPERIMENTAL SETUP In this section, we demonstrate the effectiveness of XDQN through experiments on real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In all experiments we utilize a Gra- dient Boosting Regressor [36] as the mimic learner, so as to exploit its boosting ability to learn effectively by exploiting instances gener- ated by the deep Q-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Although most decision tree algorithms, being rule-based models, are naturally interpretable models [3, 19], this is not the case for a Gradient Boosting Regressor, since the boosting structure makes it very difficult to provide explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' However, following the work in [38], we are able to enrich the Gra- dient Boosting Regressor mimic learner with the ability to provide explainability as follows: Given a state-action pair as an input of the mimic learner, we can measure the contribution of each state feature to the predicted Q-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Therefore, our mimic learner is expected not only to mimic effectively the behavior of the DRL controller, but also, to give local and global explanations on its decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Overall, we are interested in comparing the performance of XDQN with that of DQN in real-world environments where the latter has been state-of-the-art, and also designing appropriate experimental setups, aiming at studying XDQN interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In so doing, we evaluate XDQN on real-world operational multi-agent experimen- tal scenarios, pertaining to the demand-capacity balancing (DCB) problem of air traffic management (ATM), which we describe next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='1 Real-world demand-capacity problem setting The current ATM system is based on time-based operations resulting in DCB [17] problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To solve the DCB issues at the pre-tactical stage of operations, the ATM system opts for methods that generate delays and costs for the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In ATM, the airspace consists of a set of 3D sectors where each one these is characterized by a specific capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' This is the number of flights that cross the sector during a specific period (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' of 20 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The main challenge of dealing with the DCB problem in ATM is to reduce the number of cases where the demand of airspace use exceeds its capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' These cases are called hotspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Recent work has transformed the DCB challenge to a multi-agent RL problem by formulating the setting as a multi-agent MDP [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Algorithm 1 eXplainable Deep Q-Network (XDQN) 1: Initialize replay buffer 𝐵 with capacity N 2: Initialize 𝜃 and 𝜙𝑋 3: Initialize timestep count 𝑐 = 0 4: for episode 1, M do 5: Augment 𝑐 = 𝑐 + 1 6: Initialize state 𝑠1 7: With probability 𝜖 select a random action 𝑎𝑡, otherwise 𝑎𝑡 = 𝑎𝑟𝑔𝑚𝑎𝑥𝑎𝑄(𝑠𝑡,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃) 8: Execute action 𝑎𝑡 and observe next state 𝑠𝑡+1 and reward 𝑟𝑡 9: Store transition (𝑠𝑡,𝑎𝑡,𝑠𝑡+1,𝑟𝑡) in 𝐵 10: Sample a minibatch of transitions (𝑠𝑖,𝑎𝑖,𝑠𝑖+1,𝑟𝑖) from 𝐵 11: if 𝑠𝑖+1 not terminal then 12: Set 𝑌𝑋𝐷𝑄𝑁 𝑖 = 𝑟𝑖 + 𝛾 max𝑎∈𝐴 𝑄 (𝑠𝑖+1,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜙𝑋 ) 13: else 14: Set 𝑌𝑋𝐷𝑄𝑁 𝑖 = 𝑟𝑖 15: end if 16: Perform a gradient descent step on � 𝑌𝑋𝐷𝑄𝑁 𝑖 − 𝑄(𝑠𝑖,𝑎𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃) �2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 𝜃 17: if 𝑐 mod 𝑇𝑢 = 0 then 18: Initialize 𝜙𝑋 19: Sample a minibatch of transitions (𝑠𝑖,𝑎𝑖,𝑠𝑖+1,𝑟𝑖) from 𝐵 that were stored at most 𝑐 − 𝐾 timesteps before 20: Perform mimic learning update on (𝑄(𝑠,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜙𝑋 ) − 𝑄(𝑠,𝑎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃))2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='t 𝜙𝑋 21: end if 22: end for We follow the work and the experimental setup of [15–17, 31, 32] and encourage the reader to see the problem formulation [17] in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In this setting, we consider a society of agents, where each agent is a flight (related to a specific aircraft) that needs to coordi- nate its decisions, so as to resolve hotspots that occur, jointly with other society agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Agents’ local states comprise 81 state variables related to: (a) the delay (in the range of 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=', maxDelay) set by the referring agent, (b) the number of hotspots in which the agent is involved in, (c) the sectors that it crosses, (d) the minutes that the agent is within each sector it crosses, (e) the periods in which the agent joins in hotspots in sectors, and (f) the minute of the day that the agent takes off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The tuple containing all agents’ local states is the joint global state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Q-learning [33] agents has been shown to achieve remarkable performance on this task [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In our experiments, all agents share parameters and replay buffer and act independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' A DCB scenario comprises multiple flights crossing various airspace sectors in a time horizon of 24h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' This time horizon is segre- gated into simulation time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' At each simulation time step (equal to 10 minutes of real time), given only the local state, each agent se- lects an action which is related to its preference to add ground delay regulating its flight, in order to resolve hotspots in which it partici- pates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The set of local actions for each agent contains |maxDelay+1| actions, at each simulation time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We use maxDelay = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The joint (global) action is a tuple of local actions selected by the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Similarly, we consider local rewards and joint (global) rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The local reward is related to the cost per minute within a hotspot, the total duration of the flight (agent) in hotspots as well as to the delay that a flight has accumulated up to the simulation timestep [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2 Evaluation Metrics and Methods For the evaluation of the proposed method, first, we make use of two known evaluation metrics: (a) play performance [19] of the online deep Q-network, and (b) fidelity [35] of the mimic learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Play performance measures how well the deep Q-network performs with the mimic learner estimating its temporal difference targets, while fidelity measures how well the mimic learner matches the predictions of the online deep Q-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' As far as play performance is concerned, we aim at minimizing the number of hotspots, the average delay per flight and the number of delayed flights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' As for fidelity, we use two metric scores: (a) the mean absolute error (MAE) and (b) the accuracy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Given a minibatch of states, we calculate the MAE of this minibatch for any action as the mean absolute difference between the Q-values estimated by the mimic learner and the Q-values estimated by the deep Q-network for that action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' More formally, for a minibatch of states 𝐷𝑠, the MAE𝑖 of action 𝑎𝑖 is denoted as: 𝑀𝐴𝐸𝑖 = 1 |𝐷𝑠 | ∑︁ 𝑠 ∈𝐷𝑠 |𝑄(𝑠,𝑎𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜙𝑋 ) − 𝑄(𝑠,𝑎𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='𝜃)| (9) It is worth noting that minimizing the MAE of the mimic learner is very important for training XDQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Since deep Q-network updates its parameters 𝜃 by using the mimic model to provide the target Q-values, large MAEs can lead deep Q-network to overestimate bad states and understimate the good ones, and thus, find very diverging policies that completely fail to solve the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To calculate the accuracy score, again given a minibatch of states, for each state we compare the action selected by the mimic model and the online Q-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Accuracy measures the percentage of the Scenario DQN XDQN Final Hotspots Average Delay Delayed Flights Final Hotspots Average Delay Delayed Flights 20190705 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='04 1556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='19 1618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='54 20190708 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='4 1387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='73 1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='58 20190714 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='72 1645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='46 1849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='49 Table 1: Comparison of testing performance of DQN and XDQN on the three experimental ATM scenarios predictions of the two estimators that agree with each other, consid- ering that both models select the action with the highest estimated Q-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Second, we design appropriate experiments and illustrate XDQN’s local and global interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We focus on providing aggregated interpretations, focusing on the contribution of features to local deci- sions and to the overall policy: This, as suggested by ATM operators, is beneficial towards understanding decisions, helping them to in- crease their confidence to the solutions proposed, and mastering the inherent complexity in such a multi-agent setting, as solutions may be due to complex phenomena that are hard to be traced [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Specifically, in this work, local explainability measures state fea- tures’ importance on a specific instance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' a single state-action pair), demonstrating which features contribute to the selection of a particular action over the other available ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Global explain- ability aggregates feature importance on particular action selections over many different instances and aims to explain the overall policy of mimic learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Third, we demonstrate global explainability of the DRL model through the whole training process, addressing the question of how a DRL model learns to solve the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='3 Experimental Scenarios and Settings Experiments were conducted on three in total scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Each of these scenarios corresponds to a date in 2019 with heavy traffic in the Spanish airspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In particular, the date scenarios, on which we as- sess our models, are 20190705, 20190708 and 20190714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' However, to bootstrap the training process we utilize a deep Q-network pre- trained in various scenarios, also including 20190705 and 20190708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In the training process, the deep Q-network is further trained ac- cording to the method we propose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The experimental scenarios were selected based on the number of hotspots and the average delay generated in the ATM system within the duration of the day, which shows the difficulty of the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We note that for each scenario we ran five separate experiments and average results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Table 2 presents information on the three experimental scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In particular, the flights column indicates the total number of flights (represented by agents) during the specific day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The initial hotspots column indicates the number of hotspots appearing in the initial state of the scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The flights in hotspots column indicates the number of flights in at least one of the initial hotspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Note that all three scenarios display populations of agents of similar size, with 20190708 having the smaller population and the least initial hotspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Scenario Flights Initial Hotspots Flights in Hotspots 20190705 6676 100 2074 20190708 6581 79 1567 20190714 6773 92 2004 Table 2: The three experimental Air Traffic Management (ATM) scenarios Figure 1: Episodic reward in the three evaluated ATM scenarios 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='4 Implementation Details In our implementation setting we utilize a deep multilayer perceptron as the Q-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In particular, we use an 𝜖-greedy policy, which at the start of exploration has 𝜖 equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='9 decaying by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='01 every 15 episodes until reaching the minimum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The total number of episodes are set to 1600 and the update target frequency is set to 9 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In the exploitation mode, we set 𝜖 equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We set the maximum depth of the Gradient Boosting Regressor equal to 45 and the number of minimum samples for a split equal to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We also use the mean squared error as the splitting criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To train a single decision tree for all different actions, we create a non binary splitting rule of the root based on the action size of the task, so that the state-action pairs sharing the same action match the same subtree of the splitting root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Empirically, we set the memory capacity of the experience replay for the mimic learner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' the hyperparameter 𝐾, equal to the 1/20 of the product of three other hyperparameters, namely the total number of timesteps per episode (set to 1440), the update target frequency (set to 9) and the number of agents (set to 7000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Thus, 𝐾 is set to 4536000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='5 Evaluation of play performance Table 1 demonstrates the performance of DQN and XDQN on the three experimental scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The final hotspots column indicates the number of unresolved hotspots in the final state: It must be noted 20190708 20190714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='8 20190705 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='7 Reward 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='4 0 250 500 750 1000 1250 1500 1750 EpisodesAction (Delay Option) XDQN mimic models X0705 X0708 X0714 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='291 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='766 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='971 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='942 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='928 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='002 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='575 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='497 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='740 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='823 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='292 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='525 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='461 Table 3: Evaluation of the average Mean Absolute Errors (MAE) of the trained mimic models over all mimic updates that these hotspots may have emerged due to delays assigned to flights and may be different than the hotspots at the beginning of each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The average delay per flight column shows the total minutes of delay imposed, divided by the number of flights in the specific scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The delayed flights column indicates the number of flights affected by more than four minutes of delay, as it is done by operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We observe that XDQN performs similar to DQN in all three evaluated metric scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In particular, DQN slightly outperforms XDQN in terms of the final hotspots and average delay in all three scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Nonetheless, XDQN achieves to decrease the number of the delayed flights in one scenario, while it demonstrates competitive performance on the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Figure 1 shows the episodic reward of XDQN over time: XDQN manages to reach convergent behavior in all scenarios by retaining high episodic rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='6 Evaluation of fidelity As discussed in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2, for the fidelity evaluation, we mea- sure the mean absolute error (MAE) and the accuracy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Given the DCB experimental scenarios, we train three different mimic mod- els;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' namely X0705, X0708 and X0714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Table 3 reports the average MAE for each decided action over all mimic learning updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We observe that all errors range in relatively small quantities, given that in testing, the absolute Q-values hovered around 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' As we highlighted above, this is very important for stabilizing the training process of XDQN, since we need very accurate mimic Q-value pre- dictions, ideally equal to the ones generated by the deep Q-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To further assess the fidelity of XDQN mimic learner, Table 4 il- lustrates the average accuracy scores over all mimic learning updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Since a Gradient Boosting Regressor mimic learner is a boosting algorithm, it produces sequential decision trees that can successfully seperate the state space and approximate well the predictions of the deep Q-network function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We observe that the mimic learner and the deep Q-network agree with each other to a very good extent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' namely from approximately 81% to 91%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Therefore, we expect the mimic learner to be able to accumulate the knowledge from the deep Q-network with high fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Scenario Accuracy (%) 20190705 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='45 20190708 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='89 20190714 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='88 Table 4: The accuracy scores of mimic models 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='7 Local and Global Explainability In the DCB setting, it is important for the operator to understand how the system reaches decisions on regulations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' assignment of delays to flights): This, as already pointed out, should be done at a level of abstraction that would allow them to increase their confidence to the solutions proposed, mastering the inherent complexity of the setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Therefore, we are mainly interested in receiving explanations about which state features contribute to the selection of delay actions over the no-delay action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' action equal to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' First, we demonstrate the ability of the mimic learner to provide local explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' As already said, local explainability involves showing which state features contribute to the selection of a par- ticular action over the other available ones in a specific state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To this aim, we work on pairs of actions - let 𝑎1 and 𝑎2 - and calculate the differences of feature contributions in selecting 𝑎1 and 𝑎2 in a single state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To highlight only the most significant differences, we focus only on those features whose differences are above a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Empirically, we set this threshold equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Figure 2 illustrates local explainability on a given state in which action "2" was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Figure 2 provides the differences of feature contributions to the estimation of Q-values when selecting action "0" against selecting action "2" (denoted by “0-2").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We observe that the features that contributed more to the selection of the delay action "2" were those with index 32 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The sector in which the last hotspot occurs), 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' the sector in which the first hotpot occurs) and 62 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' the minutes that the flight spends crossing the last sector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Finally, we demonstrate XDQN global explainability by aggre- gating the importance of features on particular action selections over many different state-action instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In particular, we are interested in measuring the state feature contributions to the selection of delay actions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' actions in the range [1, 10]) over the no-delay action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Feature Index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Feature Meaning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='ACD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Delay the corresponding flight has accumulated up to this point ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Positive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Total number of hotspots the corresponding flight participates in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Positive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='The sector in which the second hotspot the corresponding flight participates occurs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Positive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='The minute of day the flight takes off given the delay (CTOT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Negative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='The minutes the flight remains in the first sector it crosses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Negative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='The minutes the flight remains in the fifth sector it crosses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Negative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Table 5: Demonstration of the most significant state features in terms of average contribution difference (ACD) in selecting the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='no-delay action versus a delay action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' A positive ACD means that the corresponding state feature on average contributes more to the selection of the no-delay action “0".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' On the contrary, a negative ACD means that the corresponding state feature on average contributes to the selection of a delay action “1 - 10".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Figure 2: Illustration of significant differences of feature con- tributions to Q-value in selecting action "0" and action "2" in a single state, in which action "2" was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Positive differ- ences mean that the respective state features have a greater con- tribution to Q-value when action "0" is selected, rather than when action "2" is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Negative differences have the oppo- site meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' action "0") in the overall policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To this aim, we work on all possible pairs of actions, with one action always being the no-delay action and the other one being a delay action, and average the differences of feature contributions to estimating the Q-value in selecting those actions over many different state-action instances with the same se- lected delay action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Table 5 shows the most significant state features in terms of average contribution difference (ACD) in selecting the no-delay action versus a delay action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To select those features, we initially filter the most significant ones, namely the features whose absolute ACD is greater than a threshold, for each action in the range [1, 10]) over the no-delay action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' action "0"), and present the three most common features with positive and negative ACD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We observe that features with index 0, 1 and 3 contribute more to the selection of the no-delay action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' On the contrary, features with indexes 64, 63 and 68 contribute more to the selection of a no-delay action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Last but not least, we demonstrate how global explainability evolves through the training process, addressing the question of how a DRL model learns to solve the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To this aim, we measure the absolute average feature contribution (AAFC) to Q- value at different training episodes for the selection of each action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Figure 3 illustrates the evolution of global explainability for selecting Figure 3: Illustration of the evolution of features’ contribu- tions for selecting the no-delay action ("0") and a delay one ("2") through 5 representative training episodes (360th, 720th, 1100th, 1400th and 1600th) in terms of absolute average feature contribution (AAFC) to Q-value for the eight features with high- est AAFC values in the final model (episode 1600) in the selec- tion of the aforementioned actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' the no-delay action and a delay action through 5 representative training episodes (360th,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 720th,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 1100th,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 1400th and 1600th),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='terms of AAFC to Q-value for the eight features with highest AAFC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='values in the final model (episode 1600) in the selection of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='69 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='State Feature30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 360 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Average Contribution to Q-value for Action 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 720 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 1100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='State Feature30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 360 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Average Contribution to Q-value for Action 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 720 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 1100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='Episode 1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='State Featureaforementioned actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' We observe that for both evaluated actions most of the features show an increasing/decreasing trend in their average contribution to Q-value over time, such as those with indices 0, 1 and 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' It is worth noting that although the features with indexes 0 and 1 have been highlighted as the most significant for the selection of the no-delay action, they have also significant but less contribution to a delay action as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 5 RELATED WORK Explainability in Deep Reinforcement Learning (DRL) is an emer- gent area whose necessity is related to the fact that DRL agents solve sequential tasks, acting in the real-world, in operational settings where safety, criticality of decisions and the necessity for trans- parency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' explainability with respect to real-world pragmatic con- straints [35]) is the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' However, DRL methods use closed-boxes whose functionality is intertwined and are not interpretable: This may hinder DRL methods explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In this paper we address this problem by proposing an interpretable DQN method comprising two models which are trained jointly: An interpretable mimicking model and a deep policy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The later offers training samples to the mimicking one and the former interpretable model offers target action values for the other to improve its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' At the end of the training process, the mimicking model has the capacity to pro- vide high-fidelity interpretations to the decisions of the deep policy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' This is a specific example for interpreting DRL methods, according to the interpretable box design paradigm: This paradigm follows the conjecture (stated for instance in [28]) that there is high probability that the accuracy of closed boxes can be approximated by well designed interpretable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In this work, following this paradigm, we train an interpretable model via mimicking, in par- allel to the online Q network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Distillation could be another option [29], but in this work we explore mimicking as a process to train inherently interpretable models, such as decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' There are many proposals for interpreting deep NNs models, through distillation and mimicking approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' These approaches differ in several dimensions: (a) the targeted representation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=', decision trees in DecText [6], logistic model trees (LMTs) in ref- erence [9], or Gradient Boosting Trees in reference [7]), (b) to the different splitting rules used towards learning a comprehensive repre- sentation, (c) to the actual method used for building the interpretable model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=', [9] uses the LogiBoost method, reference [6] proposes the DecText method, while the approach proposed in reference [7] proposes a pipeline with an external classifier, (d) on the way of generating samples to expand the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' These methods can be used towards interpreting constituent individual DRL models employing (deep) NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The interested reader is encouraged to read a thorough review on these methods provided in [4, 12, 24, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' For DRL, authors in [19] introduce Linear Model U-trees (LMUTs) to approximate predictions for DRL agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' An LMUT is learned by an on-line algorithm that is well-suited for an active play setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The use of LMUTs is compared against using CART, M5 with regression tree, Fast Incremental Model Tree (FIMT) and with Adaptive Filters (FIMT-AF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' The use of decision trees as interpretable policy models trained through mimicking has been also investigated in [21], in con- junction to using a causal model representing agent’s objectives and opportunity chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' However, the decision tree in this work is used to infer the effects of actions approximating the causal model of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Similarly to what we do here, the decision tree policy model is trained concurrently with the RL policy model, assuming a model-free RL algorithm and exploiting state-action samples using an experience replay buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In [8] authors illustrate how Soft Deci- sion Trees (SDT) [10] can be used in spatial settings as interpretable policy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' SDT are hybrid classification models of binary trees of predetermined depth, and neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' However their inherent interpretability is questioned given their structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Other approaches train interpretable models other than trees, such as the Abstracted Policy Graphs (APGs) proposed in [34], assuming a well-trained policy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' APGs can offer interpretable representations of poli- cies, concisely summarizing them, so that individual decisions can be explained in the context of expected future transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Approaches following the interpretable box design paradigm also use use attention models for visual agents [1, 23], and interpretable policy models in a rather direct way [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' In contrast to the above mentioned approaches, XDQN can be applied to any setting with arbitrary state features, where the in- terpretable model formed using Gradient Boosting Regressors is trained jointly to a deep one through mimicking in an active play setting, following the DQN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' It is worth noting that experi- mentally, instead of Gradient Boosting Regressors, we also tested naturally interpretable Linear Trees (such as LMUTs [19]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' de- cision trees with linear models in their leaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' However, such ap- proaches completely failed to solve the task, demonstrating quite low play performance with very large mean absolute errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' As far as explanations are concerned, we opted for features’ con- tributions to the Q-values, in a rather aggregated way, using the residue of each Gradient Boosting Regressor node, as done in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' This approach, as shown in [38], reports advantages over using well known feature importance calculation methods, avoiding linearity as- sumptions made by LIME [26] and bias in areas where features have high variance, and also avoiding taking all tree paths into account in case of outliers, as done by SHAP [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' 6 CONCLUSION AND FUTURE WORK In this work, we address the challenging issue of training inter- pretable policy models for solving real-world problems, such as the multi-agent demand-capacity balancing problem pertaining to air traffic management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' To this aim, we have trained interpretable deep Q-learning models through mimic learning without requiring the ex- istence of already well-trained deep Q-networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Experimentally, we have shown that the proposed interpretable XDQN method, utilizing a Gradient Boosting Regressor as the mimic learner, performs on a par with DQN in terms of play performance whereas demonstrating high fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Further work on XDQN is to design, evaluate and compare var- ious explainable mimic models that can effectively substitute the target Q-Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Moreover, the proposed mimicking paradigm is generic, and can be naturally extended to many well-known DRL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' Thus, future steps should also aim at benchmarking our methodology utilizing state-of-the-art DRL in various experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE1T4oBgHgl3EQfQwM3/content/2301.03043v1.pdf'} +page_content=' ACKNOWLEDGMENTS Acknowledgements will appear in the final version of this manu- script.' 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b/xdFAT4oBgHgl3EQfAByy/content/tmp_files/2301.08396v1.pdf.txt @@ -0,0 +1,2149 @@ +arXiv:2301.08396v1 [math-ph] 20 Jan 2023 +Generalized Lie Symmetries and Almost Regular Lagrangians: A +Link Between Symmetry and Dynamics +Achilles D. Speliotopoulos +Department of Physics, University of California, Berkeley, CA 94720 USA∗ +(Dated: January 23, 2023) +1 + +Abstract +The generalized Lie symmetries of almost regular Lagrangians are studied, and their impact on +the evolution of dynamical systems is determined. It is found that if the action has a generalized +Lie symmetry, then the Lagrangian is necessarily singular; the converse is not true, as we show +with a specific example. +It is also found that the generalized Lie symmetry of the action is a +Lie subgroup of the generalized Lie symmetry of the Euler-Lagrange equations of motion. The +converse is once again not true, and there are systems for which the Euler-Lagrange equations +of motion have a generalized Lie symmetry while the action does not, as we once again show +through a specific example. Most importantly, it is shown that each generalized Lie symmetry of +the action contributes one arbitrary function to the evolution of the dynamical system. The number +of such symmetries gives a lower bound to the dimensionality of the family of curves emanating +from any set of allowed initial data in the Lagrangian phase space. Moreover, if second- or higher- +order Lagrangian constraints are introduced during the application of the Lagrangian constraint +algorithm, these additional constraints could not have been due to the generalized Lie symmetry +of the action. +I. +INTRODUCTION +The symmetries of the Euler-Lagrange equations of motion were recently used to study +the constrained dynamics of singular Lagrangians [1]. The focus was on almost regular La- +grangians [2–5], and it was found that for these Lagrangians the Euler-Lagrange equations of +motion admit a generalized Lie symmetry (also known as a local gauge symmetry). The gen- +erators Sym of this symmetry group GrSym were determined in the Lagrangian phase space +approach to Lagrangian mechanics, and were found to lie in the kernel of the Lagrangian +two-form ΩL. While it is well-known that the solutions XE of the energy equation, +0 = dE − iXEΩL, +(1) +is not unique for almost regular Lagrangians, it was shown in [1] that the action of Sym on +a general solution to this equation—and in particular, on the second-order, Lagrangian +∗ Also at Physical Science and Engineering Division, Diablo Valley College, Pleasant Hill, CA 94523, USA; +ads@berkeley.edu +2 + +vector field (SOLVF)—will result in a vector field that is no longer a solution of Eq. (1). +Thus, not all solutions of the energy equation have GrSym as a symmetry group. It is, +however, possible to construct solutions to Eq. (1) for whom Sym does generate a group +of symmetry transformations [1]. +These vector fields are called second-order, Euler- +Lagrange vector fields (SOELVFs). As the evolution of the dynamical system for sin- +gular Lagrangians must lie on Lagrangian constraint surfaces [5], a Lagrangian constraint +algorithm for SOELVFs was also introduced in [1] to construct such solutions to the energy +equation. It was then shown that these SOELVFs, along with the dynamical structures in +the Lagrangian phase space needed to describe and determine the motion of the dynamical +system, are projectable to the Hamiltonian phase space. In particular, the primary Hamil- +tonian constraints can be constructed from vectors that lie in the kernel of ΩL, and the +Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of +the total Hamiltonian (we follow the terminology found in [6]; see also [7–9]) obtained using +constrained Hamiltonian mechanics. Importantly, the end result of this stability analysis +gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the La- +grangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics +for almost regular Lagrangians were thereby shown to be equivalent. +While [1] focused on the generalized Lie symmetries of the Euler-Lagrange equations of +motion and whether the dynamical structures constructed in the Lagrangian phase space are +projectable to the Hamiltonian phase space, in this paper the focus is on the symmetries of +the action itself and the impact these symmetries have on the evolution of dynamical systems. +This impact is found to be quite broad, surprisingly restrictive, and unexpectedly subtle. +Indeed, even the seemingly reasonable expectation that any generalized Lie symmetry of the +Euler-Lagrange equations of motion should be a reflection of the symmetries of the action +itself is not borne out. +We find that if the action has a generalized Lie symmetry, then its Lagrangian is nec- +essarily singular; the converse need not be true, as we show through a specific example. +We also find that the generators of the generalized Lie symmetry of the action form a Lie +sub-algebra of the generators of the generalized Lie symmetry of the Euler-Lagrange equa- +tion of motion; once again, the converse is not true. We give an example of a dynamical +system for which the Euler-Lagrange equations of motion has a generalized Lie symmetry, +while its action does not. Most importantly, for systems where the Lagrangian is almost +3 + +regular and for which the two-form ΩL has constant rank, we show that each generalized +Lie symmetry of the action contributes one arbitrary constant to the SOELVF. The dimen- +sionality of the space of solutions to the energy equation that have GrSym as a symmetry +group is thus at least as large as the number of generalized Lie symmetries of the action. +Moreover, if second- or higher-order Lagrangian constraints are introduced during the ap- +plication of the Lagrangian constraint algorithm, these additional constraints cannot be due +to the generalized Lie symmetry of the action. +Symmetries of Lagrangian systems have been studied before. However, such analyses have +been focused on time-dependent Lagrangians [10–17]; on systems of first-order evolution +equations [18–22]; or on general solutions of Eq. (1) [23] (see also [24]). Importantly, the +great majority of these studies have been done using first-order prolongations on first-order +jet bundles with a focus on the Lie symmetries of first-order evolution equations. +Our +interest is in the symmetries of the action, which naturally leads us to consider generalized +Lie symmetries and second-order prolongations. To our knowledge, such symmetry analysis +of the action has not been done before. (The framework for kth-order prolongations on +kth-order jet bundles have been introduced before [16, 17, 23, 25, 26], but they were not +applied to the action or to the Euler-Lagrange equations of motion.) +The rest of the paper is arranged as follows. In Section II the conditions under which +the action for a dynamical system, and the conditions under which the Euler-Lagrange +equations of motion for this action, have a generalized Lie symmetry are determined. To +compare the conditions for each, the analysis for the two are done separately, with each +self-contained. In Section III properties of the Lagrangian phase space are reviewed, and +the notation used here established. The generators of the generalized Lie symmetry group +for the Euler-Lagrange equations of motion were determined in [1], and a summary of the +results found therein that are needed here is given. In Section IV the generators of the +generalized Lie symmetry group for the action is found within the Lagrangian phase space +approach, and their relation to the generators for the symmetry group of the Euler-Lagrange +equations of motion is determined. +The impact of the symmetries of the action on the +SOELVF is then analyzed by applying the Lagrangian constraint algorithm introduced in +[1] to these SOELVF. The results obtained in this paper is then applied to three different +dynamical systems in Section V. In particular, an example of a dynamical system that +has no generalized Lie symmetries and yet is still singular, and another example where the +4 + +action has no symmetries and yet the Euler-Lagrange equations of motion do, are given. +Concluding remarks can be found in Section VI. +II. +GENERALIZED LIE SYMMETRIES AND LAGRANGIAN MECHANICS +In this section we determine the conditions under which the action of a dynamical system, +and the conditions under which the Euler-Lagrange equations of motion for this system, has +a generalized Lie symmetry. While the determination for both is done within Lagrangian +mechanics, the analysis for the action is completed separately from that of the equations +of motion—with each self-contained—so that the two conditions can be compared. We will +later show that every generator of the generalized Lie symmetry of the action is a generator +of a generalized Lie symmetry of the Euler-Lagrange equations of motion. Interestingly, the +converse is not true. +A. +Symmetries of the Action +We begin with Lagrangian mechanics, and an analysis of the generalized Lie symmetry +[27] of the action +S := +� t2 +t1 +L (q(t), ˙q(t)) dt, +for a dynamical system on a D-dimensional configuration space Q. Here, L (q(t), ˙q(t)) is +the Lagrangian along a path q(t) = +� +q1(t), . . . , qD(t) +� +on Q with end points given by Q1 := +q(t1), Q2 := q(t2). These points are chosen at the same time the choice of S is made, and +are fixed. +As L (q(t), ˙q(t)) depends on both the position q(t) and the velocity ˙q(t) of the path, we +consider a generalized Lie symmetry that is generated by +gL := ρL(q, ˙q) · ∂ +∂q , +where ρL(q, ˙q) does not depend explicitly on time. Evolution along the path gives the total +time derivative +d +dt := ˙q · ∂ +∂q + ¨q · ∂ +∂ ˙q . +(2) +This in turn gives ˙ρL := dρL/dt, and the second-order prolongation vector [27], +pr gL := ρL · ∂ +∂q + ˙ρL · ∂ +∂ ˙q + ¨ρL · ∂ +∂¨q , +(3) +5 + +on the second-order jet space M(2) = {(t, q, ˙q, ¨q)} where this pr gL ∈ TM(2). +Under this generalized Lie symmetry, the action varies by +δS = +� t2 +t1 +pr gL +� +L(q(t), ˙q(t)) +� +dt, +with the requirement that ρL(q(t1), ˙q(t1)) = 0 = ρL(q(t2), ˙q(t2)). Then after an integration +by parts, +δS = +� t2 +t1 +ρL · +�∂L +∂q − d +dt +�∂L +∂ ˙q +�� +dt. +(4) +It is important to realize that the action may be evaluated along any path on Q. As such, +if gL generates a symmetry of the action, then Eq. (4) must vanish for all paths q(t) on Q, +and not just for those that minimize the action. +To make connection with the Lagrangian phase space approach used in the rest of the +paper, we make use of +E (q, ˙q) := ˙qa∂L (q, ˙q) +∂ ˙qa +− L (q, ˙q) , +along with +Mab (q, ˙q) := ∂2L (q, ˙q) +∂ ˙qa∂ ˙qb , +and +Fab (q, ˙q) := ∂2L (q, ˙q) +∂ ˙qa∂qb +− ∂2L (q, ˙q) +∂ ˙qb∂qa , +to express Eq. (4) as +δS = − +� t2 +t1 +ρa +L +�∂E +∂qa + Fab(q, ˙q) ˙qb + Mab(q, ˙q)¨qb +� +dt. +(5) +Here, Latin indices run from 1 to D, and Einstein’s summation convention is used. We then +arrive at our first result. +Lemma 1 An action S of a dynamical system has a generalized Lie symmetry generated by +gL if and only if there exists a ρL ∈ ker Mab such that +0 = ρa +L(q, ˙q) +�∂E +∂qa + Fab(q, ˙q) ˙qb +� +, +(6) +on TQ. +Proof. If gL generates a generalized Lie symmetry of S, then Eq. (5) must vanish for all +paths on Q. For an arbitrary path on Q the curvature of the path ¨q will not depend on either +the q(t) or the ˙q(t) for the path, however. As such, for δS = 0, it must be that ρa +LMab¨qb = 0 +for any choice of ¨q, and thus ρa +L ∈ ker Mab. +The remaining terms in Eq. (5) gives the +condition Eq. (6). +6 + +The set of all vector fields gL that satisfy Lemma 1 is denoted by gL, while pr gL := +{pr gL | gL ∈ gL} is the set of their prolongations. This pr gL is involutive [27], and the +conditions under which pr gL generates a generalized Lie symmetry group are given in [27]. +We see from Lemma 1 that if the action has a generalized Lie symmetry, then the +Lagrangian is necessarily singular, and as such the Lagrangian two-form ΩL will not have +maximum rank. +It is also important to note that while equations of the form Eq. (6) +often appear in the Lagrangian phase space description of mechanics [1], they appear as +Lagrangian constraints, conditions that must be imposed for evolution under the Euler- +Lagrange equations to be well defined. Here, Eq. (6) is not a constraint. Rather, because +the action must have this symmetry for all possible paths on Q, and since the set of all +possible paths cover Q, Eq. (6) is a condition on ρL that must be satisfied identically on all +of TQ—and thus, on the Lagrangian phase space—for gL to be a generator of the symmetry +group. We will see that not all the vectors in ker Mab satisfy the identity Eq. (6), however, +and thus not all of these vectors will generate a generalized Lie symmetry of the action. +B. +Symmetries of the Euler-Lagrange Equations of Motion +While in Section II A the focus was on arbitrary paths on the configuration space Q and +the symmetries of the action, in this section the focus is on the trajectories that minimizes +the action and the generalized Lie symmetries of them. These trajectories are solutions of +the Euler-Lagrange equations of motion, and for almost regular Lagrangians such solutions +form a family of curves. It is, in fact, the presence of this family of curves that gives rise to +the generalized Lie symmetry. The treatment here follows closely to that given in [1]. +For almost regular Lagrangians the solutions of the Euler-Lagrange equations of motion +Mab(q, ˙q)¨qb = −∂E +∂qa − Fab(q, ˙q) ˙qb, +(7) +are not unique. While for these Lagrangians the rank of Mab (q, ˙q) = D − N0—with N0 = +dim (ker Mab(q, ˙q))—is constant, this rank is not maximal, and thus Eq. (7) does not have +a unique solution for ¨q. Instead, for a chosen set of initial data (q0 = q(t0), ˙q0 = ˙q(t0)), the +solution to Eq. (7) results in a family of solutions that evolve from this (q0, ˙q0). As with the +paths in Section II A, these solutions are related to one another through a generalized Lie +symmetry [27]. +7 + +Following [27], the collection of functions +∆a(q, ˙q, ¨q) := ∂E(q, ˙q) +∂qa ++ Fab(q, ˙q) ˙qb + Mab(q, ˙q)¨qb, +(8) +defines a set of surfaces ∆a(q, ˙q, ¨q) = 0 on M(2), while the family of solutions to Eq. (7) +O (q0, ˙q0) := +� +q (t) | ∆a(q, ˙q, ¨q) = 0 with q (t0) = q0, ˙q (t0) = ˙q0 +� +, +that evolve from the same initial data (q0, ˙q0) gives the collection of trajectories that lie on +these surfaces. Indeed, for any two such solutions qa(t) and Qa(t) there exists a z(q, ˙q) ∈ +ker Mab(q, ˙q) such that ¨Qa − ¨qa = za. Importantly, because za depends on both q and ˙q, +the symmetry group that maps one member of O to another must be a generalized Lie +symmetry. We therefore take the generator of this symmetry group to be +g := ρ(q, ˙q) · ∂ +∂q , +with the corresponding the second-order prolongation vector for g being, +pr g := ρ · ∂ +∂q + ˙ρ · ∂ +∂ ˙q + ¨ρ · ∂ +∂¨q , +with this pr g ∈ TM(2). As with the above, the total time derivative is given by Eq. (2), +but unlike the analysis in Section II A, the evolution of the path—and indeed, for all the +trajectories in O(q0, ˙q0)—here is given by the Euler-Lagrange equations of motion. +The action of this prolongation on ∆a on the ∆a = 0 surface gives, +pr g [∆a(q, ˙q, ¨q)] = −∂¨qb +∂qaMbc(q, ˙q)ρc + d +dt +� +Fab(q, ˙q)ρb + Mab(q, ˙q) ˙ρb� +. +Since N0 > 0, ¨q is not unique on this surface, and yet g must generate the same symmetry +group for all the trajectories in O(q0, ˙q0). +Necessarily, ρ(q, ˙q) ∈ ker Mab(q, ˙q). +It then +follows that pr g[∆a(q, ˙q, ¨q)] = 0 if and only if (iff) there are constants ba such that ba = +Fabρb + Mab ˙ρb. The solutions in O(q0, ˙q0) all have the same initial data, however, and thus +necessarily ρ(q0, ˙q0) = 0 = ˙ρ(q0, ˙q0). We conclude that ba = 0. The following result, first +proved in [1], then follows. +Lemma 2 If g is a generalized infinitesimal symmetry of ∆a, then ρa(q, ˙q) ∈ ker Mab(q, ˙q), +and ˙ρa(q, ˙q) is a solution of +0 = Fab(q, ˙q)ρb(q, ˙q) + Mab(q, ˙q) ˙ρb(q, ˙q). +(9) +8 + +As before, we denote the set of all vector fields g that satisfy Lemma 2 by g, while +pr g := {pr g | g ∈ g} is the set of their prolongations. Once again pr g is involutive, and +the conditions under which pr g generates a generalized Lie symmetry group are given in +[27]. Note, however, that while ρ = 0 and ˙ρ = z for any z ∈ ker Mab(q, ˙q) is a solution of +Eq. (9), we require that ˙ρ = dρ/dt; these solutions cannot be generators of the generalized +Lie symmetry. Next, if ˙ρ is a solution of Eq. (9), then ˙ρa + z is a solution of Eq. (9) as well, +and thus these solutions are not unique. This, along with the previous observation, leads us +to generators that are constructed from equivalence classes of prolongations. Finally, Eq. (8) +gives for any z ∈ ker Mab(q, ˙q), +0 = za +�∂E +∂qa + Fab(q, ˙q) ˙qb +� +, +(10) +on the solution surface ∆a(q, ˙q, ¨q) = 0. If Eq. (10) does not hold identically, it must be +imposed, leading to Lagrangian constraints [5]. More importantly, because each q(t) ∈ O(u0) +must lie on the Lagrangian constraint submanifold, any symmetry transformation of q(t) +generated by pr g must give a path Q(t) that also lies on the constraint submanifold. +Not all vectors in pr g will be generators of the generalized Lie symmetry group for +O(u0). Determining which of these vectors are, and the relationship between the generators +of symmetries of the Euler-Lagrange equations of motion and those of the action, is best +done within the Lagrangian phase space framework. To accomplish this, we will need the +following generalization of Lemma 2. +Consider the vector +k := c · ∂ +∂q + ˙c · ∂ +∂ ˙q , +with a c ∈ ker Mab(q, ˙q) along with the quantity +la := Fabcb(q, ˙q) + Mab ˙cb(q, ˙q). +After an integration by parts, +la = cb(q, ˙q) +� +Fab(q, ˙q) − d +dt +∂2L +∂ ˙qa∂ ˙qb +� +, += cb(q, ˙q) +� +Fab(q, ˙q) − +�d +dt , ∂ +∂ ˙qa +� ∂L +∂ ˙qb − ∂ +∂ ˙qa +�d +dt +∂L +∂ ˙qb +�� +. +Using Eq. (2) we have +�d +dt , ∂ +∂ ˙qa +� ∂L +∂ ˙qb = − ∂2L +∂qa∂ ˙qb − ∂¨qc +∂ ˙qa +∂2L +∂ ˙qc∂ ˙qb. +9 + +As q(t) is a solution of the Euler-Lagrange equations of motion, we find that +la = cb(q, ˙q) +� +Fab(q, ˙q) + +∂2L +∂qa∂ ˙qb − +∂2L +∂ ˙qa∂qb ++ ∂¨qc +∂ ˙qa +∂2L +∂ ˙qc∂ ˙qb +� +. +This last expression vanishes after the definition of Fab(q, ˙q) is used along with the require- +ment that c ∈ ker Mab(q, ˙q). We then have the following result. +Lemma 3 For any vector +k = c · ∂ +∂q + ˙c · ∂ +∂ ˙q , +such that c ∈ ker Mab, +0 = Fabcb(q, ˙q) + Mab ˙cb(q, ˙q). +. +III. +GENERATORS OF THE GENERALIZED LIE SYMMETRY FOR THE EULER- +LAGRANGE EQUATIONS OF MOTION +The generators of the generalized Lie symmetry for both the Euler-Lagrange equations +of motion and the action are best found using the Lagrangian phase space approach to +mechanics. This phase space and its concomitant mathematical structure provide the tools +needed to determine both the generators of the symmetry and the solutions to the energy +equation on which they act. For the Euler-Lagrange equations of motion this determination +was done in [1]. In this section we will review the Lagrangian phase space approach, establish +the notation used in this paper, and summarize the results obtained in [1] that are needed +here. (We will also take the opportunity to correct typographical errors made in [1].) Proofs +of the majority of the assertions listed in this section will not be given; the reader is instead +referred to [1] where the proofs and the context of their development can be found. +A. +The Lagrangian Phase space +For a configuration space Q the Lagrangian phase space PL is the tangent space +PL = TQ, with the coordinates on PL denoted as u = (q1, . . . , qD, v1, . . . vD). Integral flows +on PL, t ∈ [t0, ∞) → u(t) ∈ PL [28], for a set of initial data u0 = (q0, v0) are given as +solutions to +du +dt := X(u), +10 + +where X is a smooth vector field in TPL = T(TQ). The two tangent spaces TQ and TPL +have the bundle projections: τQ : TQ → Q and τTQ : T(TQ) → TQ. They can be used +to construct two other projection maps: τQ ◦ τTQ : T(TQ) → Q and the prolongation of +τTQ to T(TQ) (see [3] and [28]). This prolongation is the map TτQ : T(TQ) → TQ, and is +defined by requiring that the two maps τQ ◦ τTQ and τQ ◦ TτQ map any point in T(TQ) to +the same point in Q. The vertical subbundle [TPL]v of T(TQ) is [TPL]v = ker TτQ [3]; +a Xv ∈ [TuPL]v above a point u ∈ PL is called a vertical vector field. The horizontal +subbundle [TPL]q of T(TQ) is [TPL]q = Image TτQ; a Xq ∈ [TuPL]q is called a horizontal +vector field. Consequently, each X ∈ TuPL consists of a Xq ∈ [TuPL]q and a Xv ∈ [TuPL]v +with X = Xq + Xv. In terms of local coordinates, +Xq := Xqa ∂ +∂qa, +and +Xv := Xva ∂ +∂va . +Of special interest is the second order Lagrangian vector field XL. This vector field is the +particular solution of Eq. (1) for which TτQ ◦ XL is the identity on TQ (see [28]). In terms +of local coordinates +XL = va ∂ +∂qa + Xva ∂ +∂va. +The space of one-forms on PL is the cotangent space T∗PL. For a one-form α ∈ T∗ +uPL, +and a vector field X ∈ TuPL, the dual prolongation map T∗τQ is defined as +⟨α|TτQX⟩ = ⟨T∗τQα|X⟩, +after a useful adaptation of Dirac’s bra and ket notation. In addition, for a general k-form +ω in the k-form bundle Λk (PL), +ω (x) : Y1 ⊗ · · · ⊗ Yk → ⟨ω (x)| Y1 ⊗ · · · ⊗ Yk⟩ ∈ R, +with Yj ∈ TuPL for j = 1, . . . , k. The vertical one-form subbundle [T∗PL]v of T∗PL +is [T∗PL]v := ker T∗τQ; a αv ∈ [T∗ +uPL]v is called a vertical one-form. The horizontal +one-form subbundle [T∗PL]q of T∗PL is [T∗PL]q = Image T∗PL; a αq ∈ [T∗ +uQ]q is called +a horizontal one-form. +Each one-form ϕ ∈ T∗ +uPL consists of a ϕq ∈ [T∗ +uPL]q and a +ϕv ∈ [T∗ +uPL]q such that ϕ = ϕq + ϕv. In terms of local coordinates ϕq := ϕqa dqa and +ϕv := ϕvadva. +Following [3, 4], the Lagrangian two-form is defined as ΩL := −ddJL, where dJ is the +vertical derivative (see [3]). This two-form can be expressed as ΩL := ΩF + ΩM such that +11 + +for any X, Y ∈ TuPL. +ΩF(X, Y) := ΩL(TτQX, TτQY), +and is thus the horizontal two-form of ΩL. As ΩM(X, Y) = ΩL(X, Y) − ΩF(X, Y), ΩM +is then a mixed two-form of ΩL. In terms of local coordinates, +ΩL = −dθL, +where +θL := ∂L +∂va dqa, +while +ΩF := 1 +2Fabdqa ∧ dqb, and ΩM := Mabdqa ∧ dvb. +For regular Lagrangians XL is the unique solution of Eq. (1). For almost regular La- +grangians, on the other hand, this solution is not unique, but instead depends on +ker ΩL (u) := {K ∈ TuPL | iKΩL = 0} . +From Section II we expect this kernel to play a role in determining the generators of the +generalized Lie symmetry of both the Euler-Lagrange equations of motion and the action. +Indeed, consider the natural isomorphism iso : (t, q, ˙q, ¨q) ∈ M(2) → (t, q, v, Xva +L ) defined in +[1], and the prolongation pr g of a generator g ∈ g of a generalized Lie symmetry of the +Euler-Lagrange equations of motion. This pr g contains the vector +k = ρ · ∂ +∂q + ˙ρ · ∂ +∂ ˙q . +The collection of all such vectors has been shown to be involutive (see [1]). The isomorphism +maps iso : k → k′ where +k′ = ρ · ∂ +∂q + ˙ρ · ∂ +∂v . +Then k′ ∈ TPL, and from Lemma 2, k′ ∈ ker ΩL(u) as well. A similar result holds for the +generators in gL after Lemma 1 and Lemma 3 are used. +The two-form ΩL gives the lowering map Ω♭ +L : TuPL → T∗ +uPL, with Ω♭ +LX := iXΩL. This +map consists of Ω♭ +L = Ω♭ +F + Ωv♭ +M + Ωq♭ +M, with Ω♭ +F : X ∈ TuPL → [T∗ +uPL]q; Ωq♭ +M : X ∈ TuPL → +[T∗ +uPL]q; and Ωv♭ +M : X ∈ TuPL → [T∗ +uPL]v. In terms of local coordinates, Ω♭ +FX = FabXqadqb, +Ωq♭ +MX = −MabXvadqb, and Ωv♭ +MX = MabXqadvb. +For almost regular Lagrangians ker Ωv♭ +M = C ⊕[TuPL]v while ker Ωq♭ +M = [TuPL]q ⊕G. Here +C := {C ∈ [TqPL]q | iCΩM = 0} , +12 + +and +G := {G ∈ [TqPL]v | iGΩM = 0} . +As Mab(u) has constant rank on PL, there exists a basis, +� +z(n) (u) = +� +z1 +(n) (u) , . . . , zD +(n) (u) +� +| Mab (u) zb +(n) (u) = 0, n = 1, . . . , N0 +� +, +for ker Mab (u) at each u ∈ PL. Spans of both +C = span +� +Uq +(n) = z(n) · ∂ +∂q , n = 1, . . . , N0 +� +, and +G = span +� +Uv +(n) = z(n) · ∂ +∂v , n = 1, . . . , N0 +� +, +can then be constructed. Importantly, G is involutive [5], and when the rank of ΩL(u) is +constant on PL, ker ΩL(u) is involutive as well. +Corresponding to Uq +(n) and Uv +(n) we have the one-forms Θ(m) +q +and Θ(m) +v +where ⟨Θ(m) +q +|Uq +(n)⟩ = +δ(m) +(n) and ⟨Θ(m) +v +|Uv +(n)⟩ = δ(m) +(n) . Then [TuPL]q = C ⊕ C⊥ and [TuPL]v = G ⊕ G⊥, where +C⊥ := +� +X ∈ [TuPL]q | +� +Θ(n) +q +�� X +� += 0, n = 1, . . . , N0 +� +, and +G⊥ := +� +X ∈ [TuPL]v | +� +Θ(n) +v +�� X +� += 0, n = 1, . . . , N0 +� +. +The vectors that lie in ker ΩL(u) can be determined by using the reduced matrix Fnm := +za +(n)Fabzb +(m) to define +C := +� +C ∈ C +���� +N0 +� +m=1 +¯FnmC +(m) = 0 +� +⊂ C. +Then, +Theorem 4 The vectors K = Kq + Kv ∈ ker ΩL are given by, +Kq = C, +Kv = G+�C, +where, C ∈ C, G ∈ G, and �C ∈ G⊥ is the unique solution of Mab �Cb = −FabC +b. +We found in [1] that dim (ker ΩL (u)) = N0 + ¯D, where +¯D := dim C ≤ N0 (see [1] for +proof). However, the results of Lemma 3 show that we can construct from any vector +Uq ∈ C a vector that lies in the ker ΩL(u), and as dim (C) = N0, it follows that dim +(ker ΩL(u)) = 2N0. +13 + +B. +First-order Lagrangian constraints +For singular Lagrangians solutions of the energy equation XE are not unique. It is well +known that they also do not, in general, exist throughout PL, but are instead confined to a +submanifold of the space given by Lagrangian constraints. +With XE = Xq +E + Xv +E, it is convenient to use the one form Ψ +Ωq♭ +MXv +E = Ψ. +constructed from the energy equation. The first-order constraint functions are then +γ[1] +n := +� +Ψ| Uq +(n) +� += 0 for n = 1, . . . , N0. In terms of local coordinates, +γ[1] +n = Uqa +(n) +� ∂E +∂qa + Fabvb +� +. +They may also be expressed [3, 4] as γ[1] +n += ⟨dE|P(n)⟩ = P(n)E for any basis {P(n)} of +ker ΩL(u) for which ⟨Θ(m) +q +|P(n)⟩ = δ(m) +(n) . In general, γ[1] +n +̸= 0 on PL. Instead, the con- +dition γ[1] +n += 0 must be imposed, and this in turn defines a set of submanifolds of PL +given by the collection C[1] +L +:= +� +γ[1] +1 , . . . , γ[1] +N0 +� +. +The collection of these surfaces, P[1] +L +:= +� +u ∈ PL | γ[1] +n (u) = 0 , n = 1, . . . , N0 +� +is called the first-order Lagrangian constraint +submanifold, and has dim P[1] +L = 2D−I[1]. Here I[1] is the number of independent functions +in C[1] +L with I[1] = rank +� +dγ[1] +n +� +≤ N0. +The constraint one-form +β[XE] := dE − iXEΩL, +was introduced in [1] with the condition β[XE] = 0 giving both the solution of the energy +equation and the submanifold P[1] +L . As ⟨β|Uq +(n)⟩ = γ[1] +n , this β[XE] can also be expressed as +β[XE] = +N0 +� +n=1 +γ[1] +n Θ(n) +q . +(11) +C. +The Generalized Lie Symmetry Group for the Euler-Lagrange Equations of +motion +The generalized Lie symmetry group for O(u0) is determined using +ker ΩL(u) := {P ∈ ker ΩL(u) | [G, P] ∈ [TuPL]v ∀ G ∈ G}, +(12) +14 + +along with the following collection of functions on PL, +F := {f ∈ C∞on PL | Gf = 0 ∀ G ∈ G}. +This ker ΩL(u) is also involutive. +The following results were proved in [1]. +Lemma 5 Let X ∈ TuPL and G ∈ G such that [G, X] ∈ ker ΩL(u). Then [G, X] ∈ [TuPL]v +iff [G, X] ∈ G. +It then follows that [G, P] ∈ G for all P ∈ ker ΩL(u). +As G is involutive and as G ⊂ +ker ΩL(u), G ⊂ ker ΩL(u) as well, and thus G is an ideal of ker ΩL(u). +Lemma 6 There exists a choice of basis for ker ΩL(u) that is also a basis of ker ΩL(u). +As G is an ideal of ker ΩL(u), we may define for any P1, P2 ∈ ker ΩL(u) the equivalence +relation: P1 ∼ P2 iff P1 − P2 ∈ G. The equivalence class, +[P] := {Y ∈ ker ΩL(u) | Y ∼ P}, +(13) +can be constructed along with the quotient space ker ΩL(u)/G. (For the sake of notational +clarity we will suppress the square brackets for equivalence classes when there is no risk of +confusion.) This space is a collection of vectors that lie in the kernel of ΩL, but with the +vectors in G removed; ker ΩL(u)/G thereby addresses the first two observations listed at the +end of Section II B. +We now turn our attention to the third observation. Because the integral flow uX(t) of +any solution X of the energy equation must lie on P[1] +L , a symmetry transformation of uX(t) +must result in an integral flow uY(t) of another solution Y of the energy equation, which +must also lie on P[1] +L . Implementing this condition is done through β[XE]. +As ⟨β[XE]|G⟩ = ⟨dE|G⟩ = GE = 0 for all G ∈ G on P[1] +L , the Lie derivative LG of β +along G is, +LGβ[XE] = +N0 +� +n=1 +� +Gγ[1] +n +� +Θ(n) +q . +Given a P(n) ∈ ker ΩL(u) such that P(n) = Uq +(n) + �U(n) + G′ with G′ ∈ G, Gγ[1] +n += +[G, P(n)]E + P(n)GE. But G is an ideal of ker ΩL(u), and thus Gγ[1] +n = 0 on the first-order +constraint manifold. It follows that LGβ = 0 on P[1] +L . The collection of vectors, +Sym := +� +P ∈ ker ΩL(u)/G | LPβ[XE] = d⟨β[XE]|P⟩ on P[1] +L +� +, +15 + +is therefore well defined, and is involutive. It follows that P ∈ Sym iff ⟨dβ[XE]|P ⊗ X⟩ = 0 +for all X ∈ TPL. We are then able to construct from each P ∈ Sym a one-parameter +subgroup σP(ǫ, x) defined as the solution to +dσP +dǫ := P (σP) , +where σP(0, u) = u for u ∈ PL. The collection of such subgroups with give the Lie group +GrSym. +D. +Euler-Lagrange Solutions of the Energy Equation +We denote the set of general solutions to the energy equation as +Sol := {XE ∈ TuPL | iXEΩL = dE on P[1] +L }. +If u(t) is the integral flow of a vector in Sol whose projection onto Q corresponds to a solution +of the Euler-Lagrange equations of motion, then GrSym must map one of such flows into +another one. However, while LGXL = [G, XL] ∈ ker ΩL(u), in general LGXL /∈ G. The +action of σP on the flow uXL will in general result in a flow uY generated by a Y that is +not a SOLVF. It need not even be a solution of the energy equation. By necessity, general +solutions of the energy equation must be considered, leading us to consider the collection of +solutions +Sol := {XEL ∈ Sol | [G, XEL] ∈ [TuPL]v +∀G ∈ G}. +This collection generates the family of integral flows +OEL(u0) := +� +u(t) +���� +du +dt = XEL(u), XEL ∈ Sol, and u(t0) = u0 +� +. +Importantly, if P ∈ Sym, then +i[XE,P]ΩL = iPdβ[XE] = 0. +As such, we find that +Lemma 7 [XEL, P] ∈ ker ΩL(u) for all P ∈ Sym. +It then follows that +16 + +Theorem 8 GrSym forms a group of symmetry transformations of OEL(u0). +Proof of both assertions can be found in [1]. +The generators of the generalized Lie symmetry for OEL(u0) are thus given by Sym. The +corresponding solutions to the Euler-Lagrange equations that have this symmetry are given +by Sol, and a vector XEL ∈ Sol is called a second-order, Euler-Lagrange vector field +(SOELVF). It has the general form, +XEL = XL + +N0 +� +m=1 +um(u) +� +P(m) +� +, +(14) +where um(u) ∈ F and {[P(n)], n = 1, . . . , N0} is a choice of basis for ker ΩL(u)/G. The +vector field XL is constructed from the second order Lagrangian vector field XL and vectors +in ker ΩL(u) by requiring XL ∈ Sol. This construction is described in [1]; we will only need +the existence of such a vector field in this paper. +IV. +GENERALIZED LIE SYMMETRIES OF THE ACTION AND ITS IMPACT +ON DYNAMICS +We now turn our attention to the generators of the generalized Lie symmetry of the +action, and the impact this symmetry has on the evolution of dynamical systems. +A. +The Generalized Lie Symmetry of the Action +In determining the conditions (as listed in Lemma 1) under which the action admits a +generalized Lie symmetry, the understanding that the action must have this symmetry for +all possible paths on Q played an essential role. By necessity, these conditions could only be +placed on ρL, and not on ˙ρL; unlike ρL, ˙ρL depends explicitly on the evolution of a particular +path, while the symmetry must hold for all paths. We note, however, that the family OEL +of trajectories determined by the Euler-Lagrange equations of motion also consists of paths +on Q, and as such the generalized Lie symmetry of the action is also a symmetry of OEL. +Importantly, how these trajectories evolve with time is known, and as such, the ˙ρL for a +given ρL is also known for these trajectories. With this understanding, and after comparing +Lemma 1 and the results of Lemma 3 with Lemma 2, we conclude that the generators +17 + +of the generalized Lie symmetry of the action must also be generators of the generalized Lie +symmetry of the Euler-Lagrange equations of motion. This leads us to consider the following +collection of vectors. +SymL = {P ∈ ker ΩL(u)/G | γ[1] +P = ⟨β|P⟩ = 0 on PL}. +We will also need NSymL = dim (SymL) in the following. +Lemma 9 SymL ⊂ Sym. +Proof. +Let {P(l), l = 1, . . . , N0} be a basis of ker ΩL(u)/G such that P(l) ∈ SymL for +l = 1, . . . , NSymL. We may choose the basis of C such that ⟨Θ(m) +q +|P(l)⟩ = δ(m) +(l) . Then for any +P(n) ∈ Sym, we see from Eq. (11) that, +⟨dβ|P(n)⊗Y⟩ = +N0 +� +m=1 +� +⟨dγ[1] +m |P(n)⟩⟨Θ(m) +q +|Y⟩ − ⟨dγ[1] +m |Y⟩⟨Θ(m) +q +|P(n)⟩ + γ[1] +m ⟨dΘ(m) +q +|P(n) ⊗ Y⟩ +� +, +for any Y ∈ TPL. The last term vanishes on the first-order constraint manifold P[1] +L , while +for the second term, ⟨dγ[1] +m |Y⟩⟨Θ(m) +q +|P(n)⟩ = ⟨dγ[1] +n |Y⟩δ(m) +(n) . But as P(n) ∈ SymL, γ[1] +n = 0 on +PL, and this term vanishes as well. Finally, for the first term, ⟨dγ[1] +m |P(n)⟩ = P(n)P(m)E = +[P(n), P(m)]E +P(m)P(n)E. But γ[1] +n = P(n)E = 0 on PL, while ker Ω(u) is involutive. There +then exists a P(nm) ∈ ker Ω(u) such that P(nm) = [P(n), P(m)]. As P(nm)E := γ[1] +(nm), this +γ[1] +(nm) must be a linear combination of first-order constraint functions, and they also vanish +on P[1] +L . It then follows that ⟨dβ|P(n) ⊗ Y⟩ = 0 on P[1] +L , and P(n) ∈ Sym. +If P1, P2 ∈ SymL, then γ[1] +[P1,P2] = P1P2E − P2P1E = P1γP2 − P2γP1 = 0, and thus +SymL is involutive. +Then for each P ∈ SymL we once again have the one-parameter +subgroup σ +SymL +P +(ǫ, u) define as the integral flow of +dσSymL +P +dǫ +:= P, +with σ +SymL +P +(0, u) = u for u ∈ PL. The collection of such subgroups gives the Lie group +GrSymL. +As SymL ⊂ Sym, GrSymL is a Lie subgroup of GrSym. +It then follows from +Theorem 8 that GrSymL also forms a group of symmetry transformations of OEL(u0). As +the family OEL(u0) of trajectories are paths on Q, and as the symmetry transformation of +the action must be the same for all paths on Q, it also follows that, +Theorem 10 GrSymL forms the group of symmetry transformations of the action S. +18 + +B. +Symmetries and Dynamics +While OEL(u0) gives the family of integral flows on which both GrSym and GrSymL act, a +general flow in OEL(u0) need not be confined to P[1] +L , and yet this is the submanifold on which +the solutions XEL ∈ mathcalSol of the energy equations exist. In such cases it is necessary +to jointly choose a SOELVF XEL and a submanifold of P[1] +L on which the resultant flow uXEL +will be confined. This is done through the implementation of a constraint algorithm, one of +which was proposed in [1]. In that paper the product of this algorithm was the most that +could be said about the general structure of SOELVFs that have integral flow fields which +lie on P[1] +L . Here, with the results obtained in Section IV A, we can say much more, and +we will see that the presence of a generalized Lie symmetry of the action greatly restricts +the structure of the SOELVFs that such systems can have. +Following [1], we introduce for a XEL ∈ Sol the notation +X +[1] +EL := XEL, X +[1] +L := XL, P[1] +(n) := P(n), um +[1] := um, N[1] +0 +:= N0, +when the constraint algorithm is implemented, with the superscript [1] denoting the first +iteration of this algorithm. (This notation is only used in this section.) In addition, we +choose P[1] +(n) ∈ SymL for n = 1, . . . , NSymL. +For the integral flow field of XEL to lie on P[1] +L , +LXELβ = 0, +(15) +which reduces to LXELγ[1] +n = 0 on P[1] +L . This is called the constraint condition. As both +un +[1], γ[1] +n ∈ F, +� +P[1] +(n) +� +γ[1] +m = P(n)γ[1] +m , and after making use of the general form of a SOELVF +given in Eq. (14), Eq. (15) reduces to +N0 +� +m=1 +Γ[1] +nmum +[1] = − +� +dγ[1] +n +���X +[1] +L +� +, with Γ[1] +nm := +� +dγ[1] +n +���P[1] +(m) +� +. +(16) +Since +� +dγ[1] +n +���P[1] +(m) +� += P[1] +(m)P[1] +(n)E = [P[1] +(m), P[1] +(n)]E + Γ[1] +mn. But ker ΩL(u) is involutive, and +thus [P[1] +(m), P[1] +(n)]E is a linear combination of first-order Lagrangian constraints. As these +constraints vanishes on P[1] +L , Γ[1] +nm = Γ[1] +mn on the first-order constraint manifold. +Next, when n = 1, . . . , NSymL, P[1] +(n) ∈ SymL, and γ[1] +n += 0. +Thus, Γ[1] +nm = 0 when +n = 1, . . . , NSymL, and as Γ[1] +nm is a symmetric matrix on P[1] +L , Γ[1] +mn = 0 for these values of n +as well. Thus while Γ[1] +nm is a N0 × N0 matrix, the only nonzero components of this matrix +19 + +lie in the +� +N0 − NSymL +� +× +� +N0 − NSymL +� +submatrix ¯Γ[1] +n m := +� +dγ[1] +n+NSymL +����P[1] +m+NSymL +� +where n, m = 1, . . . , N0 − NSymL. As +� +dγ[1] +n +���X +[1] +L +� += 0 as well when n = 1, . . . , NSymL, +Eq. (16) reduces to +N0−NSymL +� +m=1 +¯Γ[1] +n ¯mu +m+NSymL +[1] += − +� +dγ[1] +n+NSymL +���X +[1] +L +� +. +(17) +It is then readily apparent that the NSymL arbitrary functions um +[1] for m = 1, . . . , NSymL +are not determined at this iteration, while r[1] = rank ¯Γ[1] +n m of the um +[1] for m > NSymL are. +There are then N[2] +0 +:= N[1] +0 − r[1] second-order Lagrangian constraint functions +γ[2] +n[2] := +� +dγ[1] +n[2] +���X +[1] +L +� +, n[2] = 1, · · · , N[2] +0 , +with the conditions γ[2] +n[2] = 0 imposed if necessary. +In general there will be I[2] := +rank +� +dγ[1] +n[1], dγ[2] +n[2] +� +independent functions in C[2] +L +:= C[1] ∪ +� +γ[2] +n[2] | n[2] = 1, . . . , N[2] +0 +� +, +and P[1] +L is reduced to the second-order constraint submanifold, +P[2] +L := +� +u ∈ P[1] +L +��� γ[2] +[n2](u) = 0, n[2] = 1, . . . , N[2] +0 +� +, +where dim P[2] +L = 2D−I[2]. At this point, there are two possibilities. If I[2] = I[1] or I[2] = 2D, +the iterative process stops, and no new Lagrangian constraints are introduced. If not, the +process continues. +For the second iteration in the constraint algorithm, we choose a basis +� +P[2] +(n) +� +for +ker ΩL(u)/G and the arbitrary functions +� +um +[2] +� +such that for m = 1, . . . , N[2] +0 , um +[2] are linear +combinations of um +[1] that lie in the kernel Γ[1] +nm. We once again require that P[2] +(n) ∈ SymL for +n = 1, . . . , NSymL. Then +X +[2] +EL = X +[2] +L + +N[2] +0 +� +m=1 +um +[2] +� +P[2] +(m) +� +, +with +X +[2] +L = X +[1] +L + +N[1] +0 +� +m=N[2] +0 +1 +um +[2] +� +P[2] +(m) +� +. +Here, the functions um +[2] for m = N[2] +0 + 1, . . . , N[1] +0 +have been determined through the con- +straint analysis of γ[1] +n . +As shown in [1], Gum +[1] = 0. Similarly, Gγ[2] +n = L[G,XEL]dγ[2] +n = 0. Clearly γ[2] +n ∈ F and +we may require um +[2] ∈ F as well. It then follows that +� +P[2] +(n) +� +γ[2] +m = P[2] +(n)γ[2] +m , and imposing +20 + +Eq. (15) on γ[2] +n , gives +N[2] +0 +� +m=1 +Γ[2] +nmum +[2] = − +� +dγ[2] +n +���X +[2] +L +� +, where Γ[2] +nm := +� +dγ[2] +n +���P[2] +(m) +� +, n = 1, . . . , N[2] +0 . +(18) +Once again, Γ[2] +nm = Γ[2] +mn, but now on the constraint manifold P[2] +L . Moreover, since γ[2] +n += +γ[1] +n = 0 for n = 1, . . . , NSymL, Γ[2] +nm = 0 = Γ[2] +mn, and +� +dγ[2] +n +���X +[2] +L +� += 0. There is once again a +reduction of Eq. (18), and we are left with +N[2] +0 −NSymL +� +m=1 +Γ +[2] +n mu +m+NSymL +[2] += −⟨dγ[2] +n+NSymL|X +[2] +L ⟩. +where ¯Γ[2] +n m := +� +dγ[2] +n+NSymL +����P[2] +(m+NSymL) +� +. As before, the NSymL arbitrary functions um +[2] +are not determined, while r[2] := rank ¯Γ[2] +n m of the remaining um +[2] for m > NSymL are. There +are now N[3] +0 += N[2] +0 − r[2] third-order Lagrangian constraint functions, +γ[3] +n[3] = +� +dγ[2] +n[3] +���X +[2] +L +� +, n[3] = 1, . . . , N[3] +0 , +with the conditions γ[3] +n[3] = 0 imposed if necessary. With +I[3] := rank +� +dγ[1] +n[1], dγ[2] +n[2], dγ[3] +n[3] +� +, +independent functions in C[3] +L := C[2] +L ∪ +� +γ[3] +n[3], n[3] = 1, . . . , N[3] +0 +� +, we now have the third- +order constraint submanifold, +P[3] +L := +� +u ∈ P[2] +L +��� γ[3] +n[3](u) = 0, n[3] = 1, . . . , N[3] +0 +� +. +Once again, the process stops when I[3] = I[2] or I[3] = 2D. However, if I[2] < I[3] < 2D, the +process continues until at the nF-iteration when either I[nF ] = I[nF ]−1 or I[nF ] = 2D. +Following [1], the end result of this algorithm is +1. A submanifold P[nF ] +L +⊂ PL on which dynamics takes place. +2. A collection C[nF ] +L +⊂ F of constraint functions of order 1 to nF. +3. A second-order, Euler-Lagrange vector field +X +[nF ] +EL = X +[nF ] +L ++ +N +[nF ] +0� +m=1 +um +[nF ](u) +� +P[nF ] +(m) +� +, +21 + +with N[nF ] +0 +≥ NSymL arbitrary functions um +[nF ](u) ∈ F for m = 1, . . . , N[nF ] +0 +, and +X +[nF ] +L += X +[1] +L + +N[1] +0 +� +m=N +[nF ] +0 ++1 +um +[nF ](u) +� +P[nF ] +(m) +� +, +where the N[1] +0 +− N[nF ] +0 +functions um +[nF ](u) ∈ F, m = N[nF ] +0 ++ 1, . . . , N[1] +0 , have been +uniquely determined through the constraint algorithm. +We assume that the rank of Γ[l] +nm is constant on PL for each l = 1, . . . , nF, and that P[nF ] +L +is +non-empty. +The end result of the constraint algorithm X +[nF ] +EL is still a SOELVF, and we define the +collection of such vector fields as +SolP[nF ] +L +:= {XEL ∈ Sol | LXELβ = 0}. +Importantly, dim SolP +[nF ] +L +≥ NSymL. +V. +THE GENERALIZED LIE SYMMETRIES OF THREE DYNAMICAL SYS- +TEMS +Three examples of dynamical systems with almost regular Lagrangians were introduced +in [1]. In that paper the focus of these examples was on the explicit construction of the +dynamical structures needed to describe and predict motion in the Lagrangian phase space, +and to show that these structures are projectable to the Hamiltonian phase space. We return +to these examples here, but with the focus now being on the generalized Lie symmetries of +each, and the application of the results we have found in this paper. In particular, we are in +interested in the dimensionality of the symmetry groups for each of the systems as compared +to the dimensionality of SolP +[nF ] +L +of each. A summary of our results can be found in Table I +A. +A Lagrangian With and Without a Generalized Lie Symmetry +Whether the action +S1 := +� � +1 +2m +�d�q +dt +�2 +− V (qa) +� +dt, +22 + +with |q| = √qaqa and �qa := qa/|q|, a = 1, . . . , D, has a generalized Lie symmetry depends on +the choice of potential V (q). With one choice both the Lagrangian and the Euler-Lagrange +equations of motion have a generalized gauge symmetry; with a second choice the equations +of motion has a generalized Lie symmetry while the Lagrangian does not; and with a third +choice neither the action nor the equations of motion have a symmetry. Irrespective of the +choice of V (q), however, L is singular, demonstrating that while all actions with a generalized +Lie symmetry have a singular Lagrangian, not all singular Lagrangians have a generalized +Lie symmetry. +Defining Πab(q) := δab − �qa�qb, we find +ΩM = m +|q|2Πab(q)dqa ∧ dvb, +ΩF = m +|q|3 (�q · dq) ∧ (v · Π(q) · dq) . +Then C and G are spanned by Uq +(1) = �q · ∂/∂q and Uv +(1) = �q · ∂/∂v, respectively, while +ker ΩL(u) is spanned by Uv +(1) and +P(1) = �q · ∂ +∂q + 1 +|q|v · ∂ +∂v . +That dim (ker ΩL(u)/G) = 1 then follows. +The energy is +E = 1 +2 +m +|q|2v · Π(q) · v + V (q), +and there is only one first-order Lagrangian constraint, +γ[1] = Uq +(1)V, +(19) +so that β[XEL] = γ[1]Θ(1) +q , where Θ(1) +q += �q · dq. Using Eq. (19), +LP(1)β = d +� +Uq +(1)V +� +− 1 +|q|2 �q · ∂ +∂q +� +Π b +a (q)∂V +∂�qb +� +dqa. +(20) +Whether or not Sym or SymL is empty therefore depends on the symmetries of V (q), as +we would expect. +It was found in [1] that +XL = v · Π(q) · ∂ +∂q + (�q · v) +|q| +v · ∂ +∂v − |q|2 +m +∂V +∂q · Π(q) · ∂ +∂v , +and a general SOELVF is given by XEL = XL + u(u) +� +P(1) +� +, where u(u) ∈ F. +As the +constraint algorithm gives +LXELγ[1] = v · Π · ∂γ[1] +∂q ++ u(u)Uq +(1)γ[1], +(21) +23 + +whether or not u(u) (which in turn determines the dimensionality of SolP +[nf ] +L ) is determined +by the constraint condition also depends on the symmetries of V (q). +There are three cases to consider. +The symmetric potential +For P(1) to generate a generalized Lie symmetry of the Euler-Lagrange equations of +motion, +0 = +1 +|q|2 �q · ∂ +∂q +� +Π b +a (q)∂V +∂�qb +� +, +and as such the potential must satisfy +∂V +∂�qa = ∂VAS(�qa) +∂�qa +, +where VAS is a function of �qa only. It follows that P(1) generates a generalized Lie symmetry +iff V (qa) = VSph(|q|) + VAS(�qa), where VSph is a function of |q| only. For this potential, Sym +is one-dimensional, and is spanned by P(1). +The constraint condition Eq. (21) for this potential reduces to +0 = u(u)d2VSph(q) +d|q|2 +, +which must be satisfied on P[1] +L . There are two possibilities. +Case 1: +d2VSph +d|q|2 += 0. +Then VSph(|q|) = a|q| + b, but since +γ[1] = dVSph +d|q| = a, +the condition γ[1] = 0 requires a = 0. It then follows that γ[1] = 0 on PL, and thus SymL is +one-dimensional; it also is spanned by P(1). The potential is then V (q) = b + VAS(�qa), and +the Lagrangian is invariant under the transformation qa → αqa, where α is an arbitrary, +nonvanishing function on PL. This Lagrangian therefore has a local conformal symmetry. +Importantly, the function u(u) is not determined, and thus the dynamics of the particle is +given only up to an arbitrary function. Then dim (SolP +[nF ] +L +) = 1 as well, and is also spanned +by P(1). +Case 2: +d2VSph +d|q|2 +̸= 0. +24 + +In this case u(u) = 0, and the dynamics of the particle is completely determined by its +initial data; SolP +[nF ] +L += {XL}. The first-order Lagrangian constraint γ[1] does not vanish +automatically, but instead defines a surface on PL, and it follows that SymL = ∅. Indeed, +the action’s lack of a local gauge symmetry in this case can be seen explicitly. +Equation (19) reduces to +0 = �q · ∂Vsph +∂q , +and for dynamics to be possible the set of solutions +� +Ri ∈ R +����� +dVSph +d|q| +����� +Ri += 0 +� +, +must be non-empty. Dynamics are on the surfaces |q|−Ri = 0 where the potential reduces to +V (q) = VSph(Ri) + VAS(�qa). This reduced potential has the same symmetry as the potential +VAS(�qa) in Case 1, and it is for this reason that the Euler-Lagrange equations of motion +have the same generalized Lie symmetry for the two cases. This is explicitly shown in the +appendix. +In Case 1 the action has a local conformal symmetry, while in Case 2 it does not. (In +[1] it was erroneously stated that in this case the action has a global rotational symmetry.) +The Lagrangian for the two cases do not have the same invariances, resulting in one case +dynamics that are determined only up to an arbitrary u(u), and in the other case to a +u(u) = 0 and dynamics that are instead completely determined by the choice of initial data. +The asymmetric potential +For a general V the second term in Eq. (20) does not vanish, P(1) does not generate a +symmetry of the equations of motion, and Sym = {∅}. As before, γ[1] does not vanish, and +thus SymL = {∅} as well. Furthermore, as Eq.(21) results in +XE = XL − +v · Π · ∂γ[1] +∂q +q2Uq +(1)γ[1] [P(1)], +the dynamics of the particle is uniquely determined by its initial data, and SolP +[nF ] +L += {XEL} +once again consists of a single point. +25 + +B. +A Lagrangian with Local Conformal Symmetry +The action, +S2 := +� � +1 +2m +�d�q1 +dt +�2 ++ 1 +2m +�d�q2 +dt +�2 ++ λ +2 +� qa +1 +|q2| +d +dt +� q2a +|q1| +� +− qa +2 +|q1| +d +dt +� q1a +|q2| +�� � +dt, +where a = 1, . . . , d, D = 2d, describes an interacting, two particle system that is invariant +under the local conformal transformation qa +1 → α(u)qa +1 and qa +2 → α(u)qa +2. +With +ΩM = +m +|q1|2Πab(q1)dqa +1 ∧ dvb +1 + m +|q2|2Πab(q2)dqa +2 ∧ dvb +2, and +ΩF = +m +|q1|3 (�q1 · dq1) ∧ (v1 · Π(q1) · dq1) + m +|q2|3 (�q2 · dq2) ∧ (v2 · Π(q2) · dq2) − +λ +|q1||q2| [dqa +1 ∧ (Π(q2) · dq2)a + (Π(q1) · dq1)a ∧ dqa +2 − (Π(q1) · dq1)a ∧ (Π(q2) · dq2)a] − +λ +|q1|2 (�q1 · dq1) ∧ (�q2 · Π(q1) · dq1) + +λ +|q2|2 (�q2 · dq2) ∧ (�q1 · Π(q2) · dq2) , +C and G are two-dimensional, and are spanned by +Uq +(1) = �q1 · ∂ +∂q1 +, +Uq +(2) = �q2 · ∂ +∂q2 +, +and +Uv +(1) = �q1 · ∂ +∂v1 +, +Uv +(2) = �q2 · ∂ +∂v2 +, +respectively. The reduced ¯F = 0, and ker ΩL(u) is spanned by Uv +(1), Uv +(2), +P(+) = q1 · ∂ +∂q1 ++ q2 · ∂ +∂q2 ++ v1 · ∂ +∂v1 ++ v2 · ∂ +∂v2 +, +and +P(−) = q1 · ∂ +∂q1 +− q2 · ∂ +∂q2 ++ v1 · ∂ +∂v1 +− v2 · ∂ +∂v2 +− +2 λ +m +�|q1| +|q2|q2 · ∂ +∂v1 ++ |q2| +|q1|q1 · ∂ +∂v2 +� +. +As such, dim (ker ΩL)/G = 2. +The energy is +E = 1 +2 +m +|q1|2v1 · Π(q1) · v1 + 1 +2 +m +|q2|2v2 · Π(q2) · v2. +We find that γ[1] +(+) = 0 while +γ[1] +(−) = − +2λ +|q1||q2| (q2 · Π(q1) · v1 + q1 · Π(q2) · v2) , +26 + +giving, +β[XEL] = 1 +2γ[1] +(−) +� +Θ(1) +q +|q1| − Θ(2) +q +|q2| +� +. +Then SymL is one-dimensional and spanned by P(+). As expected, LP(+)β = 0. Because +LP(−)β = −4λ +m +� +1 − (�q1 · �q2)2� +� +Θ(1) +q +|q1| − Θ(2) +q +|q2| +� +, +Sym is also one-dimensional, and is also spanned by P(+). +A general SOELVF is +XEL = XL − m +8λ2 +XLγ[1] +(−) +[1 − (�q1 · �q2)] +� +P(−) +� ++ u(+)(u) +� +P(+) +� +, +(22) +where u(+)(u) ∈ F, and from [1], +XL = v1 · Π(q1) · ∂ +∂q1 ++ v2 · Π(q2) · ∂ +∂q2 ++ +��q1 · v1 +|q1| +� +v1 · Π(q1) · ∂ +∂v1 ++ +��q2 · v2 +|q2| +� +v2 · Π(q2) · ∂ +∂v2 ++ +λ +m +�|q1| +|q2|v2 · Π(q2) · Π(q1) · ∂ +∂v1 +− |q2| +|q1|v1 · Π(q1) · Π(q2) · ∂ +∂v2 +� +, +after the constraint algorithm is applied. Equation (22) is a consequence of the identity +⟨dγ[1] +(+)|XL⟩ = 0 and +− 1 +2λ⟨dγ[1] +(−)|XL⟩ = −2(�q1 · �q2)E +m + +2 +|q1||q2|v1 · Π(q1) · Π(q2) · v2 − +λ +m(�q1 · �q2) [v2 · Π(q2) · �q1 − v1 · Π(q1) · �q2] . +We see that SolP +[nF ] +L +is also one-dimensional, and is also spanned by P(+). +C. +A Lagrangian with Local Conformal and Time-reparametization Invariance +The action +S3 := sm +� � +s +�d�q +dt +�2�1/2 +dt, +where s = ±1, is invariant under both the local conformal transformations, qa → α(u)qa, +and the reparametization of time t → τ(t) where τ is a monotonically increasing function of +t. Then +ΩL = m +|q| +Pab(u) +� +sv · Π(q) · v +dqa ∧ dvb, +27 + +Action +Potential +ker ΩL/G +Sym +SymL +I[1] +SolP +[nF ] +L +VAS(ˆqa) +1 +1 +1 +0 +1 +S1 +Vsph(|q|) + VAS(ˆqa) +1 +1 +0 +1 +0 +V (qa) +1 +0 +0 +1 +0 +S2 +λ +2 +� qa +1 +|q2| +d +dt +� +q2a +|q1| +� +− qa +2 +|q1| +d +dt +� +q1a +|q2| +�� +2 +1 +1 +1 +1 +S3 +0 +2 +2 +2 +0 +2 +TABLE I. A summary of the symmetries of the three examples considered in this paper. With the +exception of the I[1] column, the numerical entries are the dimensionality of the vector spaces listed +along the first row. Notice the case where the Euler-Lagrange equations of motion has a generalized +Lie symmetry while the action itself does not. In all three examples, dim (SymL) = dim (Sol +P +[nf ] +L +). +and ΩF = 0. Here, a = 1. . . . , D, +ua = +Πab(q)vb +� +sv · Π(q) · v +, +so that u2 = s, while Pab(u) = Πab(q) − suaub. As such, ker ΩL(u) = ker ΩM(u). Both C +and G are two-dimensional, and are spanned by +Uq +(1) = �q · ∂ +∂q , +Uq +(2) = u · ∂ +∂q , +and +Uv +(1) = �q · ∂ +∂v , +Uv +(2) = u · ∂ +∂v , +respectively. It follows that dim (ker ΩL/G) = 2. +Because this system is fully constrained, E = 0. +As ΩF = 0 as well, there are no +Lagrangian constraints. It follows that SymL is two dimensional and spanned by Uq +(1) and +Uq +(2). As β = 0 as well, Sym is also two dimensional, and is also spanned by Uq +(1) and Uq +(2). +We found in [1] that XL = 0. +A general SOELVF is then XEL = u1(u) +� +Uq +(1) +� ++ +u2(u) +� +Uq +(2) +� +, with un(u) ∈ F for n = 1, 2. It follows that SolP +[nF ] +L +is also two-dimensional, +and is spanned by Uq +(1) and Uq +(2) as well. +VI. +CONCLUDING REMARKS +That each generalized Lie symmetry of the action contributes one arbitrary function +to the SOELVF for a dynamical system is known anecdotally, and is a result expected on +physical grounds. For almost regular Lagrangians, the appearance in physics of a generalized +28 + +Lie symmetry is due to a local gauge symmetry of the dynamical system, and thus to the +absence of a gauge—the length of vectors for local conformal invariance, or a measure for +time for time-reparametization invariance—for some dynamical property of the system. As +the generalized Lie symmetries of the action for an almost regular Lagrangian would have +NSymL of these gauge freedoms, it is reasonable that the absence of these gauges will result +in an equal number of arbitrary functions in the SOELVF. An equal number of terms to +fix these gauges would then be needed to determine the dynamics of the system uniquely. +But while these expectations are reasonable, up to now they have been fulfilled only on a +case-by-case basis. This is in great part because the analysis of dynamical systems with a +local gauge symmetry has traditionally been done using constrained Hamiltonian mechanics. +Such analysis relies on the canonical Hamiltonian, however, and the connection between the +canonical Hamiltonian and the symmetries of the Lagrangian is indirect at best, in contrast +to the Lagrangian approach followed here. Moreover, the process of determining the total +Hamiltonian for the system is often prescriptive, with results that are specific to the system +at hand. By focusing on the Lagrangian and on the Lagrangian phase space, we have been +able to show for all systems with an almost regular Lagrangian that has a constant rank +Lagrangian two-form, a direct link between local gauge symmetries and its dynamics. In +particular, it establishes a link between the number of gauge symmetries of the action and +the number of arbitrary functions that naturally appear in the evolution of such dynamical +systems. +As γ[1] +P = 0 for any choice of P ∈ SymL, the vectors in SymL do not contribute to +the first-order constraint manifold P[1] +L , and as such do not contribute to the Lagrangian +constraint algorithm at this order, or at any higher orders. It is for this reason that the +NSymL arbitrary functions um +[1] are not determined by the algorithm, and why these func- +tions will still contribute to XEL even after the algorithm has been completed. It also means +that if second- and higher-order Lagrangian constraints are introduced, they are accidental +and cannot be due to the local gauge symmetries of the action. Interestingly, we have yet +to find a dynamical system with a Lagrangian that is both almost-regular and has a La- +grangian two-form with constant rank where second- or higher-order Lagrangian constraints +are introduced. +This impact of generalized Lie symmetries on the dynamics of particles illustrates the +inherent differences between the analysis of the symmetries of regular Lagrangians and that +29 + +of almost regular Lagrangians. For regular Lagragians, the generator of the generalized Lie +symmetry (at times referred to as a global symmetry) gives rise to a prolongation vector, +and the action of this prolongation on the Lagrangian gives the variation of the action, δS, +under this symmetry. When the Euler-Lagrange equations of motion are thenimposed, the +conserved quantity for this symmetry along the path given by the solution of these equations +of motion is then obtained. While the generator of the generalized Lie symmetry for the +almost regular Lagrangian gL does give a prolongation vector pr gL Eq. (3), and while the +action of pr gL on L does give δS, imposing the Euler-Lagrange equations of motion on δS +in Eq. (4) gives the vacuous statement δS = 0. Instead, the requirement that δS = 0 for +all paths on Q gives the conditions that the generators of the symmetry must satisfy. This +in turn shows that the existence of these generators is due solely to the Lagrangian being +singular. These conditions then affect the dynamics of the system through γ[1] +P = 0, and in +doing so, sets a lower bound to the dimensionality of SolP +[nf ] +L . +We have found it quite difficult to construct more than one example of a dynamical +system that has an almost regular Lagrangian with both a generalized Lie symmetry and a +Lagrangian two-form with constant rank on PL. We have, on the other hand, found it quite +easy to construct examples of dynamical systems that have an almost regular Lagrangian +with a generalized Lie symmetry and a Lagrangian two-form whose rank varies across PL. +Indeed, it is the latter case that is the more prevalent one, and yet much of the results of this +paper and a good portion of the results of our previous one [1] relies on the condition that +the rank of the Lagrangian two-form be constant on PL. This is even more concerning when +we realize that these more prevalent systems are expected, by their nature, to have much +richer dynamics and mathematical structures (indeed, we have found that such systems +often require the introduction of second- or higher-order Lagrangian constraints), and yet it +is not known which of the results that have been shown to hold for systems with constant +rank Lagrangian two-forms will still hold when the rank varies across PL. Determining the +generalized Lie symmetries of these systems; showing that the passage from the Lagrangian +to the Hamiltonian phase space is possible; and finding the links between symmetry and +dynamics is a necessity for future research. +30 + +ACKNOWLEDGMENTS +This paper would not have been possible without the contributions by John Garrison, +who provided much of the underlying symmetry analysis of the action used in Section II A, +and most of the essential mathematics in Section III. Publication made possible in part by +support from the Berkeley Research Impact Initiative (BRII) sponsored by the UC Berkeley +Library. +Appendix +The Euler-Lagrangian equations of motion for the action S1 is +0 = m +|q|3Πab(q)¨qb − 2m +|q|3(�q · ˙q)Πab(q) ˙qb + ∂V +∂qa. +(A.1) +Contracting both sides of this equation with �q results in the first-order Lagrangian constraint +Eq. (19), and it is clear that dynamics is only possible on this constraint surface. Acting on +Eq. (A.1) with Πab(q) gives +0 = m +|q|3Πab(q)¨qb − 2m +|q|3(�q · ˙q)Πab(q) ˙qb + Πb +a(q)∂V +∂qb, +(A.2) +since Πac(q)Πc +b(q) = Πab(q). But in this case V (qa) = Vsph(|q|) + VAS(ˆq), and as +Πb +a(q)∂VSph +∂qb += Πb +a(q)∂|q| +∂qb V ′ +Sph(|q|) = 0, +while the identity +∂ˆqa +∂qb = Πa +b(q), +ensures that +Πb +a(q)∂VAS +∂qb += ∂VAS +∂qa , +Eq. (A.2) thereby reduces to the same equations of motion for the system as found for Case +1. It is for this reason that the two cases have same generalized Lie symmetry. +[1] A. D. Speliotopoulos, Constrained dynamics: +generalized Lie symmetries, singular La- +grangians, and the passage to Hamiltonian mechanics, J. Phys Commun., 4 065002 (2020). +31 + +[2] M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann +theory of constraints, Journal of Mathematical Physics, 19 2388–2399 (1978) 10.1063/1.523597. +[3] M. J. Gotay and J. M. Nester, Presymplectic lagrangian systems I: the constraint algorithm +and the equivalence theorem, Annales de L’Institut Henri Poincare, Section A, 30(2) 129–142 +(1979) +[4] M. J. Gotay and J. M. Nester, Presymplectic lagrangian systems II: the second-order problem, +Annales de L’Institut Henri Poincare, Section A, 32(1) 1–13 (1980). +[5] J. F. Cariñena, Theory of singular Lagrangians, Fortschritte der Physik, 38(9) 641–679 (1990) +10.1002/prop.2190380902. +[6] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, (Princeton University Press, +Princeton, New Jersey, 1992). +[7] P. A. M. Dirac, Generalized Hamiltonian dynamics, Canadian Journal of Mathematics, 2 129– +148 (1950) 10.4153/CJM-1950-012-1. +[8] M. C. Muñoz-Lecanda, Hamiltonian systems with constraints: A geometric approach, Inter- +national Journal of Theoretical Physics, 28(11) 1405–1417 (1989) 10.1007/BF00671858. +[9] L. Lusanna, Dirac-Bergmann constraints in physics: Singular Lagrangians, Hamiltonian con- +straints and the Second Noether Theorem, International Journal of Geometric Methods in +Modern Physics, 15(10) 1830004 (2018), 10.1142/S0219887818300040. +[10] G. Prince, Toward a classification of dynamical symmetries in classical mechanics, Bulletin of +the Australian Mathematical Society, 27 53–71 (1983) 10.1017/S0004972700011485. +[11] G. Prince, A complete classification of dynamical symmetries in classical mechanics, Bulletin +of the Australian Mathematical Society, 32 299–308 (1985) 10.1017/S0004972700009977. +[12] M. Crampin, Tangent bundle geometry Lagrangian dynamics, Journal of Physics A: Mathe- +matical and General Physics, 16 3755–3772 (1983) 10.1088/0305-4470/16/16/014. +[13] J. F. Cariñena, J. Fernández-Núñez and E. Martínez, A geometric approach to Noether’s +Second Theorem in time-dependent Lagrangian mechanics, Letters in Mathematical Physics, +23 51–63 (1991) 10.1007/BF01811294. +[14] J. F. Cariñena and M. F. Rañada, Noether’s theorem for singular Lagrangians, Letters on +Mathematical Physics, 15 305–311 (1988) 10.1007/BF00419588. +[15] J. F. Cariñena, E. Martínez and J. Fernández-Núñez, Noether’s theorem in time-dependent +Lagrangian mechanics, Reports on Mathematical Physics, 31 189–203 (1992) 10.1016/0034- +32 + +4877(92)90014-R. +[16] J. F. Cariñena, J. Fernández-Núñez and M. F. Rañada, Singular Lagragians affine in velocities, +Journal of Physics A: Mathematical and General Physics, 36 3789–3807 (2003) 10.1088/0305- +4470/36/13/311. +[17] J. F. Cariñena and J. Fernández-Núñez, Geometric theory of time-dependent singular La- +grangians, Fortschritte der Physik, 41(6) 517–552 (1993). +[18] J. F. Cariñnena and E. Martinez, in Summetries and Algebra Structures in Physics, Part +2: Integral Systems, Soli State Physics, and Theory of Phase Transitions, edited by V. V. +Dodonov and V. I. Man’ko, (Nova Science Publishers, New York, 1991) Chap. Generalized +Jacobi equation and inverse problem in classical mechanics, pp 84–98. +[19] G. Marmo, G. Mendella and W. M. Tulczyjew, Symmetries and constants of the motion for +dynamics in implicit form, Annales de L’Institut Henri Poincare, Section A, 57(2) 147–166 +(1992). +[20] X. Grácia and J. M. Pons, Symmetries and infinitesimal symmetries of singular differential +equations, Journal of Physics A: Mathematical and General Physics, 35 5059–5077 (2002) +10.1088/0305-4470/35/24/306. +[21] X. Grácia and R. Martín, Geometric aspects of time-dependent singular differential equa- +tions, International Journal of Geometric Methods in Modern Physics, 2(4) 597–618 (2005) +10.1142/S0219887805000697. +[22] L. Popescu, Symmetries of second order differential equations on Lie algebroids, Journal of +Geometry and Physics, 117 84–98 (2017) 10.1016/j.geomphys.2017.03.006. +[23] M. de León and D. M. de Diego, Symmetries and constants of the motion for singular +Lagrangian systems, International Journal of Theoretical Physics, 35(5) 975–1011 (1996) +10.1007/BF02302383. +[24] N. Dimakis, P. A. Terzis and T. Christodoulakis , Contact symmetries of constrained +quadratic Lagrangians, Journal of Physics: Conference Series, 670 1–6 (2016) 10.1088/1742- +6596/670/1/012021. +[25] M. Popescu, Totally singular Lagrangians and affine Hamiltonians, Balkan Journal of Geometry +and Its Applications, 14(1) 60–71 (2009) +[26] M. Popescu and P. Popescu, Totally singular Lagrangians and affine Hamiltonians of higher +order, Balkan Journal of Geometry and Its Applications, 16(2) 122–132 (2011). +33 + +[27] P. J. Olver, Applications of Lie Groups to Differential Equations, (Springer-Verlag, New York, +New York, 1993). +[28] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed, (Addison-Wesley, Reading, +Massachusetts, 1978). +34 + diff --git a/xdFAT4oBgHgl3EQfAByy/content/tmp_files/load_file.txt b/xdFAT4oBgHgl3EQfAByy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df2873edc83fed85a9e13d9e436b13f927ef3c3f --- /dev/null +++ b/xdFAT4oBgHgl3EQfAByy/content/tmp_files/load_file.txt @@ -0,0 +1,744 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf,len=743 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content='08396v1 [math-ph] 20 Jan 2023 Generalized Lie Symmetries and Almost Regular Lagrangians: A Link Between Symmetry and Dynamics Achilles D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Speliotopoulos Department of Physics, University of California, Berkeley, CA 94720 USA∗ (Dated: January 23, 2023) 1 Abstract The generalized Lie symmetries of almost regular Lagrangians are studied, and their impact on the evolution of dynamical systems is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It is found that if the action has a generalized Lie symmetry, then the Lagrangian is necessarily singular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' the converse is not true, as we show with a specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It is also found that the generalized Lie symmetry of the action is a Lie subgroup of the generalized Lie symmetry of the Euler-Lagrange equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The converse is once again not true, and there are systems for which the Euler-Lagrange equations of motion have a generalized Lie symmetry while the action does not, as we once again show through a specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Most importantly, it is shown that each generalized Lie symmetry of the action contributes one arbitrary function to the evolution of the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The number of such symmetries gives a lower bound to the dimensionality of the family of curves emanating from any set of allowed initial data in the Lagrangian phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Moreover, if second- or higher- order Lagrangian constraints are introduced during the application of the Lagrangian constraint algorithm, these additional constraints could not have been due to the generalized Lie symmetry of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' INTRODUCTION The symmetries of the Euler-Lagrange equations of motion were recently used to study the constrained dynamics of singular Lagrangians [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The focus was on almost regular La- grangians [2–5], and it was found that for these Lagrangians the Euler-Lagrange equations of motion admit a generalized Lie symmetry (also known as a local gauge symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The gen- erators Sym of this symmetry group GrSym were determined in the Lagrangian phase space approach to Lagrangian mechanics, and were found to lie in the kernel of the Lagrangian two-form ΩL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' While it is well-known that the solutions XE of the energy equation, 0 = dE − iXEΩL, (1) is not unique for almost regular Lagrangians, it was shown in [1] that the action of Sym on a general solution to this equation—and in particular, on the second-order, Lagrangian ∗ Also at Physical Science and Engineering Division, Diablo Valley College, Pleasant Hill, CA 94523, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' ads@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content='edu 2 vector field (SOLVF)—will result in a vector field that is no longer a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Thus, not all solutions of the energy equation have GrSym as a symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It is, however, possible to construct solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (1) for whom Sym does generate a group of symmetry transformations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' These vector fields are called second-order, Euler- Lagrange vector fields (SOELVFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As the evolution of the dynamical system for sin- gular Lagrangians must lie on Lagrangian constraint surfaces [5], a Lagrangian constraint algorithm for SOELVFs was also introduced in [1] to construct such solutions to the energy equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It was then shown that these SOELVFs, along with the dynamical structures in the Lagrangian phase space needed to describe and determine the motion of the dynamical system, are projectable to the Hamiltonian phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In particular, the primary Hamil- tonian constraints can be constructed from vectors that lie in the kernel of ΩL, and the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian (we follow the terminology found in [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' see also [7–9]) obtained using constrained Hamiltonian mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the La- grangian constraint algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The Lagrangian and Hamiltonian formulations of mechanics for almost regular Lagrangians were thereby shown to be equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' While [1] focused on the generalized Lie symmetries of the Euler-Lagrange equations of motion and whether the dynamical structures constructed in the Lagrangian phase space are projectable to the Hamiltonian phase space, in this paper the focus is on the symmetries of the action itself and the impact these symmetries have on the evolution of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This impact is found to be quite broad, surprisingly restrictive, and unexpectedly subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Indeed, even the seemingly reasonable expectation that any generalized Lie symmetry of the Euler-Lagrange equations of motion should be a reflection of the symmetries of the action itself is not borne out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We find that if the action has a generalized Lie symmetry, then its Lagrangian is nec- essarily singular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' the converse need not be true, as we show through a specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We also find that the generators of the generalized Lie symmetry of the action form a Lie sub-algebra of the generators of the generalized Lie symmetry of the Euler-Lagrange equa- tion of motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' once again, the converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We give an example of a dynamical system for which the Euler-Lagrange equations of motion has a generalized Lie symmetry, while its action does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Most importantly, for systems where the Lagrangian is almost 3 regular and for which the two-form ΩL has constant rank, we show that each generalized Lie symmetry of the action contributes one arbitrary constant to the SOELVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The dimen- sionality of the space of solutions to the energy equation that have GrSym as a symmetry group is thus at least as large as the number of generalized Lie symmetries of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Moreover, if second- or higher-order Lagrangian constraints are introduced during the ap- plication of the Lagrangian constraint algorithm, these additional constraints cannot be due to the generalized Lie symmetry of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Symmetries of Lagrangian systems have been studied before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' However, such analyses have been focused on time-dependent Lagrangians [10–17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' on systems of first-order evolution equations [18–22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' or on general solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (1) [23] (see also [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Importantly, the great majority of these studies have been done using first-order prolongations on first-order jet bundles with a focus on the Lie symmetries of first-order evolution equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Our interest is in the symmetries of the action, which naturally leads us to consider generalized Lie symmetries and second-order prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' To our knowledge, such symmetry analysis of the action has not been done before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (The framework for kth-order prolongations on kth-order jet bundles have been introduced before [16, 17, 23, 25, 26], but they were not applied to the action or to the Euler-Lagrange equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=') The rest of the paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In Section II the conditions under which the action for a dynamical system, and the conditions under which the Euler-Lagrange equations of motion for this action, have a generalized Lie symmetry are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' To compare the conditions for each, the analysis for the two are done separately, with each self-contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In Section III properties of the Lagrangian phase space are reviewed, and the notation used here established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The generators of the generalized Lie symmetry group for the Euler-Lagrange equations of motion were determined in [1], and a summary of the results found therein that are needed here is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In Section IV the generators of the generalized Lie symmetry group for the action is found within the Lagrangian phase space approach, and their relation to the generators for the symmetry group of the Euler-Lagrange equations of motion is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The impact of the symmetries of the action on the SOELVF is then analyzed by applying the Lagrangian constraint algorithm introduced in [1] to these SOELVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The results obtained in this paper is then applied to three different dynamical systems in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In particular, an example of a dynamical system that has no generalized Lie symmetries and yet is still singular, and another example where the 4 action has no symmetries and yet the Euler-Lagrange equations of motion do, are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Concluding remarks can be found in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' GENERALIZED LIE SYMMETRIES AND LAGRANGIAN MECHANICS In this section we determine the conditions under which the action of a dynamical system, and the conditions under which the Euler-Lagrange equations of motion for this system, has a generalized Lie symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' While the determination for both is done within Lagrangian mechanics, the analysis for the action is completed separately from that of the equations of motion—with each self-contained—so that the two conditions can be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We will later show that every generator of the generalized Lie symmetry of the action is a generator of a generalized Lie symmetry of the Euler-Lagrange equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Interestingly, the converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Symmetries of the Action We begin with Lagrangian mechanics, and an analysis of the generalized Lie symmetry [27] of the action S := � t2 t1 L (q(t), ˙q(t)) dt, for a dynamical system on a D-dimensional configuration space Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Here, L (q(t), ˙q(t)) is the Lagrangian along a path q(t) = � q1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , qD(t) � on Q with end points given by Q1 := q(t1), Q2 := q(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' These points are chosen at the same time the choice of S is made, and are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As L (q(t), ˙q(t)) depends on both the position q(t) and the velocity ˙q(t) of the path, we consider a generalized Lie symmetry that is generated by gL := ρL(q, ˙q) · ∂ ∂q , where ρL(q, ˙q) does not depend explicitly on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Evolution along the path gives the total time derivative d dt := ˙q · ∂ ∂q + ¨q · ∂ ∂ ˙q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (2) This in turn gives ˙ρL := dρL/dt, and the second-order prolongation vector [27], pr gL := ρL · ∂ ∂q + ˙ρL · ∂ ∂ ˙q + ¨ρL · ∂ ∂¨q , (3) 5 on the second-order jet space M(2) = {(t, q, ˙q, ¨q)} where this pr gL ∈ TM(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Under this generalized Lie symmetry, the action varies by δS = � t2 t1 pr gL � L(q(t), ˙q(t)) � dt, with the requirement that ρL(q(t1), ˙q(t1)) = 0 = ρL(q(t2), ˙q(t2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then after an integration by parts, δS = � t2 t1 ρL · �∂L ∂q − d dt �∂L ∂ ˙q �� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (4) It is important to realize that the action may be evaluated along any path on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As such, if gL generates a symmetry of the action, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (4) must vanish for all paths q(t) on Q, and not just for those that minimize the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' To make connection with the Lagrangian phase space approach used in the rest of the paper, we make use of E (q, ˙q) := ˙qa∂L (q, ˙q) ∂ ˙qa − L (q, ˙q) , along with Mab (q, ˙q) := ∂2L (q, ˙q) ∂ ˙qa∂ ˙qb , and Fab (q, ˙q) := ∂2L (q, ˙q) ∂ ˙qa∂qb − ∂2L (q, ˙q) ∂ ˙qb∂qa , to express Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (4) as δS = − � t2 t1 ρa L �∂E ∂qa + Fab(q, ˙q) ˙qb + Mab(q, ˙q)¨qb � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (5) Here, Latin indices run from 1 to D, and Einstein’s summation convention is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We then arrive at our first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Lemma 1 An action S of a dynamical system has a generalized Lie symmetry generated by gL if and only if there exists a ρL ∈ ker Mab such that 0 = ρa L(q, ˙q) �∂E ∂qa + Fab(q, ˙q) ˙qb � , (6) on TQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' If gL generates a generalized Lie symmetry of S, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (5) must vanish for all paths on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For an arbitrary path on Q the curvature of the path ¨q will not depend on either the q(t) or the ˙q(t) for the path, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As such, for δS = 0, it must be that ρa LMab¨qb = 0 for any choice of ¨q, and thus ρa L ∈ ker Mab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The remaining terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (5) gives the condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 6 The set of all vector fields gL that satisfy Lemma 1 is denoted by gL, while pr gL := {pr gL | gL ∈ gL} is the set of their prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This pr gL is involutive [27], and the conditions under which pr gL generates a generalized Lie symmetry group are given in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We see from Lemma 1 that if the action has a generalized Lie symmetry, then the Lagrangian is necessarily singular, and as such the Lagrangian two-form ΩL will not have maximum rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It is also important to note that while equations of the form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (6) often appear in the Lagrangian phase space description of mechanics [1], they appear as Lagrangian constraints, conditions that must be imposed for evolution under the Euler- Lagrange equations to be well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Here, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (6) is not a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Rather, because the action must have this symmetry for all possible paths on Q, and since the set of all possible paths cover Q, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (6) is a condition on ρL that must be satisfied identically on all of TQ—and thus, on the Lagrangian phase space—for gL to be a generator of the symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We will see that not all the vectors in ker Mab satisfy the identity Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (6), however, and thus not all of these vectors will generate a generalized Lie symmetry of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Symmetries of the Euler-Lagrange Equations of Motion While in Section II A the focus was on arbitrary paths on the configuration space Q and the symmetries of the action, in this section the focus is on the trajectories that minimizes the action and the generalized Lie symmetries of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' These trajectories are solutions of the Euler-Lagrange equations of motion, and for almost regular Lagrangians such solutions form a family of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It is, in fact, the presence of this family of curves that gives rise to the generalized Lie symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The treatment here follows closely to that given in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For almost regular Lagrangians the solutions of the Euler-Lagrange equations of motion Mab(q, ˙q)¨qb = −∂E ∂qa − Fab(q, ˙q) ˙qb, (7) are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' While for these Lagrangians the rank of Mab (q, ˙q) = D − N0—with N0 = dim (ker Mab(q, ˙q))—is constant, this rank is not maximal, and thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (7) does not have a unique solution for ¨q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Instead, for a chosen set of initial data (q0 = q(t0), ˙q0 = ˙q(t0)), the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (7) results in a family of solutions that evolve from this (q0, ˙q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As with the paths in Section II A, these solutions are related to one another through a generalized Lie symmetry [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 7 Following [27], the collection of functions ∆a(q, ˙q, ¨q) := ∂E(q, ˙q) ∂qa + Fab(q, ˙q) ˙qb + Mab(q, ˙q)¨qb, (8) defines a set of surfaces ∆a(q, ˙q, ¨q) = 0 on M(2), while the family of solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (7) O (q0, ˙q0) := � q (t) | ∆a(q, ˙q, ¨q) = 0 with q (t0) = q0, ˙q (t0) = ˙q0 � , that evolve from the same initial data (q0, ˙q0) gives the collection of trajectories that lie on these surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Indeed, for any two such solutions qa(t) and Qa(t) there exists a z(q, ˙q) ∈ ker Mab(q, ˙q) such that ¨Qa − ¨qa = za.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Importantly, because za depends on both q and ˙q, the symmetry group that maps one member of O to another must be a generalized Lie symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We therefore take the generator of this symmetry group to be g := ρ(q, ˙q) · ∂ ∂q , with the corresponding the second-order prolongation vector for g being, pr g := ρ · ∂ ∂q + ˙ρ · ∂ ∂ ˙q + ¨ρ · ∂ ∂¨q , with this pr g ∈ TM(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As with the above, the total time derivative is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (2), but unlike the analysis in Section II A, the evolution of the path—and indeed, for all the trajectories in O(q0, ˙q0)—here is given by the Euler-Lagrange equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The action of this prolongation on ∆a on the ∆a = 0 surface gives, pr g [∆a(q, ˙q, ¨q)] = −∂¨qb ∂qaMbc(q, ˙q)ρc + d dt � Fab(q, ˙q)ρb + Mab(q, ˙q) ˙ρb� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Since N0 > 0, ¨q is not unique on this surface, and yet g must generate the same symmetry group for all the trajectories in O(q0, ˙q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Necessarily, ρ(q, ˙q) ∈ ker Mab(q, ˙q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It then follows that pr g[∆a(q, ˙q, ¨q)] = 0 if and only if (iff) there are constants ba such that ba = Fabρb + Mab ˙ρb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The solutions in O(q0, ˙q0) all have the same initial data, however, and thus necessarily ρ(q0, ˙q0) = 0 = ˙ρ(q0, ˙q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We conclude that ba = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The following result, first proved in [1], then follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Lemma 2 If g is a generalized infinitesimal symmetry of ∆a, then ρa(q, ˙q) ∈ ker Mab(q, ˙q), and ˙ρa(q, ˙q) is a solution of 0 = Fab(q, ˙q)ρb(q, ˙q) + Mab(q, ˙q) ˙ρb(q, ˙q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (9) 8 As before, we denote the set of all vector fields g that satisfy Lemma 2 by g, while pr g := {pr g | g ∈ g} is the set of their prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Once again pr g is involutive, and the conditions under which pr g generates a generalized Lie symmetry group are given in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Note, however, that while ρ = 0 and ˙ρ = z for any z ∈ ker Mab(q, ˙q) is a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (9), we require that ˙ρ = dρ/dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' these solutions cannot be generators of the generalized Lie symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Next, if ˙ρ is a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (9), then ˙ρa + z is a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (9) as well, and thus these solutions are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This, along with the previous observation, leads us to generators that are constructed from equivalence classes of prolongations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Finally, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (8) gives for any z ∈ ker Mab(q, ˙q), 0 = za �∂E ∂qa + Fab(q, ˙q) ˙qb � , (10) on the solution surface ∆a(q, ˙q, ¨q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' If Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (10) does not hold identically, it must be imposed, leading to Lagrangian constraints [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' More importantly, because each q(t) ∈ O(u0) must lie on the Lagrangian constraint submanifold, any symmetry transformation of q(t) generated by pr g must give a path Q(t) that also lies on the constraint submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Not all vectors in pr g will be generators of the generalized Lie symmetry group for O(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Determining which of these vectors are, and the relationship between the generators of symmetries of the Euler-Lagrange equations of motion and those of the action, is best done within the Lagrangian phase space framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' To accomplish this, we will need the following generalization of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Consider the vector k := c · ∂ ∂q + ˙c · ∂ ∂ ˙q , with a c ∈ ker Mab(q, ˙q) along with the quantity la := Fabcb(q, ˙q) + Mab ˙cb(q, ˙q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' After an integration by parts, la = cb(q, ˙q) � Fab(q, ˙q) − d dt ∂2L ∂ ˙qa∂ ˙qb � , = cb(q, ˙q) � Fab(q, ˙q) − �d dt , ∂ ∂ ˙qa � ∂L ∂ ˙qb − ∂ ∂ ˙qa �d dt ∂L ∂ ˙qb �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (2) we have �d dt , ∂ ∂ ˙qa � ∂L ∂ ˙qb = − ∂2L ∂qa∂ ˙qb − ∂¨qc ∂ ˙qa ∂2L ∂ ˙qc∂ ˙qb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 9 As q(t) is a solution of the Euler-Lagrange equations of motion, we find that la = cb(q, ˙q) � Fab(q, ˙q) + ∂2L ∂qa∂ ˙qb − ∂2L ∂ ˙qa∂qb + ∂¨qc ∂ ˙qa ∂2L ∂ ˙qc∂ ˙qb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This last expression vanishes after the definition of Fab(q, ˙q) is used along with the require- ment that c ∈ ker Mab(q, ˙q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We then have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Lemma 3 For any vector k = c · ∂ ∂q + ˙c · ∂ ∂ ˙q , such that c ∈ ker Mab, 0 = Fabcb(q, ˙q) + Mab ˙cb(q, ˙q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' GENERATORS OF THE GENERALIZED LIE SYMMETRY FOR THE EULER- LAGRANGE EQUATIONS OF MOTION The generators of the generalized Lie symmetry for both the Euler-Lagrange equations of motion and the action are best found using the Lagrangian phase space approach to mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This phase space and its concomitant mathematical structure provide the tools needed to determine both the generators of the symmetry and the solutions to the energy equation on which they act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For the Euler-Lagrange equations of motion this determination was done in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In this section we will review the Lagrangian phase space approach, establish the notation used in this paper, and summarize the results obtained in [1] that are needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (We will also take the opportunity to correct typographical errors made in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=') Proofs of the majority of the assertions listed in this section will not be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' the reader is instead referred to [1] where the proofs and the context of their development can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The Lagrangian Phase space For a configuration space Q the Lagrangian phase space PL is the tangent space PL = TQ, with the coordinates on PL denoted as u = (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , qD, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' vD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Integral flows on PL, t ∈ [t0, ∞) → u(t) ∈ PL [28], for a set of initial data u0 = (q0, v0) are given as solutions to du dt := X(u), 10 where X is a smooth vector field in TPL = T(TQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The two tangent spaces TQ and TPL have the bundle projections: τQ : TQ → Q and τTQ : T(TQ) → TQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' They can be used to construct two other projection maps: τQ ◦ τTQ : T(TQ) → Q and the prolongation of τTQ to T(TQ) (see [3] and [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This prolongation is the map TτQ : T(TQ) → TQ, and is defined by requiring that the two maps τQ ◦ τTQ and τQ ◦ TτQ map any point in T(TQ) to the same point in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The vertical subbundle [TPL]v of T(TQ) is [TPL]v = ker TτQ [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' a Xv ∈ [TuPL]v above a point u ∈ PL is called a vertical vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The horizontal subbundle [TPL]q of T(TQ) is [TPL]q = Image TτQ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' a Xq ∈ [TuPL]q is called a horizontal vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Consequently, each X ∈ TuPL consists of a Xq ∈ [TuPL]q and a Xv ∈ [TuPL]v with X = Xq + Xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In terms of local coordinates, Xq := Xqa ∂ ∂qa, and Xv := Xva ∂ ∂va .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Of special interest is the second order Lagrangian vector field XL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This vector field is the particular solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (1) for which TτQ ◦ XL is the identity on TQ (see [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In terms of local coordinates XL = va ∂ ∂qa + Xva ∂ ∂va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The space of one-forms on PL is the cotangent space T∗PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For a one-form α ∈ T∗ uPL, and a vector field X ∈ TuPL, the dual prolongation map T∗τQ is defined as ⟨α|TτQX⟩ = ⟨T∗τQα|X⟩, after a useful adaptation of Dirac’s bra and ket notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In addition, for a general k-form ω in the k-form bundle Λk (PL), ω (x) : Y1 ⊗ · · · ⊗ Yk → ⟨ω (x)| Y1 ⊗ · · · ⊗ Yk⟩ ∈ R, with Yj ∈ TuPL for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The vertical one-form subbundle [T∗PL]v of T∗PL is [T∗PL]v := ker T∗τQ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' a αv ∈ [T∗ uPL]v is called a vertical one-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The horizontal one-form subbundle [T∗PL]q of T∗PL is [T∗PL]q = Image T∗PL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' a αq ∈ [T∗ uQ]q is called a horizontal one-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Each one-form ϕ ∈ T∗ uPL consists of a ϕq ∈ [T∗ uPL]q and a ϕv ∈ [T∗ uPL]q such that ϕ = ϕq + ϕv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In terms of local coordinates ϕq := ϕqa dqa and ϕv := ϕvadva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Following [3, 4], the Lagrangian two-form is defined as ΩL := −ddJL, where dJ is the vertical derivative (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This two-form can be expressed as ΩL := ΩF + ΩM such that 11 for any X, Y ∈ TuPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' ΩF(X, Y) := ΩL(TτQX, TτQY), and is thus the horizontal two-form of ΩL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As ΩM(X, Y) = ΩL(X, Y) − ΩF(X, Y), ΩM is then a mixed two-form of ΩL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In terms of local coordinates, ΩL = −dθL, where θL := ∂L ∂va dqa, while ΩF := 1 2Fabdqa ∧ dqb, and ΩM := Mabdqa ∧ dvb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For regular Lagrangians XL is the unique solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For almost regular La- grangians, on the other hand, this solution is not unique, but instead depends on ker ΩL (u) := {K ∈ TuPL | iKΩL = 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' From Section II we expect this kernel to play a role in determining the generators of the generalized Lie symmetry of both the Euler-Lagrange equations of motion and the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Indeed, consider the natural isomorphism iso : (t, q, ˙q, ¨q) ∈ M(2) → (t, q, v, Xva L ) defined in [1], and the prolongation pr g of a generator g ∈ g of a generalized Lie symmetry of the Euler-Lagrange equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This pr g contains the vector k = ρ · ∂ ∂q + ˙ρ · ∂ ∂ ˙q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The collection of all such vectors has been shown to be involutive (see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The isomorphism maps iso : k → k′ where k′ = ρ · ∂ ∂q + ˙ρ · ∂ ∂v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then k′ ∈ TPL, and from Lemma 2, k′ ∈ ker ΩL(u) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A similar result holds for the generators in gL after Lemma 1 and Lemma 3 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The two-form ΩL gives the lowering map Ω♭ L : TuPL → T∗ uPL, with Ω♭ LX := iXΩL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This map consists of Ω♭ L = Ω♭ F + Ωv♭ M + Ωq♭ M, with Ω♭ F : X ∈ TuPL → [T∗ uPL]q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Ωq♭ M : X ∈ TuPL → [T∗ uPL]q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' and Ωv♭ M : X ∈ TuPL → [T∗ uPL]v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In terms of local coordinates, Ω♭ FX = FabXqadqb, Ωq♭ MX = −MabXvadqb, and Ωv♭ MX = MabXqadvb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For almost regular Lagrangians ker Ωv♭ M = C ⊕[TuPL]v while ker Ωq♭ M = [TuPL]q ⊕G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Here C := {C ∈ [TqPL]q | iCΩM = 0} , 12 and G := {G ∈ [TqPL]v | iGΩM = 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As Mab(u) has constant rank on PL, there exists a basis, � z(n) (u) = � z1 (n) (u) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , zD (n) (u) � | Mab (u) zb (n) (u) = 0, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0 � , for ker Mab (u) at each u ∈ PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Spans of both C = span � Uq (n) = z(n) · ∂ ∂q , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0 � , and G = span � Uv (n) = z(n) · ∂ ∂v , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0 � , can then be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Importantly, G is involutive [5], and when the rank of ΩL(u) is constant on PL, ker ΩL(u) is involutive as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Corresponding to Uq (n) and Uv (n) we have the one-forms Θ(m) q and Θ(m) v where ⟨Θ(m) q |Uq (n)⟩ = δ(m) (n) and ⟨Θ(m) v |Uv (n)⟩ = δ(m) (n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then [TuPL]q = C ⊕ C⊥ and [TuPL]v = G ⊕ G⊥, where C⊥ := � X ∈ [TuPL]q | � Θ(n) q �� X � = 0, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0 � , and G⊥ := � X ∈ [TuPL]v | � Θ(n) v �� X � = 0, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The vectors that lie in ker ΩL(u) can be determined by using the reduced matrix Fnm := za (n)Fabzb (m) to define C := � C ∈ C ���� N0 � m=1 ¯FnmC (m) = 0 � ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then, Theorem 4 The vectors K = Kq + Kv ∈ ker ΩL are given by, Kq = C, Kv = G+�C, where, C ∈ C, G ∈ G, and �C ∈ G⊥ is the unique solution of Mab �Cb = −FabC b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We found in [1] that dim (ker ΩL (u)) = N0 + ¯D, where ¯D := dim C ≤ N0 (see [1] for proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' However, the results of Lemma 3 show that we can construct from any vector Uq ∈ C a vector that lies in the ker ΩL(u), and as dim (C) = N0, it follows that dim (ker ΩL(u)) = 2N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' First-order Lagrangian constraints For singular Lagrangians solutions of the energy equation XE are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It is well known that they also do not, in general, exist throughout PL, but are instead confined to a submanifold of the space given by Lagrangian constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' With XE = Xq E + Xv E, it is convenient to use the one form Ψ Ωq♭ MXv E = Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' constructed from the energy equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The first-order constraint functions are then γ[1] n := � Ψ| Uq (n) � = 0 for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In terms of local coordinates, γ[1] n = Uqa (n) � ∂E ∂qa + Fabvb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' They may also be expressed [3, 4] as γ[1] n = ⟨dE|P(n)⟩ = P(n)E for any basis {P(n)} of ker ΩL(u) for which ⟨Θ(m) q |P(n)⟩ = δ(m) (n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In general, γ[1] n ̸= 0 on PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Instead, the con- dition γ[1] n = 0 must be imposed, and this in turn defines a set of submanifolds of PL given by the collection C[1] L := � γ[1] 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , γ[1] N0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The collection of these surfaces, P[1] L := � u ∈ PL | γ[1] n (u) = 0 , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0 � is called the first-order Lagrangian constraint submanifold, and has dim P[1] L = 2D−I[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Here I[1] is the number of independent functions in C[1] L with I[1] = rank � dγ[1] n � ≤ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The constraint one-form β[XE] := dE − iXEΩL, was introduced in [1] with the condition β[XE] = 0 giving both the solution of the energy equation and the submanifold P[1] L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As ⟨β|Uq (n)⟩ = γ[1] n , this β[XE] can also be expressed as β[XE] = N0 � n=1 γ[1] n Θ(n) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (11) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The Generalized Lie Symmetry Group for the Euler-Lagrange Equations of motion The generalized Lie symmetry group for O(u0) is determined using ker ΩL(u) := {P ∈ ker ΩL(u) | [G, P] ∈ [TuPL]v ∀ G ∈ G}, (12) 14 along with the following collection of functions on PL, F := {f ∈ C∞on PL | Gf = 0 ∀ G ∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This ker ΩL(u) is also involutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The following results were proved in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Lemma 5 Let X ∈ TuPL and G ∈ G such that [G, X] ∈ ker ΩL(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then [G, X] ∈ [TuPL]v iff [G, X] ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It then follows that [G, P] ∈ G for all P ∈ ker ΩL(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As G is involutive and as G ⊂ ker ΩL(u), G ⊂ ker ΩL(u) as well, and thus G is an ideal of ker ΩL(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Lemma 6 There exists a choice of basis for ker ΩL(u) that is also a basis of ker ΩL(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As G is an ideal of ker ΩL(u), we may define for any P1, P2 ∈ ker ΩL(u) the equivalence relation: P1 ∼ P2 iff P1 − P2 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The equivalence class, [P] := {Y ∈ ker ΩL(u) | Y ∼ P}, (13) can be constructed along with the quotient space ker ΩL(u)/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (For the sake of notational clarity we will suppress the square brackets for equivalence classes when there is no risk of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=') This space is a collection of vectors that lie in the kernel of ΩL, but with the vectors in G removed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' ker ΩL(u)/G thereby addresses the first two observations listed at the end of Section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We now turn our attention to the third observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Because the integral flow uX(t) of any solution X of the energy equation must lie on P[1] L , a symmetry transformation of uX(t) must result in an integral flow uY(t) of another solution Y of the energy equation, which must also lie on P[1] L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Implementing this condition is done through β[XE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As ⟨β[XE]|G⟩ = ⟨dE|G⟩ = GE = 0 for all G ∈ G on P[1] L , the Lie derivative LG of β along G is, LGβ[XE] = N0 � n=1 � Gγ[1] n � Θ(n) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Given a P(n) ∈ ker ΩL(u) such that P(n) = Uq (n) + �U(n) + G′ with G′ ∈ G, Gγ[1] n = [G, P(n)]E + P(n)GE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' But G is an ideal of ker ΩL(u), and thus Gγ[1] n = 0 on the first-order constraint manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It follows that LGβ = 0 on P[1] L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The collection of vectors, Sym := � P ∈ ker ΩL(u)/G | LPβ[XE] = d⟨β[XE]|P⟩ on P[1] L � , 15 is therefore well defined, and is involutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It follows that P ∈ Sym iff ⟨dβ[XE]|P ⊗ X⟩ = 0 for all X ∈ TPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We are then able to construct from each P ∈ Sym a one-parameter subgroup σP(ǫ, x) defined as the solution to dσP dǫ := P (σP) , where σP(0, u) = u for u ∈ PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The collection of such subgroups with give the Lie group GrSym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Euler-Lagrange Solutions of the Energy Equation We denote the set of general solutions to the energy equation as Sol := {XE ∈ TuPL | iXEΩL = dE on P[1] L }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' If u(t) is the integral flow of a vector in Sol whose projection onto Q corresponds to a solution of the Euler-Lagrange equations of motion, then GrSym must map one of such flows into another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' However, while LGXL = [G, XL] ∈ ker ΩL(u), in general LGXL /∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The action of σP on the flow uXL will in general result in a flow uY generated by a Y that is not a SOLVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It need not even be a solution of the energy equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' By necessity, general solutions of the energy equation must be considered, leading us to consider the collection of solutions Sol := {XEL ∈ Sol | [G, XEL] ∈ [TuPL]v ∀G ∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This collection generates the family of integral flows OEL(u0) := � u(t) ���� du dt = XEL(u), XEL ∈ Sol, and u(t0) = u0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Importantly, if P ∈ Sym, then i[XE,P]ΩL = iPdβ[XE] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As such, we find that Lemma 7 [XEL, P] ∈ ker ΩL(u) for all P ∈ Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It then follows that 16 Theorem 8 GrSym forms a group of symmetry transformations of OEL(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Proof of both assertions can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The generators of the generalized Lie symmetry for OEL(u0) are thus given by Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The corresponding solutions to the Euler-Lagrange equations that have this symmetry are given by Sol, and a vector XEL ∈ Sol is called a second-order, Euler-Lagrange vector field (SOELVF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It has the general form, XEL = XL + N0 � m=1 um(u) � P(m) � , (14) where um(u) ∈ F and {[P(n)], n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0} is a choice of basis for ker ΩL(u)/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The vector field XL is constructed from the second order Lagrangian vector field XL and vectors in ker ΩL(u) by requiring XL ∈ Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This construction is described in [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' we will only need the existence of such a vector field in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' GENERALIZED LIE SYMMETRIES OF THE ACTION AND ITS IMPACT ON DYNAMICS We now turn our attention to the generators of the generalized Lie symmetry of the action, and the impact this symmetry has on the evolution of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The Generalized Lie Symmetry of the Action In determining the conditions (as listed in Lemma 1) under which the action admits a generalized Lie symmetry, the understanding that the action must have this symmetry for all possible paths on Q played an essential role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' By necessity, these conditions could only be placed on ρL, and not on ˙ρL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' unlike ρL, ˙ρL depends explicitly on the evolution of a particular path, while the symmetry must hold for all paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We note, however, that the family OEL of trajectories determined by the Euler-Lagrange equations of motion also consists of paths on Q, and as such the generalized Lie symmetry of the action is also a symmetry of OEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Importantly, how these trajectories evolve with time is known, and as such, the ˙ρL for a given ρL is also known for these trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' With this understanding, and after comparing Lemma 1 and the results of Lemma 3 with Lemma 2, we conclude that the generators 17 of the generalized Lie symmetry of the action must also be generators of the generalized Lie symmetry of the Euler-Lagrange equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This leads us to consider the following collection of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' SymL = {P ∈ ker ΩL(u)/G | γ[1] P = ⟨β|P⟩ = 0 on PL}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We will also need NSymL = dim (SymL) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Lemma 9 SymL ⊂ Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Let {P(l), l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0} be a basis of ker ΩL(u)/G such that P(l) ∈ SymL for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , NSymL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We may choose the basis of C such that ⟨Θ(m) q |P(l)⟩ = δ(m) (l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then for any P(n) ∈ Sym, we see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (11) that, ⟨dβ|P(n)⊗Y⟩ = N0 � m=1 � ⟨dγ[1] m |P(n)⟩⟨Θ(m) q |Y⟩ − ⟨dγ[1] m |Y⟩⟨Θ(m) q |P(n)⟩ + γ[1] m ⟨dΘ(m) q |P(n) ⊗ Y⟩ � , for any Y ∈ TPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The last term vanishes on the first-order constraint manifold P[1] L , while for the second term, ⟨dγ[1] m |Y⟩⟨Θ(m) q |P(n)⟩ = ⟨dγ[1] n |Y⟩δ(m) (n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' But as P(n) ∈ SymL, γ[1] n = 0 on PL, and this term vanishes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Finally, for the first term, ⟨dγ[1] m |P(n)⟩ = P(n)P(m)E = [P(n), P(m)]E +P(m)P(n)E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' But γ[1] n = P(n)E = 0 on PL, while ker Ω(u) is involutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' There then exists a P(nm) ∈ ker Ω(u) such that P(nm) = [P(n), P(m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As P(nm)E := γ[1] (nm), this γ[1] (nm) must be a linear combination of first-order constraint functions, and they also vanish on P[1] L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It then follows that ⟨dβ|P(n) ⊗ Y⟩ = 0 on P[1] L , and P(n) ∈ Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' If P1, P2 ∈ SymL, then γ[1] [P1,P2] = P1P2E − P2P1E = P1γP2 − P2γP1 = 0, and thus SymL is involutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then for each P ∈ SymL we once again have the one-parameter subgroup σ SymL P (ǫ, u) define as the integral flow of dσSymL P dǫ := P, with σ SymL P (0, u) = u for u ∈ PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The collection of such subgroups gives the Lie group GrSymL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As SymL ⊂ Sym, GrSymL is a Lie subgroup of GrSym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It then follows from Theorem 8 that GrSymL also forms a group of symmetry transformations of OEL(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As the family OEL(u0) of trajectories are paths on Q, and as the symmetry transformation of the action must be the same for all paths on Q, it also follows that, Theorem 10 GrSymL forms the group of symmetry transformations of the action S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 18 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Symmetries and Dynamics While OEL(u0) gives the family of integral flows on which both GrSym and GrSymL act, a general flow in OEL(u0) need not be confined to P[1] L , and yet this is the submanifold on which the solutions XEL ∈ mathcalSol of the energy equations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In such cases it is necessary to jointly choose a SOELVF XEL and a submanifold of P[1] L on which the resultant flow uXEL will be confined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This is done through the implementation of a constraint algorithm, one of which was proposed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In that paper the product of this algorithm was the most that could be said about the general structure of SOELVFs that have integral flow fields which lie on P[1] L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Here, with the results obtained in Section IV A, we can say much more, and we will see that the presence of a generalized Lie symmetry of the action greatly restricts the structure of the SOELVFs that such systems can have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Following [1], we introduce for a XEL ∈ Sol the notation X [1] EL := XEL, X [1] L := XL, P[1] (n) := P(n), um [1] := um, N[1] 0 := N0, when the constraint algorithm is implemented, with the superscript [1] denoting the first iteration of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (This notation is only used in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=') In addition, we choose P[1] (n) ∈ SymL for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , NSymL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For the integral flow field of XEL to lie on P[1] L , LXELβ = 0, (15) which reduces to LXELγ[1] n = 0 on P[1] L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This is called the constraint condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As both un [1], γ[1] n ∈ F, � P[1] (n) � γ[1] m = P(n)γ[1] m , and after making use of the general form of a SOELVF given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (14), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (15) reduces to N0 � m=1 Γ[1] nmum [1] = − � dγ[1] n ���X [1] L � , with Γ[1] nm := � dγ[1] n ���P[1] (m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (16) Since � dγ[1] n ���P[1] (m) � = P[1] (m)P[1] (n)E = [P[1] (m), P[1] (n)]E + Γ[1] mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' But ker ΩL(u) is involutive, and thus [P[1] (m), P[1] (n)]E is a linear combination of first-order Lagrangian constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As these constraints vanishes on P[1] L , Γ[1] nm = Γ[1] mn on the first-order constraint manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Next, when n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , NSymL, P[1] (n) ∈ SymL, and γ[1] n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Thus, Γ[1] nm = 0 when n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , NSymL, and as Γ[1] nm is a symmetric matrix on P[1] L , Γ[1] mn = 0 for these values of n as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Thus while Γ[1] nm is a N0 × N0 matrix, the only nonzero components of this matrix 19 lie in the � N0 − NSymL � × � N0 − NSymL � submatrix ¯Γ[1] n m := � dγ[1] n+NSymL ����P[1] m+NSymL � where n, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N0 − NSymL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As � dγ[1] n ���X [1] L � = 0 as well when n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , NSymL, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (16) reduces to N0−NSymL � m=1 ¯Γ[1] n ¯mu m+NSymL [1] = − � dγ[1] n+NSymL ���X [1] L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (17) It is then readily apparent that the NSymL arbitrary functions um [1] for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , NSymL are not determined at this iteration, while r[1] = rank ¯Γ[1] n m of the um [1] for m > NSymL are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' There are then N[2] 0 := N[1] 0 − r[1] second-order Lagrangian constraint functions γ[2] n[2] := � dγ[1] n[2] ���X [1] L � , n[2] = 1, · · · , N[2] 0 , with the conditions γ[2] n[2] = 0 imposed if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In general there will be I[2] := rank � dγ[1] n[1], dγ[2] n[2] � independent functions in C[2] L := C[1] ∪ � γ[2] n[2] | n[2] = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[2] 0 � , and P[1] L is reduced to the second-order constraint submanifold, P[2] L := � u ∈ P[1] L ��� γ[2] [n2](u) = 0, n[2] = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[2] 0 � , where dim P[2] L = 2D−I[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' At this point, there are two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' If I[2] = I[1] or I[2] = 2D, the iterative process stops, and no new Lagrangian constraints are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' If not, the process continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For the second iteration in the constraint algorithm, we choose a basis � P[2] (n) � for ker ΩL(u)/G and the arbitrary functions � um [2] � such that for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[2] 0 , um [2] are linear combinations of um [1] that lie in the kernel Γ[1] nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We once again require that P[2] (n) ∈ SymL for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , NSymL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then X [2] EL = X [2] L + N[2] 0 � m=1 um [2] � P[2] (m) � , with X [2] L = X [1] L + N[1] 0 � m=N[2] 0 +1 um [2] � P[2] (m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Here, the functions um [2] for m = N[2] 0 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[1] 0 have been determined through the con- straint analysis of γ[1] n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As shown in [1], Gum [1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Similarly, Gγ[2] n = L[G,XEL]dγ[2] n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Clearly γ[2] n ∈ F and we may require um [2] ∈ F as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It then follows that � P[2] (n) � γ[2] m = P[2] (n)γ[2] m , and imposing 20 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (15) on γ[2] n , gives N[2] 0 � m=1 Γ[2] nmum [2] = − � dγ[2] n ���X [2] L � , where Γ[2] nm := � dγ[2] n ���P[2] (m) � , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[2] 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (18) Once again, Γ[2] nm = Γ[2] mn, but now on the constraint manifold P[2] L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Moreover, since γ[2] n = γ[1] n = 0 for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , NSymL, Γ[2] nm = 0 = Γ[2] mn, and � dγ[2] n ���X [2] L � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' There is once again a reduction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (18), and we are left with N[2] 0 −NSymL � m=1 Γ [2] n mu m+NSymL [2] = −⟨dγ[2] n+NSymL|X [2] L ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' where ¯Γ[2] n m := � dγ[2] n+NSymL ����P[2] (m+NSymL) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As before, the NSymL arbitrary functions um [2] are not determined, while r[2] := rank ¯Γ[2] n m of the remaining um [2] for m > NSymL are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' There are now N[3] 0 = N[2] 0 − r[2] third-order Lagrangian constraint functions, γ[3] n[3] = � dγ[2] n[3] ���X [2] L � , n[3] = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[3] 0 , with the conditions γ[3] n[3] = 0 imposed if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' With I[3] := rank � dγ[1] n[1], dγ[2] n[2], dγ[3] n[3] � , independent functions in C[3] L := C[2] L ∪ � γ[3] n[3], n[3] = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[3] 0 � , we now have the third- order constraint submanifold, P[3] L := � u ∈ P[2] L ��� γ[3] n[3](u) = 0, n[3] = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[3] 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Once again, the process stops when I[3] = I[2] or I[3] = 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' However, if I[2] < I[3] < 2D, the process continues until at the nF-iteration when either I[nF ] = I[nF ]−1 or I[nF ] = 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Following [1], the end result of this algorithm is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A submanifold P[nF ] L ⊂ PL on which dynamics takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A collection C[nF ] L ⊂ F of constraint functions of order 1 to nF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A second-order, Euler-Lagrange vector field X [nF ] EL = X [nF ] L + N [nF ] 0� m=1 um [nF ](u) � P[nF ] (m) � , 21 with N[nF ] 0 ≥ NSymL arbitrary functions um [nF ](u) ∈ F for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[nF ] 0 , and X [nF ] L = X [1] L + N[1] 0 � m=N [nF ] 0 +1 um [nF ](u) � P[nF ] (m) � , where the N[1] 0 − N[nF ] 0 functions um [nF ](u) ∈ F, m = N[nF ] 0 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , N[1] 0 , have been uniquely determined through the constraint algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We assume that the rank of Γ[l] nm is constant on PL for each l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , nF, and that P[nF ] L is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The end result of the constraint algorithm X [nF ] EL is still a SOELVF, and we define the collection of such vector fields as SolP[nF ] L := {XEL ∈ Sol | LXELβ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Importantly, dim SolP [nF ] L ≥ NSymL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' THE GENERALIZED LIE SYMMETRIES OF THREE DYNAMICAL SYS- TEMS Three examples of dynamical systems with almost regular Lagrangians were introduced in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In that paper the focus of these examples was on the explicit construction of the dynamical structures needed to describe and predict motion in the Lagrangian phase space, and to show that these structures are projectable to the Hamiltonian phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We return to these examples here, but with the focus now being on the generalized Lie symmetries of each, and the application of the results we have found in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In particular, we are in interested in the dimensionality of the symmetry groups for each of the systems as compared to the dimensionality of SolP [nF ] L of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A summary of our results can be found in Table I A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A Lagrangian With and Without a Generalized Lie Symmetry Whether the action S1 := � � 1 2m �d�q dt �2 − V (qa) � dt, 22 with |q| = √qaqa and �qa := qa/|q|, a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , D, has a generalized Lie symmetry depends on the choice of potential V (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' With one choice both the Lagrangian and the Euler-Lagrange equations of motion have a generalized gauge symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' with a second choice the equations of motion has a generalized Lie symmetry while the Lagrangian does not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' and with a third choice neither the action nor the equations of motion have a symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Irrespective of the choice of V (q), however, L is singular, demonstrating that while all actions with a generalized Lie symmetry have a singular Lagrangian, not all singular Lagrangians have a generalized Lie symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Defining Πab(q) := δab − �qa�qb, we find ΩM = m |q|2Πab(q)dqa ∧ dvb, ΩF = m |q|3 (�q · dq) ∧ (v · Π(q) · dq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then C and G are spanned by Uq (1) = �q · ∂/∂q and Uv (1) = �q · ∂/∂v, respectively, while ker ΩL(u) is spanned by Uv (1) and P(1) = �q · ∂ ∂q + 1 |q|v · ∂ ∂v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' That dim (ker ΩL(u)/G) = 1 then follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The energy is E = 1 2 m |q|2v · Π(q) · v + V (q), and there is only one first-order Lagrangian constraint, γ[1] = Uq (1)V, (19) so that β[XEL] = γ[1]Θ(1) q , where Θ(1) q = �q · dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (19), LP(1)β = d � Uq (1)V � − 1 |q|2 �q · ∂ ∂q � Π b a (q)∂V ∂�qb � dqa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (20) Whether or not Sym or SymL is empty therefore depends on the symmetries of V (q), as we would expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It was found in [1] that XL = v · Π(q) · ∂ ∂q + (�q · v) |q| v · ∂ ∂v − |q|2 m ∂V ∂q · Π(q) · ∂ ∂v , and a general SOELVF is given by XEL = XL + u(u) � P(1) � , where u(u) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As the constraint algorithm gives LXELγ[1] = v · Π · ∂γ[1] ∂q + u(u)Uq (1)γ[1], (21) 23 whether or not u(u) (which in turn determines the dimensionality of SolP [nf ] L ) is determined by the constraint condition also depends on the symmetries of V (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' There are three cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The symmetric potential For P(1) to generate a generalized Lie symmetry of the Euler-Lagrange equations of motion, 0 = 1 |q|2 �q · ∂ ∂q � Π b a (q)∂V ∂�qb � , and as such the potential must satisfy ∂V ∂�qa = ∂VAS(�qa) ∂�qa , where VAS is a function of �qa only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It follows that P(1) generates a generalized Lie symmetry iff V (qa) = VSph(|q|) + VAS(�qa), where VSph is a function of |q| only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For this potential, Sym is one-dimensional, and is spanned by P(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The constraint condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (21) for this potential reduces to 0 = u(u)d2VSph(q) d|q|2 , which must be satisfied on P[1] L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' There are two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Case 1: d2VSph d|q|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then VSph(|q|) = a|q| + b, but since γ[1] = dVSph d|q| = a, the condition γ[1] = 0 requires a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It then follows that γ[1] = 0 on PL, and thus SymL is one-dimensional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' it also is spanned by P(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The potential is then V (q) = b + VAS(�qa), and the Lagrangian is invariant under the transformation qa → αqa, where α is an arbitrary, nonvanishing function on PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This Lagrangian therefore has a local conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Importantly, the function u(u) is not determined, and thus the dynamics of the particle is given only up to an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then dim (SolP [nF ] L ) = 1 as well, and is also spanned by P(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Case 2: d2VSph d|q|2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 24 In this case u(u) = 0, and the dynamics of the particle is completely determined by its initial data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' SolP [nF ] L = {XL}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The first-order Lagrangian constraint γ[1] does not vanish automatically, but instead defines a surface on PL, and it follows that SymL = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Indeed, the action’s lack of a local gauge symmetry in this case can be seen explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Equation (19) reduces to 0 = �q · ∂Vsph ∂q , and for dynamics to be possible the set of solutions � Ri ∈ R ����� dVSph d|q| ����� Ri = 0 � , must be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Dynamics are on the surfaces |q|−Ri = 0 where the potential reduces to V (q) = VSph(Ri) + VAS(�qa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This reduced potential has the same symmetry as the potential VAS(�qa) in Case 1, and it is for this reason that the Euler-Lagrange equations of motion have the same generalized Lie symmetry for the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This is explicitly shown in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In Case 1 the action has a local conformal symmetry, while in Case 2 it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (In [1] it was erroneously stated that in this case the action has a global rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=') The Lagrangian for the two cases do not have the same invariances, resulting in one case dynamics that are determined only up to an arbitrary u(u), and in the other case to a u(u) = 0 and dynamics that are instead completely determined by the choice of initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The asymmetric potential For a general V the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (20) does not vanish, P(1) does not generate a symmetry of the equations of motion, and Sym = {∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As before, γ[1] does not vanish, and thus SymL = {∅} as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Furthermore, as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (21) results in XE = XL − v · Π · ∂γ[1] ∂q q2Uq (1)γ[1] [P(1)], the dynamics of the particle is uniquely determined by its initial data, and SolP [nF ] L = {XEL} once again consists of a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 25 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A Lagrangian with Local Conformal Symmetry The action, S2 := � � 1 2m �d�q1 dt �2 + 1 2m �d�q2 dt �2 + λ 2 � qa 1 |q2| d dt � q2a |q1| � − qa 2 |q1| d dt � q1a |q2| �� � dt, where a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , d, D = 2d, describes an interacting, two particle system that is invariant under the local conformal transformation qa 1 → α(u)qa 1 and qa 2 → α(u)qa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' With ΩM = m |q1|2Πab(q1)dqa 1 ∧ dvb 1 + m |q2|2Πab(q2)dqa 2 ∧ dvb 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' and ΩF = m |q1|3 (�q1 · dq1) ∧ (v1 · Π(q1) · dq1) + m |q2|3 (�q2 · dq2) ∧ (v2 · Π(q2) · dq2) − λ |q1||q2| [dqa 1 ∧ (Π(q2) · dq2)a + (Π(q1) · dq1)a ∧ dqa 2 − (Π(q1) · dq1)a ∧ (Π(q2) · dq2)a] − λ |q1|2 (�q1 · dq1) ∧ (�q2 · Π(q1) · dq1) + λ |q2|2 (�q2 · dq2) ∧ (�q1 · Π(q2) · dq2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' C and G are two-dimensional,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' and are spanned by Uq (1) = �q1 · ∂ ∂q1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Uq (2) = �q2 · ∂ ∂q2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' and Uv (1) = �q1 · ∂ ∂v1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Uv (2) = �q2 · ∂ ∂v2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The reduced ¯F = 0, and ker ΩL(u) is spanned by Uv (1), Uv (2), P(+) = q1 · ∂ ∂q1 + q2 · ∂ ∂q2 + v1 · ∂ ∂v1 + v2 · ∂ ∂v2 , and P(−) = q1 · ∂ ∂q1 − q2 · ∂ ∂q2 + v1 · ∂ ∂v1 − v2 · ∂ ∂v2 − 2 λ m �|q1| |q2|q2 · ∂ ∂v1 + |q2| |q1|q1 · ∂ ∂v2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As such, dim (ker ΩL)/G = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' The energy is E = 1 2 m |q1|2v1 · Π(q1) · v1 + 1 2 m |q2|2v2 · Π(q2) · v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We find that γ[1] (+) = 0 while γ[1] (−) = − 2λ |q1||q2| (q2 · Π(q1) · v1 + q1 · Π(q2) · v2) , 26 giving, β[XEL] = 1 2γ[1] (−) � Θ(1) q |q1| − Θ(2) q |q2| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then SymL is one-dimensional and spanned by P(+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As expected, LP(+)β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Because LP(−)β = −4λ m � 1 − (�q1 · �q2)2� � Θ(1) q |q1| − Θ(2) q |q2| � , Sym is also one-dimensional, and is also spanned by P(+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A general SOELVF is XEL = XL − m 8λ2 XLγ[1] (−) [1 − (�q1 · �q2)] � P(−) � + u(+)(u) � P(+) � , (22) where u(+)(u) ∈ F, and from [1], XL = v1 · Π(q1) · ∂ ∂q1 + v2 · Π(q2) · ∂ ∂q2 + ��q1 · v1 |q1| � v1 · Π(q1) · ∂ ∂v1 + ��q2 · v2 |q2| � v2 · Π(q2) · ∂ ∂v2 + λ m �|q1| |q2|v2 · Π(q2) · Π(q1) · ∂ ∂v1 − |q2| |q1|v1 · Π(q1) · Π(q2) · ∂ ∂v2 � , after the constraint algorithm is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Equation (22) is a consequence of the identity ⟨dγ[1] (+)|XL⟩ = 0 and − 1 2λ⟨dγ[1] (−)|XL⟩ = −2(�q1 · �q2)E m + 2 |q1||q2|v1 · Π(q1) · Π(q2) · v2 − λ m(�q1 · �q2) [v2 · Π(q2) · �q1 − v1 · Π(q1) · �q2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We see that SolP [nF ] L is also one-dimensional, and is also spanned by P(+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A Lagrangian with Local Conformal and Time-reparametization Invariance The action S3 := sm � � s �d�q dt �2�1/2 dt, where s = ±1, is invariant under both the local conformal transformations, qa → α(u)qa, and the reparametization of time t → τ(t) where τ is a monotonically increasing function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Then ΩL = m |q| Pab(u) � sv · Π(q) · v dqa ∧ dvb, 27 Action Potential ker ΩL/G Sym SymL I[1] SolP [nF ] L VAS(ˆqa) 1 1 1 0 1 S1 Vsph(|q|) + VAS(ˆqa) 1 1 0 1 0 V (qa) 1 0 0 1 0 S2 λ 2 � qa 1 |q2| d dt � q2a |q1| � − qa 2 |q1| d dt � q1a |q2| �� 2 1 1 1 1 S3 0 2 2 2 0 2 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A summary of the symmetries of the three examples considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' With the exception of the I[1] column, the numerical entries are the dimensionality of the vector spaces listed along the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Notice the case where the Euler-Lagrange equations of motion has a generalized Lie symmetry while the action itself does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In all three examples, dim (SymL) = dim (Sol P [nf ] L ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' and ΩF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Here, a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' , D, ua = Πab(q)vb � sv · Π(q) · v , so that u2 = s, while Pab(u) = Πab(q) − suaub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As such, ker ΩL(u) = ker ΩM(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Both C and G are two-dimensional, and are spanned by Uq (1) = �q · ∂ ∂q , Uq (2) = u · ∂ ∂q , and Uv (1) = �q · ∂ ∂v , Uv (2) = u · ∂ ∂v , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It follows that dim (ker ΩL/G) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Because this system is fully constrained, E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As ΩF = 0 as well, there are no Lagrangian constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It follows that SymL is two dimensional and spanned by Uq (1) and Uq (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As β = 0 as well, Sym is also two dimensional, and is also spanned by Uq (1) and Uq (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We found in [1] that XL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' A general SOELVF is then XEL = u1(u) � Uq (1) � + u2(u) � Uq (2) � , with un(u) ∈ F for n = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It follows that SolP [nF ] L is also two-dimensional, and is spanned by Uq (1) and Uq (2) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' CONCLUDING REMARKS That each generalized Lie symmetry of the action contributes one arbitrary function to the SOELVF for a dynamical system is known anecdotally, and is a result expected on physical grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For almost regular Lagrangians, the appearance in physics of a generalized 28 Lie symmetry is due to a local gauge symmetry of the dynamical system, and thus to the absence of a gauge—the length of vectors for local conformal invariance, or a measure for time for time-reparametization invariance—for some dynamical property of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As the generalized Lie symmetries of the action for an almost regular Lagrangian would have NSymL of these gauge freedoms, it is reasonable that the absence of these gauges will result in an equal number of arbitrary functions in the SOELVF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' An equal number of terms to fix these gauges would then be needed to determine the dynamics of the system uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' But while these expectations are reasonable, up to now they have been fulfilled only on a case-by-case basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This is in great part because the analysis of dynamical systems with a local gauge symmetry has traditionally been done using constrained Hamiltonian mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Such analysis relies on the canonical Hamiltonian, however, and the connection between the canonical Hamiltonian and the symmetries of the Lagrangian is indirect at best, in contrast to the Lagrangian approach followed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Moreover, the process of determining the total Hamiltonian for the system is often prescriptive, with results that are specific to the system at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' By focusing on the Lagrangian and on the Lagrangian phase space, we have been able to show for all systems with an almost regular Lagrangian that has a constant rank Lagrangian two-form, a direct link between local gauge symmetries and its dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' In particular, it establishes a link between the number of gauge symmetries of the action and the number of arbitrary functions that naturally appear in the evolution of such dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' As γ[1] P = 0 for any choice of P ∈ SymL, the vectors in SymL do not contribute to the first-order constraint manifold P[1] L , and as such do not contribute to the Lagrangian constraint algorithm at this order, or at any higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It is for this reason that the NSymL arbitrary functions um [1] are not determined by the algorithm, and why these func- tions will still contribute to XEL even after the algorithm has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It also means that if second- and higher-order Lagrangian constraints are introduced, they are accidental and cannot be due to the local gauge symmetries of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Interestingly, we have yet to find a dynamical system with a Lagrangian that is both almost-regular and has a La- grangian two-form with constant rank where second- or higher-order Lagrangian constraints are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This impact of generalized Lie symmetries on the dynamics of particles illustrates the inherent differences between the analysis of the symmetries of regular Lagrangians and that 29 of almost regular Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' For regular Lagragians, the generator of the generalized Lie symmetry (at times referred to as a global symmetry) gives rise to a prolongation vector, and the action of this prolongation on the Lagrangian gives the variation of the action, δS, under this symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' When the Euler-Lagrange equations of motion are thenimposed, the conserved quantity for this symmetry along the path given by the solution of these equations of motion is then obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' While the generator of the generalized Lie symmetry for the almost regular Lagrangian gL does give a prolongation vector pr gL Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (3), and while the action of pr gL on L does give δS, imposing the Euler-Lagrange equations of motion on δS in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (4) gives the vacuous statement δS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Instead, the requirement that δS = 0 for all paths on Q gives the conditions that the generators of the symmetry must satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This in turn shows that the existence of these generators is due solely to the Lagrangian being singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' These conditions then affect the dynamics of the system through γ[1] P = 0, and in doing so, sets a lower bound to the dimensionality of SolP [nf ] L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We have found it quite difficult to construct more than one example of a dynamical system that has an almost regular Lagrangian with both a generalized Lie symmetry and a Lagrangian two-form with constant rank on PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' We have, on the other hand, found it quite easy to construct examples of dynamical systems that have an almost regular Lagrangian with a generalized Lie symmetry and a Lagrangian two-form whose rank varies across PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Indeed, it is the latter case that is the more prevalent one, and yet much of the results of this paper and a good portion of the results of our previous one [1] relies on the condition that the rank of the Lagrangian two-form be constant on PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' This is even more concerning when we realize that these more prevalent systems are expected, by their nature, to have much richer dynamics and mathematical structures (indeed, we have found that such systems often require the introduction of second- or higher-order Lagrangian constraints), and yet it is not known which of the results that have been shown to hold for systems with constant rank Lagrangian two-forms will still hold when the rank varies across PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Determining the generalized Lie symmetries of these systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' showing that the passage from the Lagrangian to the Hamiltonian phase space is possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' and finding the links between symmetry and dynamics is a necessity for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' 30 ACKNOWLEDGMENTS This paper would not have been possible without the contributions by John Garrison, who provided much of the underlying symmetry analysis of the action used in Section II A, and most of the essential mathematics in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Publication made possible in part by support from the Berkeley Research Impact Initiative (BRII) sponsored by the UC Berkeley Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Appendix The Euler-Lagrangian equations of motion for the action S1 is 0 = m |q|3Πab(q)¨qb − 2m |q|3(�q · ˙q)Πab(q) ˙qb + ∂V ∂qa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content='1) Contracting both sides of this equation with �q results in the first-order Lagrangian constraint Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (19), and it is clear that dynamics is only possible on this constraint surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' Acting on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content='1) with Πab(q) gives 0 = m |q|3Πab(q)¨qb − 2m |q|3(�q · ˙q)Πab(q) ˙qb + Πb a(q)∂V ∂qb, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content='2) since Πac(q)Πc b(q) = Πab(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' But in this case V (qa) = Vsph(|q|) + VAS(ˆq), and as Πb a(q)∂VSph ∂qb = Πb a(q)∂|q| ∂qb V ′ Sph(|q|) = 0, while the identity ∂ˆqa ∂qb = Πa b(q), ensures that Πb a(q)∂VAS ∂qb = ∂VAS ∂qa , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content='2) thereby reduces to the same equations of motion for the system as found for Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' It is for this reason that the two cases have same generalized Lie symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFAT4oBgHgl3EQfAByy/content/2301.08396v1.pdf'} +page_content=' [1] A.' metadata={'source': 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Variational Feature Aggregation +Jiaming Han1,2*, Yuqiang Ren3, Jian Ding1,2, Ke Yan3†, Gui-Song Xia1,2† +1NERCMS, School of Computer Science, Wuhan University +2State Key Lab. LIESMARS, Wuhan University +3YouTu Lab, Tencent +{hanjiaming, jian.ding, guisong.xia}@whu.edu.cn, {condiren, kerwinyan}@tencent.com +Abstract +As few-shot object detectors are often trained with abundant +base samples and fine-tuned on few-shot novel examples, the +learned models are usually biased to base classes and sensi- +tive to the variance of novel examples. To address this issue, +we propose a meta-learning framework with two novel fea- +ture aggregation schemes. More precisely, we first present a +Class-Agnostic Aggregation (CAA) method, where the query +and support features can be aggregated regardless of their cat- +egories. The interactions between different classes encourage +class-agnostic representations and reduce confusion between +base and novel classes. Based on the CAA, we then propose +a Variational Feature Aggregation (VFA) method, which en- +codes support examples into class-level support features for +robust feature aggregation. We use a variational autoencoder +to estimate class distributions and sample variational features +from distributions that are more robust to the variance of sup- +port examples. Besides, we decouple classification and re- +gression tasks so that VFA is performed on the classifica- +tion branch without affecting object localization. Extensive +experiments on PASCAL VOC and COCO demonstrate that +our method significantly outperforms a strong baseline (up to +16%) and previous state-of-the-art methods (4% in average). +Code will be available at: https://github.com/csuhan/VFA +Introduction +This paper studies the problem of few-shot object detection +(FSOD), a recently-emerged challenging task in computer +vision (Yan et al. 2019; Kang et al. 2019). Different from +generic object detection (Girshick et al. 2014; Redmon et al. +2016; Ren et al. 2017), FSOD assumes that we have abun- +dant samples of some base classes but only a few exam- +ples of novel classes. Thus, a dynamic topic is how to im- +prove the recognition capability of FSOD on novel classes +by transferring the knowledge of base classes to novel ones. +In general, FSOD follows a two-stage training paradigm. +In stage-I, the detector is trained with abundant base sam- +ples to learn generic representations required for the ob- +ject detection task, such as object localization and classifica- +tion. In stage-II, the detector is fine-tuned with only K shots +(K=1, 2, 3, . . . ) novel examples. Despite the great success +*Work done during internship at Tencent YouTu Lab. +†Corresponding author. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +plain FCs +(previous) +VAEs +(Ours) +Appearance variations of support images +������������������������ = Avg(������������1, ������������2, … ) is sensitive to example’s variations +������������������������~������������(������������, ������������) is robust to few-shot examples +������������(������������, ������������) +������������������������ +������������3 +������������4 +������������2 +������������1 +������������������������ +������������2 +������������3 +������������4 +������������1 +������������������������ +������������2 +������������4 +������������1 +������������(������������, ������������) +������������������������ +������������4 +������������2 +������������1 +rm. ������������3 +rm. ������������3 +Figure 1: Comparisons of different support feature encoding +methods. Previous methods use plain fully-connected (FC) +layers to encode support features and obtain class prototypes +by averaging these features: xp = Avg(x1, x2, . . . ). In con- +trast, our method uses variational autoencoders (VAEs) pre- +trained on abundant base examples to estimate the distribu- +tions of novel classes. Since intra-class variance is shared +across classes and can be modeled with common distribu- +tions (Lin et al. 2018), we use a shared VAE to transfer the +distributions of base classes to novel classes. Finally, we can +sample class prototypes xp from the distributions N(µ, σ) +that are robust to the variance of few-shot examples. rm.: +remove. +of this paradigm, the learned models are usually biased to +base classes due to the imbalance between base and novel +classes. As a result, the model will confuse novel objects +with similar base classes. See Fig. 5 (top) for an instance, +the novel class, cow, has high similarities with several base +classes such as dog, horse and sheep. Besides, the model is +sensitive to the variance of novel examples. Since we only +have K shots examples per class, the performance highly de- +pends on the quality of the support sets. As shown in Fig. 1, +appearance variations are common in FSOD. Previous meth- +ods (Yan et al. 2019) consider each support example as a +single point in the feature space and average all features as +class prototypes. However, it is difficult to estimate the real +class centers with a few examples. +In this paper, we propose a meta-learning framework to +address this issue. Firstly, we build a strong meta-learning +baseline based on Meta R-CNN (Yan et al. 2019), which +arXiv:2301.13411v1 [cs.CV] 31 Jan 2023 + +even outperforms a representative two-stage fine-tuning ap- +proach TFA (Wang et al. 2020). By revisiting the feature ag- +gregation module in meta-learning frameworks, we propose +Class-Agnostic Aggregation (CAA) and Variational Feature +Aggregation (VFA) to reduce class bias and improve the ro- +bustness to example’s variances, respectively. +Feature aggregation is a crucial design in FSOD, which +defines how query and support examples interact. Previous +works such as Meta R-CNN adopt a class-specific aggrega- +tion scheme (Fig. 2 (a)), i.e., query features are aggregated +with support features of the same class, ignoring cross-class +interactions. In contrast, we propose CAA (Fig. 2 (b)) which +allows feature aggregation between different classes. Since +CAA encourages the model to learn class-agnostic represen- +tations, the bias towards base classes is reduced. Besides, the +interactions between different classes simultaneously model +class relations so that novel classes will not be confused with +base classes. +Based on CAA, we propose VFA which encodes sup- +port examples into class-level support features. Our mo- +tivation is that intra-class variance (e.g. appearance varia- +tions) is shared across classes and can be modeled with +common distributions (Lin et al. 2018). So we can use base +classes’ distributions to estimate novel classes’ distributions. +We achieve this by modeling each class as a common dis- +tribution with variational autoencoders (VAEs). We firstly +train the VAE on abundant base examples and then fine-tune +it on few-shot novel examples. By transferring the learned +intra-class variance to novel classes, our method can esti- +mate novel classes’ distributions with only a few examples +(Fig. 1). Finally, we sample support features from distri- +butions and aggregate them with query features to produce +more robust predictions. +We also propose to decouple classification and regression +tasks so that our feature aggregation module can focus on +learning translation-invariant features without affecting ob- +ject localization. We conduct extensive experiments on two +FSOD datasets, PASCAL VOC (Everingham et al. 2010) +and COCO (Lin et al. 2014) to demonstrate the effectiveness +of our method. We summarize our contributions as follows: +• We build a strong meta-learning baseline Meta R- +CNN++ and propose a simple yet effective Class- +Agnostic Aggregation (CAA) method. +• We propose Variational Feature Aggregation (VFA), +which transforms instance-wise features into class-level +features for robust feature aggregation. To our best +knowledge, we are the first to introduce variational fea- +ture learning into FSOD. +• Our method significantly improves the baseline Meta R- +CNN++ and achieves a new state-of-the-art for FSOD. +For example, we outperform the strong baseline by +9%∼16% and previous best results by 3%∼7% on the +Novel Set 1 of PASCAL VOC. +Related Work +Generic Object Detection. Object detection has witnessed +significant progress in the past decade, which can be roughly +divided into two groups: one-stage and two-stage detectors. +One-stage detectors predict bounding boxes and class la- +bels by presetting dense anchor boxes (Redmon et al. 2016; +Liu et al. 2016; Lin et al. 2017), points (Law and Deng +2018; Zhou, Wang, and Kr¨ahenb¨uhl 2019), or directly out- +put sparse predictions (Carion et al. 2020; Chen et al. 2021). +Two-stage detectors (Girshick et al. 2014; Girshick 2015; +Ren et al. 2017) first generate a set of object proposals +with Region Proposal Network (RPN) and then perform +proposal-wise classification and regression. However, most +generic detectors are trained with abundant samples and not +designed for data-scarce scenarios. +Few-Shot Object Detection. Early attempts (Kang et al. +2019; Yan et al. 2019; Wang, Ramanan, and Hebert 2019) +in FSOD adopt meta-learning architectures. FSRW (Kang +et al. 2019) and Meta R-CNN (Yan et al. 2019) aggregate +image/RoI-level query features with support features gener- +ated by a meta learner. Following works explore different +designs of meta-learning architectures, e.g., feature aggre- +gation scheme (Xiao and Marlet 2020; Fan et al. 2020; Hu +et al. 2021; Zhang et al. 2021; Han et al. 2021) and feature +space augmentation (Li et al. 2021a; Li and Li 2021). Differ- +ent from meta-learning, Wang et al. propose a simple two- +stage fine-tuning approach, TFA (Wang et al. 2020). TFA +shows that only fine-tuning the last layers can significantly +improve the FSOD performance. Due to the simple structure +of TFA, a line of works (Sun et al. 2021; Zhu et al. 2021; +Qiao et al. 2021; Cao et al. 2021) following TFA are pro- +posed. In this work, we build a strong meta-learning base- +line that even surpasses the fine-tuning baseline TFA. Then +we revisit the feature aggregation scheme and propose two +novel feature aggregation methods, CAA and VFA, achiev- +ing a new state-of-the-art in FSOD. +Variational Feature Learning. Given an input image/fea- +ture, we can transform it into a distribution with VAEs. By +sampling features from the distribution, we can model intra- +class variance that defines the class’s character. The varia- +tional feature learning paradigm has been used in various +tasks, e.g., zero/few-shot learning (Zhang et al. 2019; Xu +et al. 2021; Kim et al. 2019), metric learning (Lin et al. +2018) and disentanglement learning (Ding et al. 2020). In +this work, we use VAEs trained on abundant base examples +to estimate novel classes’ distributions with only a few ex- +amples. Besides, we also propose a consistency loss to make +the model produce class-specific distributions. To our best +knowledge, we are the first to introduce variational feature +learning into FSOD. +Background and Meta R-CNN++ +Preliminaries +Problem Definition. We follow the FSOD settings in previ- +ous works (Yan et al. 2019; Wang et al. 2020). Assume we +have a dataset D = {(x, y), x ∈ X, y ∈ Y } with a set of +classes C, where x is the input image and y = {ci, bi}N +i=1 +is the corresponding class label c and bounding box b an- +notations. We then split the dataset into base classes Cb and +novel classes Cn where Cb ∪ Cn = C and Cb ∩ Cn = ∅. +Generally, we have abundant samples of Cb and K shots +samples of Cn (K=1, 2, 3, ...). The goal is to detect objects + +setting +TFA +Meta R-CNN∗ +Meta R-CNN++ +param freeze +✓ + +✓ +✓ +✓ +cosine cls. +✓ + + +✓ +✓ +last layer init. +copy +rand +rand +rand +copy +bAP (stage-I) +80.8 +72.8 +77.6 +77.6 +77.6 +bAP (stage-II) +79.6 +47.4 +64.9 +68.2 +76.8 +nAP +39.8 +20.7 +42.0 +40.5 +41.6 +Table 1: Difference analysis between Meta R-CNN and +TFA. The results are evaluated under the 1 shot setting of +PASCAL VOC Novel Set 1. stage-I and stage-II: base train- +ing and fine-tuning stages. ∗: Our re-implemented results. +of Cn with only K shots annotated instances. Existing few- +shot detectors usually adopt a two-stage training paradigm: +base training and few-shot fine-tuning, where the representa- +tions learned from Cb are transferred to detect novel objects +in the fine-tuning stage. +Meta-Learning Based FSOD. We take Meta R-CNN (Yan +et al. 2019) for an example. As shown in Fig. 3, the main +framework is a siamese network with a query feature en- +coder FQ, a support feature encoder FS, a feature aggre- +gator A and a detection head FD. Typically, FQ and FS +share most parameters and A refers to the channel-wise +product operation. Meta R-CNN follows the episodic train- +ing paradigm (Vinyals et al. 2016). Each episode is com- +posed of a set of support images and binary masks of an- +notated objects, {xi, Mi}N +i=1, where N is the number of +training classes. Specifically, we first feed the support set +{xi, Mi}N +i=1 to FS to generate class-specific support fea- +tures {Si}i∈C, and the query image to FQ to generate a set +of RoI features {Qm} (m is the index of RoIs). Then we +aggregate each Qm and Si with the feature aggregator A. +Finally, the aggregated features �Qm +i are fed to the detection +head FD to produce final predictions. +Meta R-CNN++: Stronger Meta-Learning Baseline +Meta-learning has proved a promising approach, but the +fine-tuning based approach receives more and more atten- +tion recently due to its superior performance. Here we aim +to bridge the gap between the two approaches. We choose +Meta R-CNN and TFA as baselines and explore how to build +a strong FSOD baseline with meta-learning. +Although both methods follow a two-stage training +paradigm, TFA optimizes the model with advanced tech- +niques in the fine-tuning stage: (a) TFA freezes most net- +work parameters, and only trains the last classification and +regression layers so that the model will not overfit to few- +shot examples. (b) Instead of randomly initializing the clas- +sification layer, TFA copies pre-trained weights of base +classes and only initializes the weights of novel classes. (c) +TFA adopts cosine classifier (Gidaris and Komodakis 2018) +rather than a linear classifier. +Considering the success of TFA, we build Meta R- +CNN++, which follows the architecture of Meta R-CNN +but aligns most hyper-parameters with TFA. Here we ex- +plore different design choices to mitigate the gap between +������������������������ +������������������������ +������������ +ℒ +support images +query image +same class +������������������������ +������������������������ +������������ +ℒ +support images +query image +arbitrary +classes +(a) Class-specific aggregation +(b) Class-agnostic aggregation +Figure 2: Illustration of two feature aggregation methods. +Si/Qi: support and query features of class i. A: feature ag- +gregation. L: loss functions. +the two approaches, shown in Tab. 1. (a) Parameter freeze. +By adopting the same parameter freezing strategy, Meta R- +CNN++ significantly outperforms Meta R-CNN and even +achieves higher novel AP than TFA. (b) Cosine classifier. +Different from TFA, Meta R-CNN++ with the cosine classi- +fier does not surpass the linear classifier in nAP (41.6 vs. +42.0), but its performance on base classes is better than +the linear classifier (68.2 vs. 64.9). (c) Alleviate base for- +getting. We follow TFA and copy the pre-trained classifier +weights of base classes. We find Meta R-CNN++ can also +maintain the performance on base classes (76.8 vs. 77.6). +The above experiments indicate that meta-learning re- +mains a promising approach for FSOD as long as we care- +fully handle the fine-tuning stage. Therefore, we choose +Meta R-CNN++ as our baseline in the following sections. +The Proposed Approach +Class-Agnostic Aggregation +Feature aggregation is an important module in meta-learning +based FSOD (Kang et al. 2019; Yan et al. 2019). Many +works adopt a class-specific aggregation (CSA) scheme. +Let us assume that a query image has an object of class +CQ = {i} and the corresponding RoI features {Qm +i }. In the +training phase, as shown in Fig. 2 (a), CSA aggregates each +RoI feature Qm +i +with the support features Si of the same +class: �Qm +ii = A(Qm +i , Si). In the testing phase, CSA aggre- +gates the RoI feature with support features of all classes: +�Qm +ij = A(Qm +i , Sj), j ∈ C, and each support feature Sj +is to predict objects of its corresponding class. Notably, +if the query image contains multiple classes, CSA aggre- +gates the query features with each support feature in CQ: +�Qm +ij = A(Qm +i , Sj), j ∈ CQ. But CSA still follows the class- +specific way, as support features not belonging to CQ will +never be aggregated with the query feature. +As discussed before, the learned models are usually bi- +ased to base classes due to the imbalance between base +and novel classes. Therefore, we revisit CSA and propose +a simple yet effective Class-Agnostic Aggregation (CAA). +See Fig. 2 (b) for an instance, CAA allows feature ag- +gregation between different classes, which encourages the +model to learn class-agnostic representations and thereby re- +duces the class bias. Besides, the interactions between differ- +ent classes can simultaneously model class relations so that + +query image +ℱ������������ +ℱ������������ +support images +������������ +������������ +������������ +������������ +������������ +������������(������������, ������������) +Variational Feature Aggregation +ℱ������������������������������������ +ℱ������������������������������������ +������������′ +������������ +sampling +ℒ������������������������������������ +ℒ������������������������������������������������ +ℒ������������������������������������ +ℒ������������������������ +query feature +support +feature +ℱ������������ +Figure 3: Overview of our framework. FQ and FS denote query and support feature extractors, respectively. Fenc and Fdec are +the variational feature encoder and decoder. FD: the detection head. A: feature aggregation. Note that we do not visualize RPN +and the regression branch for simplicity. +novel classes will not confuse with base classes. Formally, +for each RoI feature Qm +i of class i ∈ C and a set of support +features {Sj}j∈C, we randomly select a support feature Sj∗ +of class j∗ to aggregate with the query feature, +�Qm +ij∗ = A(Qm +i , Sj∗), j∗ ∈ C. +(1) +Then we feed the aggregated feature �Qm +ij∗ to the detection +head FD to output classification scores p = FD( �Qm +ij∗), +which is supervised with the label of class i. Note that CAA +is used for training; the testing phase still follows CSA. +Variational Feature Aggregation +Prior works usually encode support examples into single fea- +ture vectors that are difficult to represent the whole class dis- +tribution. Especially when the data is scarce and example’s +variations are large, we cannot make an accurate estimation +of class centers. Inspired by recent progress in variational +feature learning (Lin et al. 2018; Zhang et al. 2019; Xu et al. +2021), we transform support features into class distributions +with VAEs. Since the estimated distribution is not biased to +specific examples, features sampled from the distribution are +robust to the variance of support examples. Then we can +sample class-level features for robust feature aggregation. +The framework of VFA is shown in Fig. 3. +Variational Feature Learning. Formally, we aim to trans- +form the support feature S into a class distribution N, and +sample the variational feature z from N for feature aggre- +gation. We optimize the model in a similar way to VAEs, +but our goal is to sample the latent variable z instead of the +reconstructed feature S +′. Following the definition of VAEs, +we assume z is generated from a prior distribution p(z) and +S is generated from a conditional distribution p(S|z). As the +process is hidden and z is unknown, we model the posterior +distribution with variational inference. More specifically, +we approximate the true posterior distribution p(z|S) with +another distribution q(z|S) by minimizing the Kullback- +Leibler (KL) divergence: +DKL(q(z|S)||p(z|S)) = +� +q(z|S) log q(z|S) +p(z|S), +(2) +which is equivalent to maximizing the evidence lower bound +(ELBO): +ELBO = Eq(z|S)[log p(S|z))] − DKL(q(z|S)||p(z)). +(3) +Here we assume the prior distribution of z is a centered +isotropic multivariate Gaussian, p(z) = N(0, I), and set the +posterior distribution q(z|S) to be a multivariate Gaussian +with diagonal covariance: q(z|S) = N(µ, σ). The parame- +ters µ and σ can be implemented by a feature encoder Fenc: +µ, σ = Fenc(S). Then we obtain the variational feature +z with the reparameterization trick (Kingma and Welling +2013): z = µ + σ · ϵ, where ϵ ∼ N(0, I). The first term of +Eq. 3 can be simplified to a reconstruction loss Lrec which +is usually defined as the L2 distance between the input S and +the reconstructed target S +′, +Lrec = ∥S − S +′∥ = ∥S − Fdec(z)∥, +(4) +where Fdec denotes a feature decoder. As for the second +term of Eq. 3, we directly minimize the KL divergence of +q(z|S) and p(z), +LKL = DKL(q(z|S)||p(z)), +(5) +which forces the variation feature z to follow a normal dis- +tribution. +By optimizing the two objectives, Lrec and LKL, we trans- +form the support feature S into a distribution N. Then we +can sample the variational feature z from N. Since z still +lacks class-specific information, we apply a consistency loss +Lcons to the reconstructed feature S +′, which is defined as the +cross-entropy between S +′ and its class label c, +Lcons = LCE(FS +′ +cls(S +′), c), +(6) + +Method / Shots +Backbone +Novel Set 1 +Novel Set 2 +Novel Set 3 +Avg. +1 +2 +3 +5 +10 +1 +2 +3 +5 +10 +1 +2 +3 +5 +10 +FSRW (Kang et al. 2019) +YOLOv2 +14.8 15.5 26.7 33.9 47.2 +15.7 15.3 22.7 30.1 40.5 +21.3 25.6 28.4 42.8 45.9 +28.4 +MetaDet (Wang et al. 2019) +VGG16 +18.9 20.6 30.2 36.8 49.6 +21.8 23.1 27.8 31.7 43.0 +20.6 23.9 29.4 43.9 44.1 +31.0 +Meta R-CNN (Yan et al. 2019) +ResNet-101 +19.9 25.5 35.0 45.7 51.5 +10.4 19.4 29.6 34.8 45.4 +14.3 18.2 27.5 41.2 48.1 +31.1 +TFA w/ cos (Wang et al. 2020) +ResNet-101 +39.8 36.1 44.7 55.7 56.0 +23.5 26.9 34.1 35.1 39.1 +30.8 34.8 42.8 49.5 49.8 +39.9 +MPSR (Wu et al. 2020) +ResNet-101 +41.7 +- +51.4 55.2 61.8 +24.4 +- +39.2 39.9 47.8 +35.6 +- +42.3 48.0 49.7 +- +Retentive (Fan et al. 2021) +ResNet-101 +42.4 45.8 45.9 53.7 56.1 +21.7 27.8 35.2 37.0 40.3 +30.2 37.6 43.0 49.7 50.1 +41.1 +Halluc (Zhang and Wang 2021) +ResNet-101 +47.0 44.9 46.5 54.7 54.7 +26.3 31.8 37.4 37.4 41.2 +40.4 42.1 43.3 51.4 49.6 +43.2 +CGDP+FSCN (Li et al. 2021b) +ResNet-101 +40.7 45.1 46.5 57.4 62.4 +27.3 31.4 40.8 42.7 46.3 +31.2 36.4 43.7 50.1 55.6 +43.8 +CME (Li et al. 2021a) +ResNet-101 +41.5 47.5 50.4 58.2 60.9 +27.2 30.2 41.4 42.5 46.8 +34.3 39.6 45.1 48.3 51.5 +44.4 +SRR-FSD (Zhu et al. 2021) +ResNet-101 +47.8 50.5 51.3 55.2 56.8 +32.5 35.3 39.1 40.8 43.8 +40.1 41.5 44.3 46.9 46.4 +44.8 +FSOD-UP (Wu et al. 2021) +ResNet-101 +43.8 47.8 50.3 55.4 61.7 +31.2 30.5 41.2 42.2 48.3 +35.5 39.7 43.9 50.6 53.5 +45.0 +FSCE (Sun et al. 2021) +ResNet-101 +44.2 43.8 51.4 61.9 63.4 +27.3 29.5 43.5 44.2 50.2 +37.2 41.9 47.5 54.6 58.5 +46.6 +QA-FewDet (Han et al. 2021) +ResNet-101 +42.4 51.9 55.7 62.6 63.4 +25.9 37.8 46.6 48.9 51.1 +35.2 42.9 47.8 54.8 53.5 +48.0 +FADI (Cao et al. 2021) +ResNet-101 +50.3 54.8 54.2 59.3 63.2 +30.6 35.0 40.3 42.8 48.0 +45.7 49.7 49.1 55.0 59.6 +49.2 +Zhang et al. (Zhang et al. 2021) +ResNet-101 +48.6 51.1 52.0 53.7 54.3 +41.6 45.4 45.8 46.3 48.0 +46.1 51.7 52.6 54.1 55.0 +49.8 +Meta FR-CNN (Han et al. 2022) +ResNet-101 +43.0 54.5 60.6 66.1 65.4 +27.7 35.5 46.1 47.8 51.4 +40.6 46.4 53.4 59.9 58.6 +50.5 +DeFRCN (Qiao et al. 2021) +ResNet-101 +53.6 57.5 61.5 64.1 60.8 +30.1 38.1 47.0 53.3 47.9 +48.4 50.9 52.3 54.9 57.4 +51.9 +VFA (Ours) +ResNet-101 +57.7 64.6 64.7 67.2 67.4 +41.4 46.2 51.1 51.8 51.6 +48.9 54.8 56.6 59.0 58.9 +56.1 +Table 2: Results on PASCAL VOC. The results are sorted by the averaged score (Avg.). See our appendix for the generalized +FSOD results. +where FS +′ +cls denotes a linear classifier. The introduction of +Lcons transforms the learned distributions into class-specific +distributions. The support feature Si is forced to approxi- +mate a parameterized distribution N(µi, σi) of class i, so +that the sampled z can preserve class-specific information. +Variational Feature Aggregation. Since the support fea- +tures are transformed into class distributions, we can sample +features from the distribution and aggregate them with query +features. Compared with the original support feature S and +reconstructed feature S +′, the latent variable z contains more +generic features of the class (Zhang et al. 2019; Lin et al. +2018), which is robust to the variance of support examples. +Specifically, VFA follows the class-agnostic approach in +CAA but aggregates the query feature Q with a variational +feature z. Given a query feature Qi of class i and support +feature Sj of class j, we firstly approximate the class dis- +tribution N(µj, σj) and sample a variational feature zj = +µj + σj from N(µj, σj). Then we aggregate them together +with the following equation: +�Qij = A(Qi, zj) = Qi ⊙ sig(zj), +(7) +where ⊙ means channel-wise multiplication and sig is short +for the sigmoid operation. In the training phase, we ran- +domly select a support feature Sj (i.e., one support class j) +for aggregation. In the testing phase (especially K > 1), +we average K support features of class j into one ¯Sj, and +approximate the distribution N(µj, σj) with the averaged +feature, µj, σj = Fenc( ¯Sj). Instead of adopting complex +distribution estimation methods, we find the averaging ap- +proach works well in our method. +Network and Objective. VFA only introduces a light en- +coder Fenc and decoder Fdec. Fenc contains a linear layer +and two parallel linear layers to produce µ and σ, respec- +tively. Fdec consists of two linear layers to generate the re- +constructed feature S +′. We keep all layers the same dimen- +sion (2048 by default). VFA is trained in an end-to-end man- +ner with the following multi-task loss: +L = Lrpn + Lreg + Lcls + Lcons + Lrec + αLKL, +(8) +where Lrpn is the total loss of RPN, Lreg is the regression +loss, and α is a weight coefficient (α=2.5×10−4 by default). +Classification-Regression Decoupling +Generally, the detection head FD contains a shared fea- +ture extractor Fshare and two separate network Fcls and +Freg for classification and regression, respectively. In previ- +ous works, the aggregated feature is fed to FD to produce +both classification scores and bounding boxes. However, +the classification task requires translation-invariant features, +while regression needs translation-covariant features (Qiao +et al. 2021). Since support features are always translation- +invariant to represent class centers, the aggregated feature +harms the regression task. Therefore, we decouple the two +tasks in the detection head. Let Q and �Q denote the origi- +nal and aggregated query features. Previous methods take �Q +for both tasks, where the classification score p and predicted +bounding boxes b are defined as: +p = Fcls(Fshare( �Q)), b = Freg(Fshare( �Q)). +(9) +To decouple these tasks, we adopt separate feature extractors +and use the original query feature Q for regression, +p = Fcls(Fcls +share( �Q)), b = Freg(Freg +share(Q)), +(10) +where Fcls +share and Freg +share are the feature extractor for clas- +sification and regression, respectively. +Experiments and Analysis +Experimental Setting +Datasets. We evaluate our method on PASCAL VOC (Ever- +ingham et al. 2010) and COCO (Lin et al. 2014), following + +Method / Shots +10 +30 +Fine-tuning +MPSR (Wu et al. 2020) +9.8 +14.1 +TFA w/ cos (Wang et al. 2020) +10.0 +13.7 +Retentive (Fan et al. 2021) +10.5 +13.8 +FSOD-UP (Wu et al. 2021) +11.0 +15.6 +SRR-FSD (Zhu et al. 2021) +11.3 +14.7 +CGDP+FSCN (Li et al. 2021b) +11.3 +15.1 +FSCE (Sun et al. 2021) +11.9 +16.4 +FADI (Cao et al. 2021) +12.2 +16.1 +DeFRCN (Qiao et al. 2021) +18.5 +22.6 +Meta-learning +FSRW (Kang et al. 2019) +5.6 +9.1 +MetaDet (Wang, Ramanan, and Hebert 2019) +7.1 +11.3 +Meta R-CNN (Yan et al. 2019) +8.7 +12.4 +QA-FewDet (Han et al. 2021) +11.6 +16.5 +FSDetView (Xiao and Marlet 2020) +12.5 +14.7 +Meta FR-CNN (Han et al. 2022) +12.7 +16.6 +DCNet (Hu et al. 2021) +12.8 +18.6 +CME (Li et al. 2021a) +15.1 +16.9 +VFA (Ours) +16.2 +18.9 +Table 3: Results on COCO. The backbone is the same as +Tab. 2. The results are sorted by 10-shot nAP. See our ap- +pendix for the generalized FSOD results. +previous works (Kang et al. 2019; Wang et al. 2020). We use +the data splits and annotations provided by TFA (Wang et al. +2020) for a fair comparison. For PASCAL VOC, we split +20 classes into three groups, where each group contains 15 +base classes and 5 novel classes. For each novel set, we have +K={1, 2, 3, 5, 10} shots settings. For COCO, we set 60 cat- +egories disjoint with PASCAL VOC as base classes and the +remaining 20 as novel classes. We have K={10, 30} shots +settings. +Evaluation Metrics. For PASCAL VOC, we report the +Average Precision at IoU=0.5 of base classes (bAP) and +novel classes (nAP). For COCO, we report the mean AP at +IoU=0.5:0.95 of novel classes (nAP). +Implementation Details. We implement our method with +Mmdetection (Chen et al. 2019). The backbone is ResNet- +101 (He et al. 2016) pre-trained on ImageNet (Russakovsky +et al. 2015). We adopt SGD as the optimizer with batch size +32, learning rate 0.02, momentum 0.9 and weight decay 1e- +4. The learning rate is changed to 0.001 in the few-shot fine- +tuning stage. We fine-tune the model with {400, 800, 1200, +1600, 2000} iterations for K={1, 2, 3, 5, 10} shots in PAS- +CAL VOC, and {10000, 20000} iterations for K={10, 30} +shots in COCO. We keep other hyper-parameters the same +as Meta R-CNN (Yan et al. 2019) if not specified. +Main Results +PASCAL VOC. As shown in Tab. 2, VFA significantly out- +performs existing methods. VFA achieves the best (13/16) +or second-best (3/16) results on all settings. In Novel Set +1, VFA outperforms previous best results by 3.2%∼7.1%. +Our 2-shot result even surpasses previous best 10-shot re- +sults (64.6% vs. 63.4%), which indicates that our method is +Method +CRD +CAA +VFA +Shots +1 +3 +5 +Meta R-CNN++ +42.0 +56.5 +58.3 +Ours +✓ +46.0 +61.7 +62.3 +✓ +✓ +51.3 +62.8 +66.4 +✓ +✓ +✓ +57.7 +64.7 +67.2 +Table 4: Effect of different modules. +76.7 +80.2 +81.4 +84.2 +86.4 +86.9 +84.3 +87.4 +90.6 +87.2 +90.3 +91.3 +1-shot +3-shot +5-shot +75 +80 +85 +90 +95 +Recall +w/o CRD + Novel Class + Base Class +1-shot +3-shot +5-shot +75 +80 +85 +90 +95 +Recall +w/ CRD + Novel Class + Base Class +Figure 4: Comparisons of recall without/with CRD. +more robust to the variance of few-shot examples. Besides, +we notice that our gains are stable and consistent. This phe- +nomenon demonstrates that VFA is not biased to specified +class sets and can be generalized to more common scenarios. +Furthermore, VFA obtains a 56.1% average score and sur- +passes the second-best result by 4.2%, which further demon- +strates its effectiveness. +COCO. As shown in Tab. 3, VFA achieves the best nAP +among meta-learning based methods and second-best re- +sults among all methods. We notice that a fine-tuning +based method, DeFRCN (Qiao et al. 2021), outperforms our +method in nAP. To concentrate on the feature aggregation +module in meta-learning, we do not utilize advanced tech- +niques, e.g., the gradient decoupled layer (Qiao et al. 2021) +in DeFRCN. We believe the performance of VFA can be fur- +ther boosted with more advanced techniques. +Ablation Studies +We conduct a series of ablation experiments on Novel Set 1 +of PASCAL VOC. +Effect of different modules. As shown in Tab. 4, we eval- +uate the effect of different modules by gradually applying +the proposed modules to Meta R-CNN++. Although Meta +R-CNN++ is competitive enough, we show CRD improves +the performance on nAP, where the absolute gains exceed +4%. Besides, we find CRD significantly improves the re- +call on all classes (Fig. 4) and narrows the gap between base +and novel classes because it uses separate networks to learn +translation-invariant and -covariant features. Then, we ap- +ply CAA to the model and obtain further improvements. +The confusions between different classes are reduced. Fi- +nally, we build VFA and achieve a new state-of-the-art. The +1-shot performance is even comparable with 5-shot Meta R- +CNN++ in nAP, indicating that VFA is robust to the variance +of support examples especially when the data is scarce. + +bird +bus +cow +motorbike +sofa +0 +0.2 +0.4 +0.6 +0.8 +1 +Meta R-CNN++ +CAA +VFA +bird +bus +cow +motorbike +sofa +aeroplane +bicycle +boat +bottle +car +cat +chair +diningtab +dog +horse +person +pottleplant +sheep +train +tvmonitor +bird +bus +cow +motorbike +sofa +bird +bus +cow +motorbike +sofa +Figure 5: Similarity matrix visualization. We calculate co- +sine similarities of support features in the 5-shot setting of +PASCAL VOC Novel Set 1. sofa, motorbike, cow, bus and +bird are novel classes. Warmer color denotes higher similar- +ity. Zoom in for details. +1 +1.24 +1.55 +1.61 +2.85 +1.12 +1.58 +2.62 +2.95 +5.1 +10-shot +5-shot +3-shot +2-shot +1-shot +1 +2 +3 +4 +5 +relative distance + baseline + Ours +Figure 6: The distance from the estimated prototype of K- +shot examples to the real class center. For each novel class, +we take the mean feature of all training examples as its real +class center. Our 10-shot result is the reference distance, +while other results are relative distances. We only report the +averaged distance of all novel classes for simplicity. +Visual analysis of different feature aggregation. Fig. 5 +gives a visual analysis of different feature aggregation meth- +ods. Due to the imbalance between base and novel classes, +some novel classes are confused with base classes in Meta +R-CNN++ (with CSA), e.g., a novel classe, cow have higher +similarity (>0.8) with horse and sheep. In contrast, CAA +reduces class bias and confusion by learning class-agnostic +representations. The inter-class similarities are also reduced +so that a novel example will not be classified to base classes. +Finally, we use VFA to transforms support examples into +class distributions. By learning intra-class variances from +abundant base examples, we can estimate novel classes’ dis- +tributions even with a few examples. In Fig. 5 (bottom), we +can see VFA significantly improves intra-class similarities. +Robust and accurate class prototypes. In the testing phase, +detectors take the mean feature of K-shot examples as the +class prototype. As shown in Fig. 6, our estimated class pro- +totypes are more robust and accurate than the baseline. The +distances to real class centers do not increase much as the +Features +S +S +′ +µ +σ +�z +z +bAP +78.8 +78.1 +78.6 +78.3 +78.0 +78.6 +nAP +1 +55.2 +54.4 +56.6 +55.4 +53.0 +57.7 +3 +63.7 +63.6 +63.7 +64.9 +63.2 +64.7 +5 +66.6 +66.9 +66.7 +66.9 +66.3 +67.2 +avg. +61.8 +61.6 +62.3 +62.4 +60.8 +63.2 +Table 5: Comparisons of different support features. S and S +′ +are the original and reconstructed features. µ, σ, �z = µ+ϵ·σ +and z = µ + σ are latent variables. avg.: The average score. +Setting / Shots +1 +3 +5 +w/o VFA +51.3 +62.8 +66.4 +w/ VFA +w/o Lcons +53.6 +64.3 +66.7 +Lcons on S +52.9 +64.1 +67.3 +Lcons on S +′ +57.7 +64.7 +67.2 +Table 6: Effect of Lcons. w/o: without. Lcons on S/S +′: apply +Lcons to S or S +′. The results are averages of multiple runs. +shot decreases, because our method can fully leverage base +classes’ distributions to estimate novel classes’ distributions. +The prototypes sampled from distributions are robust to the +variance of support examples. While the baseline is sensitive +to the number of support examples. +Which feature to aggregate? In Tab. 5, we explore differ- +ent features for aggregation. All types of features achieve +comparable performance on base classes but vary on novel +classes. The performance of original feature S and recon- +structed feature S +′ lag behind the latent encoding µ, σ and +z. We hypothesize that the latent encoding contains more +class-generic features. Besides, �z = µ+ϵ·σ performs worst +among these features due to its indeterminate inference pro- +cess. Instead, a simplified version z = µ + σ achieves satis- +factory results, which is the default setting of VFA. +Effect of Lcons. We use a shared VAE to encode support +features but still need to preserve class-specific information. +Therefore, we add a consistency loss Lcons to produce class- +wise distributions. Tab. 6 shows that Lcons is important for +VFA. Lcons applied to S +′ forces the model to produce class- +conditional distributions so that the latent variable z can re- +train meaningful information to represent class centers. +Design of VFA. The variational feature encoder Fenc and +decoder Fdec are not sensitive to the number and dimension +of hidden layers. Please see our appendix for details. +Conclusion +This paper revisits feature aggregation schemes in meta- +learning based FSOD and proposes Class-Agnostic Aggre- +gation (CAA) and Variational Feature Aggregation (VFA). +CAA can reduce class bias and confusion between base +and novel classes; VFA transforms instance-wise support +features into class distributions for robust feature aggrega- +tion. 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Objects as +Points. arXiv preprint arXiv:1904.07850. +Zhu, C.; Chen, F.; Ahmed, U.; Shen, Z.; and Savvides, M. +2021. Semantic relation reasoning for shot-stable few-shot +object detection. In CVPR, 8782–8791. +1 +2 +3 +4 +52 +54 +56 +58 +60 +62 +64 +66 +68 +nAP(%) +layer + 1shot + 3shot + 5shot +256 +512 +1024 +2048 +52 +54 +56 +58 +60 +62 +64 +66 +68 +nAP(%) +dim + 1shot + 3shot + 5shot +Figure 7: Design of VFA. We explore different designs of +Fenc and Fdec. layer: the number of hidden layer. dim: the +number of hidden channels. +Additional Main Results. +Results on Generalized FSOD. We evaluate our method +on the Generalized FSOD benchmark (Wang et al. 2020). +The result is an average of multiple random seeds. Follow- +ing (Qiao et al. 2021), we report nAP of different methods +with 10 random seeds. Since many methods only report their +results on the traditional FSOD benchmarks, we collect as +many methods that report the G-FSOD results as possible. +PASCAL VOC: Similar to the results of our main paper, +our method performs well on PASCAL VOC. As shown in +Tab. 7, our method achieves the best (12/15) or second-best +(3/15) among all settings. Especially when the shot is low, +our method shows significant improvements. For example, +our 1-shot gains are 7.2%, 4.2% and 8.8% on the Novel +Set 1, 2 and 3, respectively. COCO: We also compare VFA +with other methods on COCO, where our method achieves +the second-best results on nAP. We notice that the gap be- +tween VFA and DeFRCN is narrowed in the G-FSOD set- +ting (0.9% vs. 2.3% on 10-shot nAP). +Additional Ablation Studies +Design of VFA. By default, the feature encoder Fenc and +decoder Fdec consist of one input layer and output layer of +1024-d. In Fig. 7, we ablate the number of input layers and +hidden channels. VFA is sensitive to these hyper-parameters +when the shot is low (up to 4% in 1 shot). The performance +becomes more stable as the shot increases, e.g., the gap be- +tween different settings is reduced to 1% in 3 and 5 shots. +VFA with/without fine-tuning. In the few-shot fine-tuning +stage, we fine-tune the variational feature encoder Fenc and +decoder Fdec by default. Tab. 9 shows that Fenc and Fdec +can work without fine-tuning. The gap between two settings, +freeze parameters vs. trainable, is relatively small (about +0.1%). The results indicate that the representation learned +from base classes can be directly transferred to novel classes +even without fine-tuning. +More analysis of different feature aggregation. In the +main paper, we give a visual analysis of different feature +aggregation methods, i.e., CSA, CAA and VFA. Here we +give a quantitative analysis of these methods, shown in Fig 8. +Compared with CSA, CAA reduces class confusion between +base and novel classes. For example, The similarity between +cow and sheep is 0.71, near the intra-class similarity of + +Method / Shots +Novel Set 1 +Novel Set 2 +Novel Set 3 +Avg. +1 +2 +3 +5 +10 +1 +2 +3 +5 +10 +1 +2 +3 +5 +10 +FRCN-ft (Ren et al. 2017) +9.9 +15.6 +21.6 +28.0 +52.0 +9.4 +13.8 +17.4 +21.9 +39.7 +8.1 +13.9 +19.0 +23.9 +44.6 +22.6 +FSRW (Kang et al. 2019) +14.2 +23.6 +29.8 +36.5 +35.6 +12.3 +19.6 +25.1 +31.4 +29.8 +12.5 +21.3 +26.8 +33.8 +31.0 +25.6 +TFA w/ cos (Wang et al. 2020) +25.3 +36.4 +42.1 +47.9 +52.8 +18.3 +27.5 +30.9 +34.1 +39.5 +17.9 +27.2 +34.3 +40.8 +45.6 +34.7 +FSDetView (Xiao and Marlet 2020) +24.2 +35.3 +42.2 +49.1 +57.4 +21.6 +24.6 +31.9 +37.0 +45.7 +21.2 +30.0 +37.2 +43.8 +49.6 +36.7 +DCNet (Hu et al. 2021) +33.9 +37.4 +43.7 +51.1 +59.6 +23.2 +24.8 +30.6 +36.7 +46.6 +32.3 +34.9 +39.7 +42.6 +50.7 +39.2 +FSCE (Sun et al. 2021) +32.9 +44.0 +46.8 +52.9 +59.7 +23.7 +30.6 +38.4 +43.0 +48.5 +22.6 +33.4 +39.5 +47.3 +54.0 +41.2 +DeFRCN (Qiao et al. 2021) +40.2 +53.6 +58.2 +63.6 +66.5 +29.5 +39.7 +43.4 +48.1 +52.8 +35.0 +38.3 +52.9 +57.7 +60.8 +49.4 +VFA (Ours) +47.4 +54.4 +58.5 +64.5 +66.5 +33.7 +38.2 +43.5 +48.3 +52.4 +43.8 +48.9 +53.3 +58.1 +60.0 +51.4 +Table 7: G-FSOD results on PASCAL VOC. The results are sorted by the averaged score (Avg.). +Method +Shots +10 +30 +TFA w/ cos (Wang et al. 2020) +9.1 +12.1 +FSDetView (Xiao and Marlet 2020) +10.7 +15.9 +FSCE (Sun et al. 2021) +11.1 +15.3 +DeFRCN (Qiao et al. 2021) +16.8 +21.2 +VFA (Ours) +15.9 +18.4 +Table 8: G-FSOD results on COCO. The results are sorted +by 10-shot nAP. +Fenc, Fdec +nAP +bAP +1 +3 +5 +1 +3 +5 +freeze +57.6 +64.5 +67.1 +71.5 +75.9 +76.7 +trainable +57.7 +64.7 +67.2 +71.6 +76.0 +76.7 +∆ +-0.1 +-0.2 +-0.1 +-0.1 +-0.1 +0.0 +Table 9: Freeze or fine-tune Fenc and Fdec in VFA. ∆: The +difference between the first and second row. +cow (0.76). While in CAA, the similarity between cow and +sheep is reduced to 0.55 and the gap of intra-class and +inter-class similarity is enlarged to 0.10 (0.65 vs. 0.55). By +applying VFA to the model, the inter-class similarity is fur- +ther reduced. For each novel class, the gap between intra- +class and inter-class similarity is enlarged to 0.12∼0.54 (the +range is 0.05∼0.3 in CSA). The results further demonstrate +that our proposed CAA and VFA learn more discriminative +and transferable features. +Visualization +We visualize the detection results in Fig. 9. In the base train- +ing stage, we pre-train the model on base classes of PAS- +CAL VOC. Then we fine-tune the model on the {1, 3, 5} +shots of Novel Set 1 and visualize the detection results. As +the support set grows, our model produces more confident +results, e.g., the scores of detected novel objects are increas- +ing from 1-shot to 5-shot. +More Training Details +Our method follows the two-stage training strategy in +FSOD (Yan et al. 2019; Kang et al. 2019), i.e., base train- +ing and few-shot fine-tuning. In the base training stage, we +build a query dataset and a support dataset, where the query +dataset contains the whole data from base classes and the +support dataset is obtained by balanced sampling from the +query dataset. We train the model on the two datasets and +update all network parameters. In the few-shot fine-tuning +stage, the support dataset is usually the same as the query +dataset with only K shot instances. We only train (a) our +Fenc and Fdec in VFA and (b) the last classification and re- +gression layers. We freeze other parameters except for the +Region Proposal Network (RPN) by default. RPN is fixed +in our PASCAL VOC experiments but not frozen in COCO +experiments because the model on COCO will not overfit to +novel classes (10∼30 shots). + +0.48 +0.4 +0.41 0.44 0.38 0.56 0.46 0.34 +0.6 +0.61 0.33 0.43 0.65 +0.4 +0.41 +0.8 +0.39 0.67 0.41 0.38 +0.36 0.26 0.47 0.33 0.43 0.27 0.31 0.31 +0.3 +0.3 +0.24 0.29 +0.3 +0.5 +0.37 0.39 0.71 0.31 +0.4 +0.39 +0.39 0.33 0.33 0.33 0.35 0.53 0.38 0.28 +0.6 +0.68 0.29 0.34 0.71 0.35 0.34 0.67 0.31 0.76 0.33 0.32 +0.36 0.43 +0.3 +0.33 0.43 0.24 0.32 0.31 0.26 0.33 0.28 +0.3 +0.29 0.35 +0.3 +0.41 +0.4 +0.33 0.73 0.38 +0.34 0.25 +0.3 +0.24 0.33 0.25 0.49 0.36 0.27 0.28 0.24 0.26 0.25 +0.3 +0.29 0.38 0.39 0.32 0.38 0.79 +0.4 +0.37 0.33 0.41 0.33 0.43 +0.4 +0.28 0.54 0.42 0.27 0.38 0.45 0.33 0.35 0.69 0.39 0.52 0.35 0.36 +0.3 +0.24 0.38 0.28 +0.4 +0.27 0.23 0.25 0.33 0.29 0.26 0.29 0.29 0.49 +0.3 +0.39 0.69 0.34 0.38 0.32 +0.27 +0.3 +0.25 0.25 0.31 0.34 0.31 0.21 +0.5 +0.58 0.21 0.29 0.55 0.32 0.28 0.52 0.34 0.65 0.28 0.29 +0.27 0.38 0.21 0.28 +0.4 +0.21 0.24 0.26 0.22 0.28 0.29 0.28 0.23 0.29 0.22 0.35 0.38 0.28 0.71 0.31 +0.28 0.19 0.21 0.18 0.27 0.22 0.48 +0.3 +0.28 0.21 0.22 0.23 0.17 0.22 0.29 0.36 0.32 0.29 0.31 0.77 +0.29 0.22 0.24 0.22 0.23 0.32 0.25 0.14 0.34 0.28 0.22 0.23 0.34 0.22 0.21 0.88 0.24 0.34 0.21 0.16 +0.3 +0.21 +0.3 +0.21 0.37 0.22 0.22 0.16 +0.3 +0.27 0.15 0.18 0.24 0.43 0.26 0.24 0.86 0.28 +0.2 +0.21 +0.33 0.27 0.23 0.26 0.29 0.29 0.27 0.18 0.49 0.56 0.23 0.23 0.54 0.28 0.25 0.34 0.28 0.68 0.26 +0.3 +0.29 0.36 0.18 0.27 0.36 +0.2 +0.2 +0.17 0.23 0.33 +0.2 +0.19 0.22 0.25 0.18 0.21 +0.2 +0.26 0.84 0.21 +0.38 0.21 0.21 0.31 0.27 0.21 0.45 +0.3 +0.24 0.31 +0.2 +0.22 0.16 +0.2 +0.26 0.16 0.21 +0.3 +0.21 0.73 +bird +bus +cow +motorbike +sofa +0.10 +0.26 +0.42 +0.58 +0.74 +0.90 +Meta R-CNN++ +CAA +VFA +bird +bus +cow +motorbike +sofa +aeroplane +bicycle +boat +bottle +car +cat +chair +diningtable +dog +horse +person +pottleplant +sheep +train +tvmonitor +bird +bus +cow +motorbike +sofa +bird +bus +cow +motorbike +sofa +Figure 8: Analysis of different feature aggregation. We follow the same setting as Fig.5 of our main paper. Differently, the +number in each cell is the averaged cosine similarity of 5 shot examples. +1-shot +3-shot +5-shot +Figure 9: Visualization of some detection results. The model is trained on the {1, 3, 5} shots of PASCAL VOC Novel Set 1 and +tested on the VOC07 test set. + +cow|0.35 +cowjo.ccowj0.46birdj0.35birdj0.81 +birdjo.72pottedplanto.89 +sofaj0.96 +chairjo.57cow|0.86 +cowj0.64 +cowl0.94personl0.67 +persono.68 +motorbike/0.98motorbikel0.84sofaj0.42busl0.85personl0.58 +1837 +4803 +personj0.40 +chair0.79person/0.6z +motorbikel0.42sofaj0.89 +bottlel0.88cowj0.97 +cow0.93cowj0.96birdj0.63 +birdj0.57 \ No newline at end of file diff --git a/zdFQT4oBgHgl3EQfzDZw/content/tmp_files/load_file.txt b/zdFQT4oBgHgl3EQfzDZw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0eaa6538b8acb41043b35c62a6093932a7e708fd --- /dev/null +++ b/zdFQT4oBgHgl3EQfzDZw/content/tmp_files/load_file.txt @@ -0,0 +1,2087 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf,len=2086 +page_content='Few-Shot Object Detection via Variational Feature Aggregation Jiaming Han1,2*, Yuqiang Ren3, Jian Ding1,2, Ke Yan3†, Gui-Song Xia1,2† 1NERCMS, School of Computer Science, Wuhan University 2State Key Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' LIESMARS, Wuhan University 3YouTu Lab, Tencent {hanjiaming, jian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='ding, guisong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='xia}@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='cn, {condiren, kerwinyan}@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='com Abstract As few-shot object detectors are often trained with abundant base samples and fine-tuned on few-shot novel examples, the learned models are usually biased to base classes and sensi- tive to the variance of novel examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' To address this issue, we propose a meta-learning framework with two novel fea- ture aggregation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' More precisely, we first present a Class-Agnostic Aggregation (CAA) method, where the query and support features can be aggregated regardless of their cat- egories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The interactions between different classes encourage class-agnostic representations and reduce confusion between base and novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Based on the CAA, we then propose a Variational Feature Aggregation (VFA) method, which en- codes support examples into class-level support features for robust feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We use a variational autoencoder to estimate class distributions and sample variational features from distributions that are more robust to the variance of sup- port examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Besides, we decouple classification and re- gression tasks so that VFA is performed on the classifica- tion branch without affecting object localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Extensive experiments on PASCAL VOC and COCO demonstrate that our method significantly outperforms a strong baseline (up to 16%) and previous state-of-the-art methods (4% in average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Code will be available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='com/csuhan/VFA Introduction This paper studies the problem of few-shot object detection (FSOD), a recently-emerged challenging task in computer vision (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Different from generic object detection (Girshick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2017), FSOD assumes that we have abun- dant samples of some base classes but only a few exam- ples of novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Thus, a dynamic topic is how to im- prove the recognition capability of FSOD on novel classes by transferring the knowledge of base classes to novel ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In general, FSOD follows a two-stage training paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In stage-I, the detector is trained with abundant base sam- ples to learn generic representations required for the ob- ject detection task, such as object localization and classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In stage-II, the detector is fine-tuned with only K shots (K=1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ) novel examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Despite the great success Work done during internship at Tencent YouTu Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' plain FCs (previous) VAEs (Ours) Appearance variations of support images ������������������������ = Avg(������������1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ������������2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' … ) is sensitive to example’s variations ������������������������~������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ������������) is robust to few-shot examples ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ������������) ������������������������ ������������3 ������������4 ������������2 ������������1 ������������������������ ������������2 ������������3 ������������4 ������������1 ������������������������ ������������2 ������������4 ������������1 ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ������������) ������������������������ ������������4 ������������2 ������������1 rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ������������3 rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ������������3 Figure 1: Comparisons of different support feature encoding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Previous methods use plain fully-connected (FC) layers to encode support features and obtain class prototypes by averaging these features: xp = Avg(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In con- trast, our method uses variational autoencoders (VAEs) pre- trained on abundant base examples to estimate the distribu- tions of novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Since intra-class variance is shared across classes and can be modeled with common distribu- tions (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2018), we use a shared VAE to transfer the distributions of base classes to novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Finally, we can sample class prototypes xp from the distributions N(µ, σ) that are robust to the variance of few-shot examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' : remove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' of this paradigm, the learned models are usually biased to base classes due to the imbalance between base and novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As a result, the model will confuse novel objects with similar base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 5 (top) for an instance, the novel class, cow, has high similarities with several base classes such as dog, horse and sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Besides, the model is sensitive to the variance of novel examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Since we only have K shots examples per class, the performance highly de- pends on the quality of the support sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 1, appearance variations are common in FSOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Previous meth- ods (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019) consider each support example as a single point in the feature space and average all features as class prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' However, it is difficult to estimate the real class centers with a few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In this paper, we propose a meta-learning framework to address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Firstly, we build a strong meta-learning baseline based on Meta R-CNN (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019), which arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='13411v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='CV] 31 Jan 2023 even outperforms a representative two-stage fine-tuning ap- proach TFA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' By revisiting the feature ag- gregation module in meta-learning frameworks, we propose Class-Agnostic Aggregation (CAA) and Variational Feature Aggregation (VFA) to reduce class bias and improve the ro- bustness to example’s variances, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Feature aggregation is a crucial design in FSOD, which defines how query and support examples interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Previous works such as Meta R-CNN adopt a class-specific aggrega- tion scheme (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2 (a)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', query features are aggregated with support features of the same class, ignoring cross-class interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In contrast, we propose CAA (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2 (b)) which allows feature aggregation between different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Since CAA encourages the model to learn class-agnostic represen- tations, the bias towards base classes is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Besides, the interactions between different classes simultaneously model class relations so that novel classes will not be confused with base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Based on CAA, we propose VFA which encodes sup- port examples into class-level support features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Our mo- tivation is that intra-class variance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' appearance varia- tions) is shared across classes and can be modeled with common distributions (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' So we can use base classes’ distributions to estimate novel classes’ distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We achieve this by modeling each class as a common dis- tribution with variational autoencoders (VAEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We firstly train the VAE on abundant base examples and then fine-tune it on few-shot novel examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' By transferring the learned intra-class variance to novel classes, our method can esti- mate novel classes’ distributions with only a few examples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Finally, we sample support features from distri- butions and aggregate them with query features to produce more robust predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We also propose to decouple classification and regression tasks so that our feature aggregation module can focus on learning translation-invariant features without affecting ob- ject localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We conduct extensive experiments on two FSOD datasets, PASCAL VOC (Everingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2010) and COCO (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2014) to demonstrate the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We summarize our contributions as follows: We build a strong meta-learning baseline Meta R- CNN++ and propose a simple yet effective Class- Agnostic Aggregation (CAA) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We propose Variational Feature Aggregation (VFA), which transforms instance-wise features into class-level features for robust feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' To our best knowledge, we are the first to introduce variational fea- ture learning into FSOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Our method significantly improves the baseline Meta R- CNN++ and achieves a new state-of-the-art for FSOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For example, we outperform the strong baseline by 9%∼16% and previous best results by 3%∼7% on the Novel Set 1 of PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Related Work Generic Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Object detection has witnessed significant progress in the past decade, which can be roughly divided into two groups: one-stage and two-stage detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' One-stage detectors predict bounding boxes and class la- bels by presetting dense anchor boxes (Redmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2017), points (Law and Deng 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Zhou, Wang, and Kr¨ahenb¨uhl 2019), or directly out- put sparse predictions (Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Two-stage detectors (Girshick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Girshick 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2017) first generate a set of object proposals with Region Proposal Network (RPN) and then perform proposal-wise classification and regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' However, most generic detectors are trained with abundant samples and not designed for data-scarce scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Few-Shot Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Early attempts (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Wang, Ramanan, and Hebert 2019) in FSOD adopt meta-learning architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' FSRW (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019) and Meta R-CNN (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019) aggregate image/RoI-level query features with support features gener- ated by a meta learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Following works explore different designs of meta-learning architectures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', feature aggre- gation scheme (Xiao and Marlet 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) and feature space augmentation (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Li and Li 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Differ- ent from meta-learning, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' propose a simple two- stage fine-tuning approach, TFA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' TFA shows that only fine-tuning the last layers can significantly improve the FSOD performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Due to the simple structure of TFA, a line of works (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) following TFA are pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In this work, we build a strong meta-learning base- line that even surpasses the fine-tuning baseline TFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Then we revisit the feature aggregation scheme and propose two novel feature aggregation methods, CAA and VFA, achiev- ing a new state-of-the-art in FSOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Variational Feature Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Given an input image/fea- ture, we can transform it into a distribution with VAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' By sampling features from the distribution, we can model intra- class variance that defines the class’s character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The varia- tional feature learning paradigm has been used in various tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', zero/few-shot learning (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019), metric learning (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2018) and disentanglement learning (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In this work, we use VAEs trained on abundant base examples to estimate novel classes’ distributions with only a few ex- amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Besides, we also propose a consistency loss to make the model produce class-specific distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' To our best knowledge, we are the first to introduce variational feature learning into FSOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Background and Meta R-CNN++ Preliminaries Problem Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We follow the FSOD settings in previ- ous works (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Assume we have a dataset D = {(x, y), x ∈ X, y ∈ Y } with a set of classes C, where x is the input image and y = {ci, bi}N i=1 is the corresponding class label c and bounding box b an- notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We then split the dataset into base classes Cb and novel classes Cn where Cb ∪ Cn = C and Cb ∩ Cn = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Generally, we have abundant samples of Cb and K shots samples of Cn (K=1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The goal is to detect objects setting TFA Meta R-CNN∗ Meta R-CNN++ param freeze ✓ \x17 ✓ ✓ ✓ cosine cls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ✓ \x17 \x17 ✓ ✓ last layer init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' copy rand rand rand copy bAP (stage-I) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 bAP (stage-II) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 nAP 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 Table 1: Difference analysis between Meta R-CNN and TFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The results are evaluated under the 1 shot setting of PASCAL VOC Novel Set 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' stage-I and stage-II: base train- ing and fine-tuning stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ∗: Our re-implemented results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' of Cn with only K shots annotated instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Existing few- shot detectors usually adopt a two-stage training paradigm: base training and few-shot fine-tuning, where the representa- tions learned from Cb are transferred to detect novel objects in the fine-tuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Meta-Learning Based FSOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We take Meta R-CNN (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019) for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 3, the main framework is a siamese network with a query feature en- coder FQ, a support feature encoder FS, a feature aggre- gator A and a detection head FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Typically, FQ and FS share most parameters and A refers to the channel-wise product operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Meta R-CNN follows the episodic train- ing paradigm (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Each episode is com- posed of a set of support images and binary masks of an- notated objects, {xi, Mi}N i=1, where N is the number of training classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Specifically, we first feed the support set {xi, Mi}N i=1 to FS to generate class-specific support fea- tures {Si}i∈C, and the query image to FQ to generate a set of RoI features {Qm} (m is the index of RoIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Then we aggregate each Qm and Si with the feature aggregator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Finally, the aggregated features �Qm i are fed to the detection head FD to produce final predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Meta R-CNN++: Stronger Meta-Learning Baseline Meta-learning has proved a promising approach, but the fine-tuning based approach receives more and more atten- tion recently due to its superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Here we aim to bridge the gap between the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We choose Meta R-CNN and TFA as baselines and explore how to build a strong FSOD baseline with meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Although both methods follow a two-stage training paradigm, TFA optimizes the model with advanced tech- niques in the fine-tuning stage: (a) TFA freezes most net- work parameters, and only trains the last classification and regression layers so that the model will not overfit to few- shot examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' (b) Instead of randomly initializing the clas- sification layer, TFA copies pre-trained weights of base classes and only initializes the weights of novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' (c) TFA adopts cosine classifier (Gidaris and Komodakis 2018) rather than a linear classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Considering the success of TFA, we build Meta R- CNN++, which follows the architecture of Meta R-CNN but aligns most hyper-parameters with TFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Here we ex- plore different design choices to mitigate the gap between ������������������������ ������������������������ ������������ ℒ support images query image same class ������������������������ ������������������������ ������������ ℒ support images query image arbitrary classes (a) Class-specific aggregation (b) Class-agnostic aggregation Figure 2: Illustration of two feature aggregation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Si/Qi: support and query features of class i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' A: feature ag- gregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' L: loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' the two approaches, shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' (a) Parameter freeze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' By adopting the same parameter freezing strategy, Meta R- CNN++ significantly outperforms Meta R-CNN and even achieves higher novel AP than TFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' (b) Cosine classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Different from TFA, Meta R-CNN++ with the cosine classi- fier does not surpass the linear classifier in nAP (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0), but its performance on base classes is better than the linear classifier (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' (c) Alleviate base for- getting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We follow TFA and copy the pre-trained classifier weights of base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We find Meta R-CNN++ can also maintain the performance on base classes (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The above experiments indicate that meta-learning re- mains a promising approach for FSOD as long as we care- fully handle the fine-tuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Therefore, we choose Meta R-CNN++ as our baseline in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The Proposed Approach Class-Agnostic Aggregation Feature aggregation is an important module in meta-learning based FSOD (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Many works adopt a class-specific aggregation (CSA) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Let us assume that a query image has an object of class CQ = {i} and the corresponding RoI features {Qm i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the training phase, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2 (a), CSA aggregates each RoI feature Qm i with the support features Si of the same class: �Qm ii = A(Qm i , Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the testing phase, CSA aggre- gates the RoI feature with support features of all classes: �Qm ij = A(Qm i , Sj), j ∈ C, and each support feature Sj is to predict objects of its corresponding class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Notably, if the query image contains multiple classes, CSA aggre- gates the query features with each support feature in CQ: �Qm ij = A(Qm i , Sj), j ∈ CQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' But CSA still follows the class- specific way, as support features not belonging to CQ will never be aggregated with the query feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As discussed before, the learned models are usually bi- ased to base classes due to the imbalance between base and novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Therefore, we revisit CSA and propose a simple yet effective Class-Agnostic Aggregation (CAA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2 (b) for an instance, CAA allows feature ag- gregation between different classes, which encourages the model to learn class-agnostic representations and thereby re- duces the class bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Besides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' the interactions between differ- ent classes can simultaneously model class relations so that query image ℱ������������ ℱ������������ support images ������������ ������������ ������������ ������������ ������������ ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ������������) Variational Feature Aggregation ℱ������������������������������������ ℱ������������������������������������ ������������′ ������������ sampling ℒ������������������������������������ ℒ������������������������������������������������ ℒ������������������������������������ ℒ������������������������ query feature support feature ℱ������������ Figure 3: Overview of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' FQ and FS denote query and support feature extractors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Fenc and Fdec are the variational feature encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' FD: the detection head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' A: feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Note that we do not visualize RPN and the regression branch for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' novel classes will not confuse with base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Formally, for each RoI feature Qm i of class i ∈ C and a set of support features {Sj}j∈C, we randomly select a support feature Sj∗ of class j∗ to aggregate with the query feature, �Qm ij∗ = A(Qm i , Sj∗), j∗ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' (1) Then we feed the aggregated feature �Qm ij∗ to the detection head FD to output classification scores p = FD( �Qm ij∗), which is supervised with the label of class i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Note that CAA is used for training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' the testing phase still follows CSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Variational Feature Aggregation Prior works usually encode support examples into single fea- ture vectors that are difficult to represent the whole class dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Especially when the data is scarce and example’s variations are large, we cannot make an accurate estimation of class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Inspired by recent progress in variational feature learning (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021), we transform support features into class distributions with VAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Since the estimated distribution is not biased to specific examples, features sampled from the distribution are robust to the variance of support examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Then we can sample class-level features for robust feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The framework of VFA is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Variational Feature Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Formally, we aim to trans- form the support feature S into a class distribution N, and sample the variational feature z from N for feature aggre- gation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We optimize the model in a similar way to VAEs, but our goal is to sample the latent variable z instead of the reconstructed feature S ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Following the definition of VAEs, we assume z is generated from a prior distribution p(z) and S is generated from a conditional distribution p(S|z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As the process is hidden and z is unknown, we model the posterior distribution with variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' More specifically, we approximate the true posterior distribution p(z|S) with another distribution q(z|S) by minimizing the Kullback- Leibler (KL) divergence: DKL(q(z|S)||p(z|S)) = � q(z|S) log q(z|S) p(z|S), (2) which is equivalent to maximizing the evidence lower bound (ELBO): ELBO = Eq(z|S)[log p(S|z))] − DKL(q(z|S)||p(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' (3) Here we assume the prior distribution of z is a centered isotropic multivariate Gaussian, p(z) = N(0, I), and set the posterior distribution q(z|S) to be a multivariate Gaussian with diagonal covariance: q(z|S) = N(µ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The parame- ters µ and σ can be implemented by a feature encoder Fenc: µ, σ = Fenc(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Then we obtain the variational feature z with the reparameterization trick (Kingma and Welling 2013): z = µ + σ · ϵ, where ϵ ∼ N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 3 can be simplified to a reconstruction loss Lrec which is usually defined as the L2 distance between the input S and the reconstructed target S ′, Lrec = ∥S − S ′∥ = ∥S − Fdec(z)∥, (4) where Fdec denotes a feature decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As for the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 3, we directly minimize the KL divergence of q(z|S) and p(z), LKL = DKL(q(z|S)||p(z)), (5) which forces the variation feature z to follow a normal dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' By optimizing the two objectives, Lrec and LKL, we trans- form the support feature S into a distribution N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Then we can sample the variational feature z from N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Since z still lacks class-specific information, we apply a consistency loss Lcons to the reconstructed feature S ′, which is defined as the cross-entropy between S ′ and its class label c, Lcons = LCE(FS ′ cls(S ′), c), (6) Method / Shots Backbone Novel Set 1 Novel Set 2 Novel Set 3 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 1 2 3 5 10 1 2 3 5 10 1 2 3 5 10 FSRW (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019) YOLOv2 14.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 Meta FR-CNN (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2022) ResNet-101 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 47.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 DeFRCN (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) ResNet-101 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 VFA (Ours) ResNet-101 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 Table 2: Results on PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The results are sorted by the averaged score (Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' See our appendix for the generalized FSOD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' where FS ′ cls denotes a linear classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The introduction of Lcons transforms the learned distributions into class-specific distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The support feature Si is forced to approxi- mate a parameterized distribution N(µi, σi) of class i, so that the sampled z can preserve class-specific information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Variational Feature Aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Since the support fea- tures are transformed into class distributions, we can sample features from the distribution and aggregate them with query features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Compared with the original support feature S and reconstructed feature S ′, the latent variable z contains more generic features of the class (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2018), which is robust to the variance of support examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Specifically, VFA follows the class-agnostic approach in CAA but aggregates the query feature Q with a variational feature z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Given a query feature Qi of class i and support feature Sj of class j, we firstly approximate the class dis- tribution N(µj, σj) and sample a variational feature zj = µj + σj from N(µj, σj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Then we aggregate them together with the following equation: �Qij = A(Qi, zj) = Qi ⊙ sig(zj), (7) where ⊙ means channel-wise multiplication and sig is short for the sigmoid operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the training phase, we ran- domly select a support feature Sj (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', one support class j) for aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the testing phase (especially K > 1), we average K support features of class j into one ¯Sj, and approximate the distribution N(µj, σj) with the averaged feature, µj, σj = Fenc( ¯Sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Instead of adopting complex distribution estimation methods, we find the averaging ap- proach works well in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Network and Objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' VFA only introduces a light en- coder Fenc and decoder Fdec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Fenc contains a linear layer and two parallel linear layers to produce µ and σ, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Fdec consists of two linear layers to generate the re- constructed feature S ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We keep all layers the same dimen- sion (2048 by default).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' VFA is trained in an end-to-end man- ner with the following multi-task loss: L = Lrpn + Lreg + Lcls + Lcons + Lrec + αLKL, (8) where Lrpn is the total loss of RPN, Lreg is the regression loss, and α is a weight coefficient (α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5×10−4 by default).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Classification-Regression Decoupling Generally, the detection head FD contains a shared fea- ture extractor Fshare and two separate network Fcls and Freg for classification and regression, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In previ- ous works, the aggregated feature is fed to FD to produce both classification scores and bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' However, the classification task requires translation-invariant features, while regression needs translation-covariant features (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Since support features are always translation- invariant to represent class centers, the aggregated feature harms the regression task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Therefore, we decouple the two tasks in the detection head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Let Q and �Q denote the origi- nal and aggregated query features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Previous methods take �Q for both tasks, where the classification score p and predicted bounding boxes b are defined as: p = Fcls(Fshare( �Q)), b = Freg(Fshare( �Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' (9) To decouple these tasks, we adopt separate feature extractors and use the original query feature Q for regression, p = Fcls(Fcls share( �Q)), b = Freg(Freg share(Q)), (10) where Fcls share and Freg share are the feature extractor for clas- sification and regression, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Experiments and Analysis Experimental Setting Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We evaluate our method on PASCAL VOC (Ever- ingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2010) and COCO (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2014), following Method / Shots 10 30 Fine-tuning MPSR (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 TFA w/ cos (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 Retentive (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 FSOD-UP (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 SRR-FSD (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 CGDP+FSCN (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021b) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 FSCE (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 FADI (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 DeFRCN (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 Meta-learning FSRW (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 MetaDet (Wang, Ramanan, and Hebert 2019) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 Meta R-CNN (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 QA-FewDet (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 FSDetView (Xiao and Marlet 2020) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 Meta FR-CNN (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2022) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 DCNet (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 CME (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021a) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 VFA (Ours) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 Table 3: Results on COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The backbone is the same as Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The results are sorted by 10-shot nAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' See our ap- pendix for the generalized FSOD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' previous works (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We use the data splits and annotations provided by TFA (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020) for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For PASCAL VOC, we split 20 classes into three groups, where each group contains 15 base classes and 5 novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For each novel set, we have K={1, 2, 3, 5, 10} shots settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For COCO, we set 60 cat- egories disjoint with PASCAL VOC as base classes and the remaining 20 as novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We have K={10, 30} shots settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For PASCAL VOC, we report the Average Precision at IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 of base classes (bAP) and novel classes (nAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For COCO, we report the mean AP at IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='95 of novel classes (nAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We implement our method with Mmdetection (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The backbone is ResNet- 101 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2016) pre-trained on ImageNet (Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We adopt SGD as the optimizer with batch size 32, learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='02, momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 and weight decay 1e- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The learning rate is changed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='001 in the few-shot fine- tuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We fine-tune the model with {400, 800, 1200, 1600, 2000} iterations for K={1, 2, 3, 5, 10} shots in PAS- CAL VOC, and {10000, 20000} iterations for K={10, 30} shots in COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We keep other hyper-parameters the same as Meta R-CNN (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019) if not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Main Results PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2, VFA significantly out- performs existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' VFA achieves the best (13/16) or second-best (3/16) results on all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In Novel Set 1, VFA outperforms previous best results by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2%∼7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Our 2-shot result even surpasses previous best 10-shot re- sults (64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4%), which indicates that our method is Method CRD CAA VFA Shots 1 3 5 Meta R-CNN++ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 Ours ✓ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 ✓ ✓ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 ✓ ✓ ✓ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 Table 4: Effect of different modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 1-shot 3-shot 5-shot 75 80 85 90 95 Recall w/o CRD Novel Class Base Class 1-shot 3-shot 5-shot 75 80 85 90 95 Recall w/ CRD Novel Class Base Class Figure 4: Comparisons of recall without/with CRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' more robust to the variance of few-shot examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Besides, we notice that our gains are stable and consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' This phe- nomenon demonstrates that VFA is not biased to specified class sets and can be generalized to more common scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Furthermore, VFA obtains a 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1% average score and sur- passes the second-best result by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2%, which further demon- strates its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 3, VFA achieves the best nAP among meta-learning based methods and second-best re- sults among all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We notice that a fine-tuning based method, DeFRCN (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021), outperforms our method in nAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' To concentrate on the feature aggregation module in meta-learning, we do not utilize advanced tech- niques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', the gradient decoupled layer (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) in DeFRCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We believe the performance of VFA can be fur- ther boosted with more advanced techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Ablation Studies We conduct a series of ablation experiments on Novel Set 1 of PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Effect of different modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 4, we eval- uate the effect of different modules by gradually applying the proposed modules to Meta R-CNN++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Although Meta R-CNN++ is competitive enough, we show CRD improves the performance on nAP, where the absolute gains exceed 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Besides, we find CRD significantly improves the re- call on all classes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 4) and narrows the gap between base and novel classes because it uses separate networks to learn translation-invariant and -covariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Then, we ap- ply CAA to the model and obtain further improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The confusions between different classes are reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Fi- nally, we build VFA and achieve a new state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The 1-shot performance is even comparable with 5-shot Meta R- CNN++ in nAP, indicating that VFA is robust to the variance of support examples especially when the data is scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' bird bus cow motorbike sofa 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 1 Meta R-CNN++ CAA VFA bird bus cow motorbike sofa aeroplane bicycle boat bottle car cat chair diningtab dog horse person pottleplant sheep train tvmonitor bird bus cow motorbike sofa bird bus cow motorbike sofa Figure 5: Similarity matrix visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We calculate co- sine similarities of support features in the 5-shot setting of PASCAL VOC Novel Set 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' sofa, motorbike, cow, bus and bird are novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Warmer color denotes higher similar- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Zoom in for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='61 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 10-shot 5-shot 3-shot 2-shot 1-shot 1 2 3 4 5 relative distance baseline Ours Figure 6: The distance from the estimated prototype of K- shot examples to the real class center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For each novel class, we take the mean feature of all training examples as its real class center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Our 10-shot result is the reference distance, while other results are relative distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We only report the averaged distance of all novel classes for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Visual analysis of different feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 5 gives a visual analysis of different feature aggregation meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Due to the imbalance between base and novel classes, some novel classes are confused with base classes in Meta R-CNN++ (with CSA), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', a novel classe, cow have higher similarity (>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8) with horse and sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In contrast, CAA reduces class bias and confusion by learning class-agnostic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The inter-class similarities are also reduced so that a novel example will not be classified to base classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Finally, we use VFA to transforms support examples into class distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' By learning intra-class variances from abundant base examples, we can estimate novel classes’ dis- tributions even with a few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 5 (bottom), we can see VFA significantly improves intra-class similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Robust and accurate class prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the testing phase, detectors take the mean feature of K-shot examples as the class prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 6, our estimated class pro- totypes are more robust and accurate than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The distances to real class centers do not increase much as the Features S S ′ µ σ �z z bAP 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 nAP 1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 3 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 64.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 Table 5: Comparisons of different support features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' S and S ′ are the original and reconstructed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' µ, σ, �z = µ+ϵ·σ and z = µ + σ are latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' : The average score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Setting / Shots 1 3 5 w/o VFA 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 w/ VFA w/o Lcons 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 Lcons on S 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 Lcons on S ′ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 Table 6: Effect of Lcons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' w/o: without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Lcons on S/S ′: apply Lcons to S or S ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The results are averages of multiple runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' shot decreases, because our method can fully leverage base classes’ distributions to estimate novel classes’ distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The prototypes sampled from distributions are robust to the variance of support examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' While the baseline is sensitive to the number of support examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Which feature to aggregate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 5, we explore differ- ent features for aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' All types of features achieve comparable performance on base classes but vary on novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The performance of original feature S and recon- structed feature S ′ lag behind the latent encoding µ, σ and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We hypothesize that the latent encoding contains more class-generic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Besides, �z = µ+ϵ·σ performs worst among these features due to its indeterminate inference pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Instead, a simplified version z = µ + σ achieves satis- factory results, which is the default setting of VFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Effect of Lcons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We use a shared VAE to encode support features but still need to preserve class-specific information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Therefore, we add a consistency loss Lcons to produce class- wise distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 6 shows that Lcons is important for VFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Lcons applied to S ′ forces the model to produce class- conditional distributions so that the latent variable z can re- train meaningful information to represent class centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Design of VFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The variational feature encoder Fenc and decoder Fdec are not sensitive to the number and dimension of hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Please see our appendix for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Conclusion This paper revisits feature aggregation schemes in meta- learning based FSOD and proposes Class-Agnostic Aggre- gation (CAA) and Variational Feature Aggregation (VFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' CAA can reduce class bias and confusion between base and novel classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' VFA transforms instance-wise support features into class distributions for robust feature aggrega- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Extensive experiments on PASCAL VOC and COCO demonstrate our effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Acknowledgement This work was partially supported by National Nature Sci- ence Foundation of China under the grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='U22B2011, No.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Vadakkepat, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' and Lee, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Few-shot object detec- tion via classification refinement and distractor retreatment.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' and Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Accurate few-shot object detection with support-query mutual guid- ance and hybrid loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In CVPR, 14424–14432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Zhang, W.' metadata={'source': 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+page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' and Kr¨ahenb¨uhl, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Objects as Points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='07850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Zhu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Ahmed, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Shen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' and Savvides, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Semantic relation reasoning for shot-stable few-shot object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In CVPR, 8782–8791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 1 2 3 4 52 54 56 58 60 62 64 66 68 nAP(%) layer 1shot 3shot 5shot 256 512 1024 2048 52 54 56 58 60 62 64 66 68 nAP(%) dim 1shot 3shot 5shot Figure 7: Design of VFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We explore different designs of Fenc and Fdec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' layer: the number of hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' dim: the number of hidden channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Additional Main Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Results on Generalized FSOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We evaluate our method on the Generalized FSOD benchmark (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The result is an average of multiple random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Follow- ing (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021), we report nAP of different methods with 10 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Since many methods only report their results on the traditional FSOD benchmarks, we collect as many methods that report the G-FSOD results as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' PASCAL VOC: Similar to the results of our main paper, our method performs well on PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 7, our method achieves the best (12/15) or second-best (3/15) among all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Especially when the shot is low, our method shows significant improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For example, our 1-shot gains are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8% on the Novel Set 1, 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' COCO: We also compare VFA with other methods on COCO, where our method achieves the second-best results on nAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We notice that the gap be- tween VFA and DeFRCN is narrowed in the G-FSOD set- ting (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3% on 10-shot nAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Additional Ablation Studies Design of VFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' By default, the feature encoder Fenc and decoder Fdec consist of one input layer and output layer of 1024-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 7, we ablate the number of input layers and hidden channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' VFA is sensitive to these hyper-parameters when the shot is low (up to 4% in 1 shot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The performance becomes more stable as the shot increases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', the gap be- tween different settings is reduced to 1% in 3 and 5 shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' VFA with/without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the few-shot fine-tuning stage, we fine-tune the variational feature encoder Fenc and decoder Fdec by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 9 shows that Fenc and Fdec can work without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The gap between two settings, freeze parameters vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' trainable, is relatively small (about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The results indicate that the representation learned from base classes can be directly transferred to novel classes even without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' More analysis of different feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the main paper, we give a visual analysis of different feature aggregation methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', CSA, CAA and VFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Here we give a quantitative analysis of these methods, shown in Fig 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Compared with CSA, CAA reduces class confusion between base and novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For example, The similarity between cow and sheep is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='71, near the intra-class similarity of Method / Shots Novel Set 1 Novel Set 2 Novel Set 3 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 1 2 3 5 10 1 2 3 5 10 1 2 3 5 10 FRCN-ft (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2017) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 21.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 VFA (Ours) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 Table 7: G-FSOD results on PASCAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The results are sorted by the averaged score (Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Method Shots 10 30 TFA w/ cos (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2020) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 FSDetView (Xiao and Marlet 2020) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 FSCE (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 DeFRCN (Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2021) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 VFA (Ours) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 Table 8: G-FSOD results on COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The results are sorted by 10-shot nAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Fenc, Fdec nAP bAP 1 3 5 1 3 5 freeze 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 trainable 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='6 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='7 ∆ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='0 Table 9: Freeze or fine-tune Fenc and Fdec in VFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' ∆: The difference between the first and second row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' cow (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' While in CAA, the similarity between cow and sheep is reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='55 and the gap of intra-class and inter-class similarity is enlarged to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='65 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' By applying VFA to the model, the inter-class similarity is fur- ther reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' For each novel class, the gap between intra- class and inter-class similarity is enlarged to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='12∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='54 (the range is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='05∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='3 in CSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' The results further demonstrate that our proposed CAA and VFA learn more discriminative and transferable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Visualization We visualize the detection results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the base train- ing stage, we pre-train the model on base classes of PAS- CAL VOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Then we fine-tune the model on the {1, 3, 5} shots of Novel Set 1 and visualize the detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' As the support set grows, our model produces more confident results, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', the scores of detected novel objects are increas- ing from 1-shot to 5-shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' More Training Details Our method follows the two-stage training strategy in FSOD (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 2019), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=', base train- ing and few-shot fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the base training stage, we build a query dataset and a support dataset, where the query dataset contains the whole data from base classes and the support dataset is obtained by balanced sampling from the query dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We train the model on the two datasets and update all network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' In the few-shot fine-tuning stage, the support dataset is usually the same as the query dataset with only K shot instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We only train (a) our Fenc and Fdec in VFA and (b) the last classification and re- gression layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' We freeze other parameters except for the Region Proposal Network (RPN) by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' RPN is fixed in our PASCAL VOC experiments but not frozen in COCO experiments because the model on COCO will not overfit to novel classes (10∼30 shots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf'} +page_content='56 0.' metadata={'source': 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